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Wavelets and other orthogonal systems Second Edition
Shen Digital Instant Download
Author(s): Shen, Xiaoping; Walter, Gilbert G
ISBN(s): 9781482285802, 1482285800
Edition: Second edition
File Details: PDF, 12.11 MB
Year: 2001
Language: english

WAVELETS
and OTHER
ORTHOGONAL
SYSTEMS
SECOND EDITION

Studies in Advanced Mathematics
Series Editor
STEVEN G. KRANTZ
Washington University in St. Louis
Editorial Board
R. Michael Beats
Rutgers University
Gerald B. Folland
University o f Washington
Dennis de Turck
University o f Pennsylvania
William Helton
University of California at San Diego
Ronald DeVore
University o f South Carolina
Norberto Salinas
University o f Kansas
Lawrence C. Evans
University o f California at Berkeley
Michael E. Taylor
University o f North Carolin
Titles Included in the Series
Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping
John J . Benedetto, Harmonic Analysis and Applications
John J . Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications
Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex
Goong Chen and Jianxin Zhou, Vibration and Damping in Distributed Systems,
Vol. 1: Analysis, Estimation, Attenuation, and Design.
Vol. 2: W K B and Wave Methods, Visualization, and Experimentation
Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces o f Analytic Functions
John P. D'Angelo, Several Complex Variables and the Geometry o f Real Hypersurfaces
Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties o f Functions
Gerald B. Folland, A Course in Abstract Harmonic Analysis
Jose Garcia-Cuerva, Eugenio Hernandez, Fernando Soria, and Jose-Luis Torrea,
Fourier Analysis and Partial Differential Equations
Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem,
2nd Edition
Alfred Gray, Modern Differential Geometry o f Curves and Surfaces with Mathematica, 2nd Edition
Eugenio Hernandez and Guido Weiss, A First Course on Wavelets
Steven G. Krantz, Partial Differential Equations and Complex Analysis
Steven G. Krantz, Real Analysis and Foundations
Kenneth L . Kuttler, Modern Analysis
Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering
Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
John Ryan, Clifford Algebras in Analysis and Related Topics
Xavier Saint Raymond, Elementary Introduction to the Theory o f Pseudodifferential Operators
John Scherk, Algebra: A Computational Introduction
Robert Strichartz, A Guide to Distribution Theory and Fourier Transforms
Andre Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones
James S. Walker, Fast Fourier Transforms, 2nd Edition
James S. Walker, Primer on Wavelets and their Scientific Applications
Gilbert G. Walter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition
Kehe Zhu, A n Introduction to Operator Algebras

GILBERT G. WALTER
Department of Mathematical Sciences
The University ofWisconsin-Milwaukee
Milwaukee, Wisconsin
XIAOPING SHEN
Department of Mathematics and Computer Sciences
Eastern Connecticut State University
Willimantic, Connecticut
WAVELETS
and OTHER
ORTHOGONAL
SYSTEMS
SECOND EDITION
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data
Walter, Gilbert G.
Wavelets and other orthogonal systems / Gilbert G.
Walter, Xiaoping Shen.— 2nd ed.
p. c m . — (Studies in advanced mathematics)
Includes bibliographical references and index.
I S B N 1-58488-227-1 (alk. paper)
1. Wavelets (Mathematics) I . Shen, Xiaoping. I I . Title. I I I . Series.
QA403 .3 .W34 2000
515,
.2433—dc21 00-050874
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety o f references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity o f all materials or for the consequences o f their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent o f CRC Press L L C does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from C R C Press L L C
for such copying.
Direct all inquiries to C R C Press L L C , 2000 N . W . Corporate Blvd., Boca Raton, Florida 33431.
T r a d e m a r k Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
© 2001 by Chapman & Hall/CRC
N o claim to original U.S. Government works
International Standard Book Number 1-58488-227-1
Library of Congress Card Number 00-050874
Printed in the United States o f America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper

V
Preface to first edition
The subject of wavelets has evolved very rapidly in the last five or
six years—so rapidly that many articles and books are already obso-
lete. However, there is one portion of wavelet theory that has reached
a plateau, that is, the subject of orthogonal wavelets. The major con-
cepts have become standard, and further development will probably be
at the margins. In one sense they are no different than other orthogonal
systems. They enable one to represent a function by a series of orthogo-
nal functions. But there are notable differences: wavelet series converge
pointwise when others don't, wavelet series are more localized and pick
up edge effects better, wavelets use fewer coefficients to represent certain
signals and images.
Unfortunately, not all is rosy. Wavelet expansions change excessively
under arbitrary translations—much worse than Fourier series. The same
is true for other operators such as convolution and differentiation.
In this book wavelets are presented in the same setting as other orthog-
onal systems, in particular Fourier series and orthogonal polynomials.
Thus their advantages and disadvantages can be seen more directly.
The level of the book is such that it should be accessible to engineering
and mathematics graduate students. It will for the most part assume a
knowledge of analysis at the level of beginning graduate real and com-
plex analysis courses. However, some of the later chapters are more
technical and will require a stronger background. The Lebesgue integral
will be used throughout. This has no practical effect on the calculation
of integrals but does have a number of theoretical advantages.
Wavelets constitute the latest addition to the subject of orthogonal
series, which are motivated by their usefulness in applications. In fact,
orthogonal series have been associated with applications from their in-
ception. Fourier invented trigonometric Fourier series in order to solve
the partial differential equation associated with heat conduction and
wave propagation. Other orthogonal series involving polynomials ap-
peared in the 19th century. These too were closely related to problems
in partial differential equations. The Legendre polynomials are used to
find solutions to Laplace's equation in the sphere, the Hermite poly-
nomials and the Laguerre polynomial for special cases of Schrodinger
wave equations. These, together with Bessel functions, are special cases

vi
of Sturm-Liouville problems, which lead to orthogonal series, which are
used to solve various partial differential equations.
The arrival of the Lebesgue integral in the early 20th century allowed
the development of a general theory of orthogonal systems. While not
oriented to applications, it allowed the introduction of new systems such
as the Haar and Walsh systems, which have proven useful in signal
processing. Also useful in this subject are the sine functions and their
translates, which form an orthogonal basis of a Paley-Wiener space.
These are related to the prolate spheroidal functions, which are solutions
both to an integral equation and a Sturm-Liouville problem.
The orthogonal sequences of wavelets, which are generalizations of
the Haar system and the sine system, have a number of unique prop-
erties. These make them useful in data compression, in image analysis,
in signal processing, in numerical analysis, and in acoustics. They are
particularly useful in digitizing data because of their decomposition and
reconstruction algorithms. They also have better convergence properties
than the classical orthogonal systems.
While the Lebesgue integral made a general theory of orthogonal sys-
tems possible, it is insufficiently general to handle many of the appli-
cations. In particular, the delta "function" or impulse function plays
a central role in signal processing but is not a square integrable func-
tion. Fortunately a theory that incorporates such things appeared in
the middle of the 20th century. This is the theory of "distributions,"
due mainly to L. Schwartz. It also is related to orthogonal systems in
that it allows representation of distributions by orthogonal systems and
also allows representations of functions by orthogonal distributions.
The body of the book is divided into 13 chapters of which the first 7
are expository and general while the remaining are more specialized and
deal with applications to other areas. Each will be concerned with the
use of or properties of orthogonal series.
In Chapter One we present two orthogonal systems that are prototypes
for wavelets. These are the Haar system and the Shannon system, which
have many, but not all, of the properties of orthogonal wavelets. They
will be preceded by a section on general orthogonal systems. This is a
standard theory that contains some results that will be useful in all of
the particular examples.
Chapter Two will give a short introduction to tempered distributions.
This is a relatively simple theory and is the only type of generalized
function needed for much of orthogonal series. Many engineers still seem
to apologize for their use of a*"delta function". There is no need to do

vii
so since these are well defined proper mathematical entities. Included
here also is the associated theory of Fourier transforms that enables one
to take Fourier transforms of things like polynomial and trigonometric
functions.
Chapter Three contains an introduction to the general theory of or-
thogonal wavelets. Their construction by a number of different schemes
is given as are a number of their properties. These include their multires-
olution property in which the terms of the series are naturally grouped
at each resolution. The decomposition and reconstruction algorithms of
Mallat, which give the coefficients at one resolution in terms of others,
are presented here. Some of these properties are extended to tempered
distributions in Chapter Five.
In Chapter Four we return to trigonometric Fourier series and discuss
more detailed properties such as pointwise convergence and summability.
These are fairly well known and many more details may be found in
Zygmund's book. A short presentation on expansion of distributions in
Fourier series is also presented.
In Chapter Five we also consider orthogonal systems in Sobolev spaces.
These can be composed of delta functions as well as ordinary functions.
In the former case we obtain an orthonormal series of delta function
wavelets.
Chapter Six is devoted to another large class of examples, the ortho-
gonal polynomials. The classical examples are defined and certain of
their properties discussed. The Hermite polynomials are naturally as-
sociated with tempered distribution; properties of this connection are
covered. Other orthogonal series are discussed in Chapter Seven.
Various kinds of convergence of orthogonal series are discussed in
Chapter Eight. In particular, pointwise convergence of wavelet series is
compared to that of other orthogonal systems. Also, the rate of conver-
gence in Sobolev spaces is determined. Gibbs' phenomenon for wavelet
series is compared to that for other series.
Chapter Nine deals with sampling theorems. These arise from many
orthogonal systems including the trigonometric and polynomial systems.
But the classical Shannon sampling theorem deals with wavelet sub-
spaces for the Shannon wavelet. This can be extended to other wavelet
subspaces as well. Both regular and irregular sampling points are con-
sidered.
In Chapter Ten we cover the relation between the translation operator
and orthogonal systems. Wavelet expansions are not very well behaved
with respect to this operator except for certain examples.

viii
Chapter Eleven deals with analytic representation based on both Fou-
rier series and wavelet. These are used to solve boundary value problems
for harmonic functions in a half-plane with specified values on the real
line.
Chapter Twelve covers probability density estimation with various
orthogonal systems. Both Fourier series and Hermite series have been
used, but wavelets come out the best.
Finally in the last chapter we cover the Karhunen-Loeve theory for
representing stochastic processes in terms of orthogonal series. An al-
ternate formulation based on wavelets is developed.
Some of this text material was presented to a graduate course of mixed
mathematics and engineering students. While not directly written as a
text, it can serve as the basis for a modern course in Special Functions
or in mathematics of signal processing. Problems are included at the
end of each chapter. For the most part these are designed to aid in the
understanding of the text material.
Acknowledgment s
Many persons helped in the preparation of the manuscript for this
book, but two deserve special mention: Joyce Miezin for her efficient
typing and ability to convert my handwriting into the correct symbols,
and Bruce O'Neill for catching many of my mathematical misprints.
G. G. Walter

ix
Preface to second edition
In the years since the first edition of this book appeared, the subject
of wavelets has continued its phenomenal growth. Much of this growth
has been associated with new applications arising out of the multiscale
properties of wavelets. Another source has been the widespread use of
threshold methods to reduce the data requirements as well as the noise in
certain signals. But in the area of wavelets as orthogonal systems, which
is the main theme of this book, the growth has not been as marked. The
principal new material has been in the area of multiwavelets, which,
however, have not found their way into as many applications as the
original theory. In addition, there seems to a resurgence of interest in
nontensor product higher dimensional wavelets, but this area still needs
some time to sort itself out.
In this new edition we have tried to correct many of the misprints and
errors in the first edition (and in the process, have probably introduced
others). We have reviewed the problems and introduced others in an
effort to make their solution possible for average graduate students. We
have also introduced a number of illustrations in an attempt to further
clarify some of the concepts and examples. The first and fourth chap-
ters remain approximately the same in this edition. The second chapter
on distribution theory has been rewritten in order to make it somewhat
more readable and self contained. Chapter three on orthogonal wavelet
theory has been expanded with some additional examples: the raised
cosine wavelets in closed form, and other Daubechies wavelets and their
derivation. In Chapter five on wavelets and distributions, a section on
impulse trains has been added. Chapter six on orthogonal polynomials
remains essentially the same, while in Chapter seven a new section on
an alternate approach to periodic wavelets has been added. In Chap-
ter eight on pointwise convergence, an additional section on positive
wavelets and their use in avoiding Gibbs' phenomenon is new. Chapter
nine has been extensively revised and, in fact, has been split into two
chapters, one devoted primarily to the Shannon sampling theorem and
its properties and the new Chapter ten which concentrates more on sam-
pling in other wavelet subspaces. New topics include irregular sampling
in wavelet subspaces, hybrid wavelet sampling, Gibbs' phenomenon for
sampling series in wavelet subspaces, and interpolating multiwavelets.

X
Chapter eleven on translation and dilation has only minor changes as
does most of Chapter twelve except for a few pages on wavelets of en-
tire analytic functions. In Chapter thirteen on statistics a number of
new topics have been added. These include positive wavelet density es-
timators, density estimators with noisy data, and threshold methods.
Some additional calculations involving some of these estimators are also
included. Chapter fourteen, which deals with stochastic processes, has
some new material on cyclostationary processes.
Acknowledgements.
The contributions of many individuals appear in this new edition. In
particular the authors wish to acknowledge the work of Youming Liu,
Hong-tae Shim, and Luchuan Cai which is covered in more detail here.
Gilbert G. Walter and Xiaoping Shen

Contents
Preface to first edition v
Preface to second edition ix
List of Figures xvii
1 Orthogonal Series 1
1.1 General theory 1
1.2 Examples 5
1.2.1 Trigonometric system 6
1.2.2 Haar system . 10
1.2.3 The Shannon system 12
1.3 Problems 15
2 A Primer on Tempered Distributions 19
2.1 Intuitive introduction 20
2.2 Test functions 22
2.3 Tempered distributions 25
2.3.1 Simple properties based on duality 27
2.3.2 Further properties 29
2.4 Fourier transforms 30
2.5 Periodic distributions 32
2.6 Analytic representations 33
2.7 Sobolev spaces 35
2.8 Problems 35
3 A n Introduction to Orthogonal Wavelet Theory 37
3.1 Multiresolution analysis 38
xi

xii
3.2 Mother wavelet 44
3.3 Reproducing kernels and a moment condition 53
3.4 Regularity of wavelets as a moment condition 55
3.4.1 More on example 3 59
3.5 Mallat's decomposition and reconstruction algorithm . . . 64
3.6 Filters 65
3.7 Problems 70
4 Convergence and Summability of Fourier Series 73
4.1 Pointwise convergence 73
4.2 Summability 79
4.3 Gibbs phenomenon 81
4.4 Periodic distributions 84
4.5 Problems 87
5 Wavelets and Tempered Distributions 91
5.1 Multiresolution analysis of tempered distributions . . . . 92
5.2 Wavelets based on distributions 95
5.2.1 Distribution solutions of dilation equations . . . . 95
5.2.2 A partial distributional multiresolution analysis . . 99
5.3 Distributions with point support 100
5.4 Approximation with impulse trains 104
5.5 Problems 107
6 Orthogonal Polynomials 109
6.1 General theory 109
6.2 Classical orthogonal polynomials 114
6.2.1 Legendre polynomials 115
6.2.2 Jacobi polynomials 119
6.2.3 Laguerre polynomials 120
6.2.4 Hermite polynomials 121
6.3 Problems 126
7 Other Orthogonal Systems 129
7.1 Self adjoint eigenvalue problems on finite intervals . . . . 130
7.2 Hilbert-Schmidt integral operators 132
7.3 An anomaly: the prolate spheroidal functions . . .134
7.4 A lucky accident? 135
7.5 Rademacher functions 140
7.6 Walsh function 142
7.7 Periodic wavelets 143

7.7.1 Periodizing wavelets 144
7.7.2 Periodic wavelets from scratch 146
7.8 Local sine or cosine basis 150
7.9 Biorthogonal wavelets 154
7.10 Problems 159
8 Pointwise Convergence of Wavelet Expansions 161
8.1 Reproducing kernel delta sequences 162
8.2 Positive and quasi-positive delta sequences 163
8.3 Local convergence of distribution expansions 169
8.4 Convergence almost everywhere 172
8.5 Rate of convergence of the delta sequence 173
8.6 Other partial sums of the wavelet expansion 177
8.7 Gibbs phenomenon 178
8.8 Positive scaling functions 181
8.8.1 A general construction 181
8.8.2 Back to wavelets 182
8.9 Problems 186
9 A Shannon Sampling Theorem in Wavelet Subspaces 187
9.1 A Riesz basis of V m 188
9.2 The sampling sequence in V m 189
9.3 Examples of sampling theorems 191
9.4 The sampling sequence in T m 195
9.5 Shifted sampling 197
9.6 Gibbs phenomenon for sampling series 199
9.6.1 The Shannon case revisited 201
9.6.2 Back to wavelets 201
9.7 Irregular sampling in wavelet subspaces 212
9.8 Problems 214
10 Extensions of Wavelet Sampling Theorems 217
10.1 Oversampling with scaling functions 218
10.2 Hybrid sampling series 223
10.3 Positive hybrid sampling 225
10.4 The convergence of the positive hybrid series 228
10.5 Cardinal scaling functions 232
10.6 Interpolating multiwavelets 240
10.7 Orthogonal finite element multiwavelets 242
10.7.1 Sobolev type norm 244
10.7.2 The mother multiwavelets 245

xiv
10.8 Problems 252
11 Translation and Dilation Invariance in Orthogonal
Systems 255
11.1 Trigonometric system 255
11.2 Orthogonal polynomials 256
11.3 An example where everything works 257
11.4 An example where nothing works 258
11.5 Weak translation invariance 259
11.6 Dilations and other operations 265
11.7 Problems 267
12 Analytic Representations V i a Orthogonal Series 269
12.1 Trigonometric series 270
12.2 Hermite series 274
12.3 Legendre polynomial series 280
12.4 Analytic and harmonic wavelets 282
12.5 Analytic solutions to dilation equations 286
12.6 Analytic representation of distributions by wavelets . . . . 287
12.7 Wavelets analytic in the entire complex plane 291
12.8 Problems 293
13 Orthogonal Series in Statistics 295
13.1 Fourier series density estimators 296
13.2 Hermite series density estimators 299
13.3 The histogram as a wavelet estimator 301
13.4 Smooth wavelet estimators of density 305
13.5 Local convergence 309
13.6 Positive density estimators based on
characteristic functions 310
13.7 Positive estimators based on positive wavelets 312
13.7.1 Numerical experiment 316
13.8 Density estimation with noisy data 318
13.9 Other estimation with wavelets 322
13.9.1 Spectral density estimation 322
13.9.2 Regression estimators 324
13.10 Threshold Methods 324
13.11 Problems 326

X V
14 Orthogonal Systems and Stochastic Processes 329
14.1 K-L expansions 329
14.2 Stationary processes and wavelets 332
14.3 A series with uncorrected coefficients 335
14.4 Wavelets based on band limited processes 341
14.5 Nonstationary processes 345
14.6 Problems 349
Bibliography 351
Index 363

List of Figures
1.1 The scaling function and mother wavelet for the Haar
system 11
1.2 The scaling function for the Shannon system 14
1.3 The mother wavelet for the Shannon system 15
2.1 A test function in the space S (the Hermite function h^(x)). 23
2.2 Some approximations to the delta function in S' 28
3.1 Typical functions in the subspaces Vb of the multiresolu-
tion analysis for the Haar scaling function 39
3.2 Typical functions in the subspaces V of the multiresolu-
3.3 Typical functions in the subspaces V2 of the multiresolu-
3.4 The scaling function of the Franklin wavelet arising from
the hat function 47
3.5 The Daubechies scaling function of example 3 47
3.6 The mother wavelet of example 3 48
3.7 The mother wavelet of Example 5 in the time domain. . . 49
3.8 The scaling function and absolute value of the mother
wavelet of Example 7 in frequency domain 51
3.9 The scaling function of Figure 3.7 in the time domain. . . 52
3.10 The reproducing kernel q(x,t) for Vo in the case of Haar
wavelets 55
3.11 Daubechies scaling function and mother wavelet (N = 4). 63
3.12 The system functions of some continuous filters: low-pass,
high-pass and band-pass 66
3.13 The system function of discrete lowpass (halfband) filter. 68
xvii

xviii
3.14 The decomposition algorithm 69
3.15 The reconstruction algorithm. . . 69
4.1 The Dirichlet kernel of Fourier series (n= 6) 77
4.2 The Fejer kernel of Fourier series (n= 6) 81
4.3 The saw tooth function 83
4.4 Gibbs phenomenon for Fourier series; approximation to
the saw tooth function using Dirichlet kernel 83
4.5 The approximation to the saw tooth function using Fejer
kernel 84
5.1 A mother wavelet with point support. The vertical bars
represent delta functions 104
5.2 A continuous function and its impulse train 105
6.1 Some Legendre polynomials (n= 2, 3, and 6) 115
6.2 Some Laguerre polynomials (alpha= l/2, n = 5, 7, and 10). 120
6.3 Some Hermite polynomials (modified by constant multi-
ples, n = 4, 5, and 7) 122
7.1 The Haar mother wavelet 141
7.2 One of the Rademacher functions 143
7.3 Two orthogonal Walsh functions 144
7.4 A bell used for a local cosine basis 151
7.5 Three elements in the local cosine basis with bell of Figure
7.4 153
7.6 Two additional elements of the local cosine basis showing
the bell 154
7.7 Two biorthogonal pairs of scaling functions with the same
MRA 156
7.8 A biorthogonal pair of scaling functions and wavelets with
compact support 159
8.1 The delta sequences from Fourier series - the Dirichelet
kernel 164
8.2 The delta sequences from Fourier series - the Fejer kernel. 165
8.3 The quasi-positive delta sequence for the Daubechies
wavelet 20 , m = 0 168
8.4 The summability function pr
(x) for Daubechies wavelet
2 0 ( x ) , m = 0 183

xix
8.5 The positive delta sequence fcrjm(x, y) for Daubechies wavelet
2(/>(x),m = 0 184
9.1 The sampling function for the Daubechies wavelet 20(0
with 7 = -± 194
9.2 The function h of Proposition 9.3 202
9.3 The partial sum of the Shannon series expansion 202
10.1 An example of the scaling function of a Meyer wavelet
at scale m= 0 and the sum of 5 terms of its sampling
expansion in the next scale m =l 221
10.2 The positive summability function for the Coiflet of de-
gree 2 with r = 0.22 227
10.3 The dual of the positive summability function in Figure
10.2 228
2
10.4 A non-negative function f(x) = e~^~[-1/2,1/2] 229
10.5 The positive hybrid series (m= 4) using Coiflet of degree
2 for the function in Figure 10.4 229
10.6 The hybrid sampling series for the function in Figure 10.4
(m = 4) using Coiflet of degree 2 230
10.7 An example of cardinal scaling function of Theorem 10.5,
type two raised cosine wavelet 239
10.8 The mother wavelet for the scaling function in
Figure 10.7 239
10.9 The scaling function and wavelet of interpolating multi-
wavelets for n = l 250
10.10 The two scaling functions of interpolating multiwavelets
for n = 2 250
10.11 The first pair of wavelets of interpolating multiwavelets
for n= 2 251
11.1 The approximation of the shifted scaling function by Haar
series at scale m =l 259
12.1 The kernel Kr(t) given in Lemma 12.1 with m — 2. . . . 271
12.2 The Hermite function h^ix) and the real part of the Her-
mite function of the second kind fi2(x + i • 0) 278
12.3 A Legendre polynomial (n = 3) 281
12.4 The real and imaginary parts of the analytic representa-
tions at y= 2 for the Legendre polynomial in Figure 12.3. 281

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Laird, (c) Mary O., b. 7. 17. i860, same: d. 4, 11. 1896: m. ^^"illiam
Florence, (d) Elizabeth, b. 3. 7. 1865. same : m. William H.
Crawford, (e) William, b. 12. 25, 1866, same : d. same. 5. 28. 1875.
(ii) Martha, b. 1830; d. 1862. in Wilson Co.: m. Hatsfield, (iii)
Thomas, b. 1835: m. McCall. and in 1890 was residing at Lebanon.
Tenn. He was during the Civil War a captain in the 24th Regiment
Tennessee Confederate Infantry, and was severely wounded at Stone
River, Tenn. (iv) John, b. 1837: d. 1872 : m. Mary Peniberton. 3474.
Josiah (farmer. Baptist), b. 10. 10. 1807. in Dividson Co., Tenn.:
removed with his parents in 1812 to Wilson Co.. that state : d. there.
1884 : bur. there, at Lebanon. He m. Melissa Casson. who d. there,
date not known. Children all I1. in that county. Issue: (i) Willie A., m.
William Billings. (ii) Edward T.. b. 10. 12. 1837: m. 6, 9, 1868, Mary
Hobbs, b. there, 9, 19, 1842: d. there, 2, 7. 1872: bur. there: a dau.
of Green Hobbs. He m. (2) 4, 2. 1874. in Warren Co.. Ky.. Ann R.
Ellis, b. there, 6, 26, 1852 : living, 1890 : a dau. of William. By his
two wives he had: (a) Lavicy. b. 5. 20, 1869. (b) Mary E., b. 11. 19,
1870: d. 7, 13, 1871. (c) Felix, b. 2, 8. 1876. (d) Thomas J., b. 10.
14, 1877: d. II. 23. 1878. (e) lulia B., b. 11, 27, 1878; d. 9, 30,
1879. (f) Lennis E.. b. 11. 15. 1880. (g) Tesse S.. b. 2. 2. 1883. (h)
.Stella J., b. 7. 16. 1885 : d. 6, 28, 1887. "( i) Mattie G., b. 8, 18,
1887. (j) .Annie P., h. 3, 12, i8go. (iii) John, who m. Sophia
Bradshaw. (iv) George, who m. Cornelia Hobbs. (v) Taylor, who m.

Abbie Hartsfield. (vi) Franklin, who m. Martha Jenkins, (vii) Levi, b.
3. 10, 1858; m. Mary Jenkins, (viii) James F. White, (ix) Harriet, who
m. James Bradshaw. (.x) Dica. m. Henry White, (xi) Samuel. 3475.
Fanny, b. 3, 25, 1810, in Davidson Co., Tenn. : d. in Carroll Co., Miss.
; m. David Stanley. Xo further record. 3476. Elizabeth, b. 10, 12,
1812, same: d. in Carroll Co., Miss. m. John Reese. X'o further
record. 3477. Rol)ert ( farmer. Baptist), b. 10. 5, 1814, same : m.
about 1848, in Wilson Co.. Julia Satterfield : a dau. of Reuben.
Robert tl. there. 1863; l)ur. near Lebanon. Children b. in Wilson Co.
Issue: (i) Pauline, b. 1849: d. there. 1865. (ii) Reuben, b. 1852: d.
there, 1873. (iii) Martha, b. 1854: m. Robertson, and in 1890 resided
at Hartsville, Tenn. (iv) Frances, b. 1856: m. Thomas Hobbs. and in

SErExni G/sxi-RArrox. 315 1890 was residing" at Tucker's
Cross Roads, Tenn. (vj James L., b. 1861 : unm., 1890. 3478. Anna.
b. 4. 17, 1817, same: d. in Wilson Co.; ni. .'^eth Phillips. No further
record. 3479. Amanda, b. 2, 8. 1819, same: d. J. 8, i8(j8, in Wilson
Co.: ni. there. Ilailey Phillips. No further record. ,^480. Dica, b. 2,
19, 1822, in Wilson Co. ; d. there. 6. 20. 1898; bur. there : ni.
William .Smart. No further record. 3481. .Sarah, b. 2. 5, 1825, same;
m. there, Ceorge Washington .Sweet, and in 1890 was residing at
Rich Pond, Kv. 1105. Jesse Harlan' (Jesse,'' John), miller, b. 5, 8,
1781, in Chester Co., Pa.: d. I. 21, 1847, in .Mbcmarle Co., 'a. ; Inir.
there, in .Morris Family liurving (irounds. Hem. 1808, in that county (
ha ing come to that county in 1806) , Abigail Morris, b. there, 11,
14, 1789; d. there, 11, 24, 1869; bur. beside the husband; a dan. of
John Morris and Nancy Perkins, farmers and natives of .Albemarle
Co., Va. 'riie chililren were each 1). in that county. Issue: 3482.
Susan Morris, b. 10, I, 1800: d. there, 6, 26, 1886: bur. there in
Morris Burying Grounds ; unm. ^483. Nancv Perkins, b. 12, 26,
1810: d. i, 30, 1882; m. Peter H. Seav, 2, 9, 1831. 3484. John
.Morris, b. 12. 26. 1812; d. 4, 2, 1884; m. Emdy Stratton. 3485.
fiugh Rice, b. 6, 24, 1816 ; d. 10, 7, 1836. in that county ; unm.
3486. William Richard (merchant), b. 10, 13, 1819; d. 11, 13, 1872;
m. 11, 28, 1844, Alary M. Harlan (No. 7952). Mary, in 1890, was
residing in the family of Thomas S. Harlan (No. 7956). Child : (i)
Willie J., b. 4. 6, 1848, in Albemarle Co., 'a. ; d. 9, 8. 1868, in
Buckingham Co.. 'a. ; in. '. .S. Moseley, also deceased. 3487.
Pleasant M.. b. 4, 10. 1823 ; d. 1,31, 1892 ; m. Rachel Griffith, 9, 2,
1858. 3488. Christoi^her I,., b. 4, 22, 1827; unm. ; residence, 1891,
Cartersville, 'a. ^489. Mary Flemming, b. 4, 29, 1831 : m. Thomas
S. Harlan ( No. 7956), =;. i41850. 3490. Thomas Jefferson, b. 8, 1
1, 1834 : m. /ilpha W. Bartlett. 9, 21, 1865. . granddaughter of
.bigail Morris Harlan relates that her grandmother was neyer
outside the state of irginia. 1107. Mary Harlan" (Thomas." John").
Methodist, b. 11, 24, 1794, in Hawkins Co., Tenn. ; d. 3, 20, 1875, i"
I'ark Co.. Ind. ; bur. in ermillion Co., same state. She m. about
1808. in Hawkins Co., Tenn., Jacob Groves (farmer, Methodist), b.

10, 17, 1794, in that county: d. 12, 9, 1843. i" V'ermillion Co., Ind.;
bur. there ; a son of John ( Iroves, farmer and native of
Pennsylvania. Jacob and Mary removed to X'ermillion Co.. Ind.. in
1830, and after his death (or in 1855) *h*^ remoyeil to Park Co.,
Ind. Issue: 3491. William C, b. 11, 14, 1817, in Hawkins Co., Tenfi. ;
m. Elizabeth E. P)Ogart, 3, 18, 1841 ; residence, 1890, Dana,
'ermillion Co., Ind. 3492. Susan, b. 2, 20. 1819, same ; d. 10. 28.
1843, i" Vermillion Co., Ind. ; bur. there : unm.

316 THE HARLAN FAMILY. 3493. John, b. II, 10, 1820,
same; d. 8, 3, 1851, in ermillion Co.; m. Sarah Ann Lowrey, 12, 2,
1845. ^'o further information. 3494. Thomas, b. 11, 10, 1820, same
; d. 12, 24, 1843, in VermilHon Co. ; unm. 3495. Eldridge Î., b. 10.
18. 1822. same; d. 3, 31, 1885; m. 4, 23, 1850, in jMissouri. No
further information. 3496. Rachel, b. 12, 30, 1827, same; living.
1890: m. William Lowrev, 11. 27. 1854. 3497. Mary Jane, b. i, 11,
1835, '" ^ ermillion Co., Ind. No further information. 1 108. John
Harlan' (Thomas,*^ John^), farmer, Methodist, b. 1796, in Hawkins
Co., Tenn. ; d. there, i860; bur. there, on his farm. He m. there,
1826, Elizabeth Forgey, b. 1805, in that county ; d. there, 1837 ; bur.
there, in New Providence Church Burjung Grounds; a dau. of James
and Margaret (Caldwell) Forgey, farmers in Hawkins Co., Tenn. The
children were each b. in Hawkins Co. Issue : 3498. Thomas P., b.
1827. 3499. ^latilda H., b. 1830; d. there: m. Joshua Smith. No
further record. 3500. Elizabeth E., b. 1832 : d. 1885, in Cameron,
Xa.. ; m. John R. Bussell. No further record. 3501. Rachel F., b.
1834; m. George Sanders. No further record. 3502. Cornelius M., b.
1836; d. there; m. 1868, in that county, Nancy Jane McClellan, b.
1845, '" Sullivan Co., Tenn. ; a dau. of William ÎcClellan (farmer), b.
1809, and Hannah Bachman, b. 1823; both in Sullivan Co., Tenn.
Cornelius M. Harlan was a farmer and dealer in livestock. Child: (i)
Maggie Belle, b. 1869, in Hawkins Co., Tenn.; unm., 1890; residence.
Ro,gersville, Tenn. mo. Rachel Harlan' (Thomas," John^),
Presbyterian, b. 11, 11, 1801, in Hawkins Co., Tenn.; d. i, 6, 1866, at
Bristol, Sullivan Co., Tenn.; bur. there, in Bristol Burying Grounds.
She m. 9 Mo., 1823, in Hawkins Co., John Gray (farmer), b. 7, 2,
1791, in Sullivan Co. ; d. there, 8 Mo., 1868, " five miles south of
Bristol " ; a son of Joseph Gray, a farmer and blacksmith, and
Elizabeth Carmack. John and Rachel had issue. 3503. Elizabeth, b. 8,
20, 1832; ni. T'lnus G. Hamniar. 11, 10, 1852. Xo further record.
3504. Eldridge H., b. i, 23, 1835 : m. there, ^laria F. Beidleman, 3,
2, 1864. No further record. 3505. Sarah Catherine, b. 10. 13, 1839;
d. there, i, 15, 1891 ; m. Robert F. Betterton, 11, 15. 18.S9. ô
further record. " Eldridge 11. Gray resides near this place, on his

father's old homestead. He is a successful farmer, and has five
children, four sons and one daughter. Elizabeth Hammer also resides
here, and has two children. Hattie and Lizzie Lee. James C. Hammer
died 1887. Catherine Betterton resided here up to her death : three
children survive her: Thomas J., anrl two daughters, Fannie and
Zella. Robert F. Betterton died about 1881." — G. T. H., Bristol.
Tenn.. 2. 10. i8qi.

SEVENTH GENERATION. 317 iiii. Cornelius Harlan"
(Thomas." John''), farmer, 1). i, 12, 1807. in Hawkins Co., 'i-nn. ;
reninved in 1832 to X'erniillion Co., Ind. ; d. there, 8, 8, 1843 ; bnr.
there in family 1). g. He m. i, 16, 1834. in .Vewport. that county,
Martha Tate, b. 8, 21, 1811, in Hawkins Co., Tenn. ; d. 10, 3, i8()8.
in Vermillion Co., Ind. ; bur. beside the husliand. The direct cause of
Cornelius Harlan's death — " Father was injured by haiiiti' a
pitchfork run thr. 1832, in ermillion Co. ; a dau. of Patton jI. and
Ann (Campbell) .Merrimon, farmers and natives of Hawkins Co.,
Tenn. They have no children; residence. 1891, Newport, Ind. 3507.
John (farmer), b. 4, 23. 1839, in 'ermillion Co. ; m. there, 1873,
Hattie -Shephard, b. in that county ; a dau. of Benjamin and
Elizabeth. No children; residence, 1892, on their farm near Dana,
'ermillion Co., Ind. 3508. F.ldridge (farmer), b. 1. 30, 1840. in
'crmillion Co.; m. 2. 14, 1864, Matilda Merrimon. b. 9, 24. 1839, in
ermillion Co.: living, 1890; a dau. of Patton M. and Ann, and sister
to the wife of his brother James. Children: (i) Laura, b. x, 30, 1865.
(ii) Callie, b. 4, 26, 1867. (iii) Thomas C, b. 6, 24. 1868. (iv) Josie
Belle, b. 6, 17, 1880; all b. in 'ermillion Co., Ind.. and .ill unm.,
1892; residence, near Newport, Ind. 35CX). Josephus, b. 8, i. 1842.
in Vermillion Co. ; m. 1888, at Dana, Ind., Emma 'alters, b. 1856,
in Edgar Co., 111. No children. He is a druggist, and in 1892 was
residing at Dana, Ind. 1114. Susanna Wiley" (Margaret." Rebecca^),
a Friend, Ij. 1778, in Chester Co., Pa. ; d. there, 10 Mo., 1812, near
Chadd's Ford, (now in) Delaware Co., Pa. ; bur. in Friends' Burying
Grounds at Birmingham Meeting House. She m. about 1708, in
Chester Co., Samuel Wilson ( farmer 1, b. there; d. there, near
Chadd's Ford ; bur. at Birmingham Meeting House ; a son of John
^^'ilson (gentleman), b. at Kendall, 'estniinster Co., England, and
Mar- Faulkner, b. in Chester Co., Pa. For this line of the Wilson
family see page 132. Issue: 3510. William, b. 1799; d. I. 25, 1S79:
m. .Sophia Stackhouse. 351 1. Mary, b. I, 2. 1801 ; d. 5, 18, 1888;
m. Benjamin Orin, 11, 23. 1826. 3512. Susan, b. 4, 29. 1811 ; d. 11,
17, 1891 ; m. Fphraim Wilson, 3 Mo., 1830. 1115. James Woodward"
(Rebecca," Hannah-'), w-eaver, farmer, b. 2. i, I7()i, in Chester Co.,

Pa.; d. there, 4. 21. 1841 : bur. there, in Friends' Burying Grounds at
Bradford fleeting House. He m. Sarah Cloud, who d. in that county
and was bur. at Bradford Meeting Hotise ; a dau. of Mordecai Cloud.
Jr.. anil .'bigail Johnson. Issue : 3513. Jane, who d. unm. 3514.
Rebecca, who m. Thomas Martin. N^o further record. 3515. Rachel,
b. 5, 31. 1802 ; m. Ellis Woodward. No further record. 3516.
Elizal)eth. who m. Taylor Strode. No further record.

318 THE HARLAX FAMILY. 111-]. Abner Woodward'
(Rebecca.'' Hannah '). farmer, b. 3, 17, 1765. in Bradford Twp.,
Chester Co.. Pa. ; m. there, about 1798, at Isaac Woodward's home,
Elizabeth (Clark) Harlan, b. 9, 13. 1766, in Chester Co., Pa. : a dau.
of John and Sarah Clark anil widow of Stephen Harlan (Xo. 352). In
2 Mo., 1808. they removed to Mill Creek Hundred, Xew Castle Co.,
Del., where Elizabeth d. 9. 18. 1844. and Abner, 4, 19. 1846, and
both were bur. in Friends' Burying Grounds at Hockessin Meeting
House in that county. Issue : .35I7- Joseph, b. 11. 12, 1799; d. 2. 5.
1875: m. Mary Klair. i, 5, 1826. 3518. Joshua, b. 6. i, 1801. in
Chester Co.: d. 12, 21, 1863, in Mill Creek Hundred : bur. there, in
Friends" Burying Grounds at Hockessin Meeting House : unm. 3519.
Rebecca, b. 5. 30, 1803, same: d. i. 3, 1874, in Mill Creek Hundred:
bur. in Wilmington and Brandywine Cemetery at Wilmington. Del.
She m. 9, 15, 1853, Joseph Worrall. Xo issue. 3520. Abner, b. 3, 9,
1805 ; m. Julia Hoopes. 12, 12. 1859. 1 1 19. Lydia Woodward ' (
Rebecca," Hannah ■') . b. 5. 6. 1769, in Chester Co.. Pa. : d. there.
3, 3, 1843, at Marshaliton : bur. there, in Friends' ISurving (irnimds
at Bradford Meeting House. She m. 1797, by John Graves. Esq..
Abraham Martin (innkeeper), a Friend, b. 9, 26, 1765, in Chester
Co.. Pa.; d. there. 4. 2. 1826. at Marshaliton : bur. there, at Bradford
Meeting House : a son of ( ieorge and Martha ( W'iddows) ^lartin.
Issue: 3521. George, b. 1.9, 1798 : d. i. 31, 1892 : m. Orpha
Marshall 3522. Joseph, b. 9, 24. 1800, in Chester Co. : d. there, i, 7,
1863. at Marshaliton : bur. there, at Bradford Meeting House : unm.
3523. .bner. b. 6, 27, 1802 : d. 3. 2j, 1877 : m. Eliza Seeds. 3524.
Martha, b. 2. 7, 1804. same: d. there. 4, zt,. 1881, at Marshaliton:
bur. there at Bradford Meeting House: unm. 1121. Joseph
Woodward' (Rebecca." Hannah"'), farmer, b. 10. 12. 1773. in West
Bradford Twp., Chester Co., Pa. : d. there, 11, 5. 1833, ^t
Marshaliton : bur. in Friends' Burying Grounds at Bradford Meeting
House. He m. 2, 21, 1809, in Chester Co., before Moses Marshall,
Esq., Elizabeth Mendenhall ( Xo. 997 ) . b. 10, 28. 1778: d. 2, 12.
1824, at Marshaliton : bur. tiiere : a dau. of I'homas Mendenhall (
Xo. 257 ) . Issue : 3525. Ruth, b. 9, 27, 1810: d. 2, 23, 1897: m.

Joseph P.. Ferrell : (2) George .Armstrong. 3526. -Margaretta, b. 5,
3. 181 2, in Chester Co. : living, 1897. at Stuart, la. : m. 1837, Milton
Rakestraw, and had: (i) JMary A., b. 9, 28, 1837. in Columbiana Co.,
Ohio: d. there, 2, 8, 1839. (ii) Benton, b. 2, 18, 1840; m. Hope
Winder, and in 1897 was farming and residing near Sherman, la. (iii)
William, b. i. 28. 1842: d. 8, 29, 1877: m. in Ohio. Jennie Laird. He
was a railroad engineer on the Chicago and Rock island Railway, and
was killed, 8, 29. 1877. in a wreck with 19 others, and in which 35
were injured. He was bur. at .Stuart, la. (iv) Elizabeth, b. 6. 12. 1845
: d. 5, 15, 1875, at Hastings, Xeb.* bur. at Stuart, la. She m. in Ohio,
.Alonzo Shelby. ( v) I-'ranklin '., b. 6, 15, 1847: m. Elizabeth Dowd.
He was a railroad engineer in 1879, and resided at Stuart, la.

SEVENTH GENERATION. 319 31527. l.avina II., 1). S, ( iXji
; d. 11, -M. iJ^'î ; "i- lolm 1"^- Kudi^crs, 12, 24, 1835. 3528. Mary
A., 1). 4, 16, iXUi; m. Samuel .ichols, u, 24. iiS_35; (2) Samuel
Elton, II. 20. 1832; (3) James McCormick, 1, y, 1870. No further
information. 3529. Minerva, b. 6, 7, 1818: d. 3, y, 1873, in
Tuscarawas Co., Ohio ; 111. loliii Kni.nht, I iMo., 1842. No further
record. 3530. Davis M., b. 3, 4, 1820, in Chester Co. ; d. 10, 2, 1880,
at .Montezuma, la. ; bur. there. He in. 8, 2y, 1848, at I'ort
Washini^noii, ( )hio, Matilda Rankin, b. 10, 1 , 1823, in Tuscarawas
Co., < )hio ; liviiii;-, 1897 ; a dau. of William Rankin (farmer), 1). at
Freejjort, Harrison Cô., ( )hio, and Christian Knight, b. near
W'heelinsj;', V. 'a. Davis ?>I. and wife went to Montezuma, la., in
1853. No children re])orted. 1 124. Jacob Woodward ' ( Rebecca,"
Hannah '), wheeKvriL;ht, b. 8, 31. 1781, in Chester Co., I'a. ; d. i,
22, 1846, in West I'.r-idford Tw]). ; bur, in friends' Burying Grounds
at liradford. (now) .Marshallton Meetint,^ House. He m. 2, 15, 1806,
in Chester Co., Pa., Lydia Woodward ( No. 1 137), b. 3, 10, 1782, in
West Bradford Twp., Chester Co. ; d. there, 3, 22. 1843 ; Imr. at I'.r.-
idford .Meetin.14 I louse ; a dau. of John Woodward and Lydia
.Martin ( No. 299) . Issue : 3531. Hannah, b. 8, 26, 1807 ; m. Aaron
Davis, 3, 8, 1832. 3532. Elisha, 1). II, 28, i8aj; d. i. 19, 1897; m.
,my b'ew, 1 i, 9, 1837. 3533. Eliza, b. 181 1, ia West Bradford Twp.;
d. there. 184,^; unm. ; bur. at Bradford Meeting' House. 3534.
Emnior, b. 12, 24, 1816 ; d. 12, 22, 1859 : m. Sarah Silvester, 12,
12. 1844. No further record. 3535. .nn. b. 1818, in (.'hester Co.; d.
there, 2, 1, 1851 ; bur. there. She m. 1;, I, 1842, at West Chester,
Elisha Baily, b. 8, 12, 1819, in Chester Co.; d. II, 22, 1849, '"
MuskinsTum Co., ( )hio ; bur. there. They moved to that county
where the children were b. .fter his death the widow and children
returned to Chester Co., Pa. Children : (i) Ruth Ann, b. 11, 2, 1842;
d. 6. 14, 1843'. ( " ) Mifflin V., b. 5, I, 1844 ; m. 3, 12, 1870. .nnie
Eriôod, b. 12, i, 1848, in Chester Co., F'a. ; a dau. of Jacob and
Mary; residence, 18110, Pottstown, Pa. (iii) Lydia, b. 6, 2y, 1847: m.
6, 10, 1875, William Ciellar ; b. 8. 27, 1847, in Ciermany. No issue,
(iv) Elisha, d. young'. 3536. ( jeorge, b. 1820, same ; d. there, 1822.

3537. Abner. b. 1823, same: unm., 1892; residence, I'ottstown, Pa.
3538. Rebecca, b. 7, 9, 1825 ; m. James Peoples, i. 2. 1851. 1126.
Joseph Martin' (Caleb.'' Hannah''), farmer, b, 12, 2-;. 1767. in West
I'radford Twp., Chester Co., Pa.; d, about 1817 in C]iper Providence
Twj),. Delaware Co.. I'a.; bur. probably at Bradford Meeting House.
He ni. 5, 8, 1788, Hannah Malin, b. 1767, in Chester, (now in)
Delaware Co., Pa. ; d. 11, 20, 1845, in Upper Providence Twp.; bur,
there; a dau. of William and Elizabeth .lalin. Children all b. in Cpper
Providence Twp., now in Delaware Co., Pa. Issue : 3539. Joshua, b.
8, 24, 1781); ni. Margaret Hizer, No further information obtained.
3540. Jehu, b. 8, 16. 1792.

320 THE HARLAN FAMILY. 3541. Elizabeth, b. 8, 25, 1796;
d. there, 2. 27, 1817 ; unni. 3542. Ruth, b. 10, 16, 1799. 3543.
Orpha, b. 3, 31, 1802; d. there at Rockdale. 1845; m. in Middletown
Twp., that count}', about 1820, William Blair (blacksmith), b. in
County of Down, Ireland ; d. 7, 24, i860, at 'alley Forge, Pa. : bur.
at Sandy Bank, near Media. Delaware Co., Pa. ; a son of William Blair
(farmer). Children : (i) Mar-, b. 8, 14, 1822 ; d. in infancy. (ii)
William, b. 3, 7, 1824; d. in infancy, (iii) Hannah, b. i, 13, 1826; m.
John Thompson, (iv) William, b. 1827. (v) Mary, b. 2, I, 1830: m.
Jeremiah Stevenson, (vi) Elizabeth, b. i. 31, 1832: d. 4. 24. 1861 ;
m. David Smiley, (vii) Rachel, b. 9, 3, 1833; d. young, (viii) Louisa,
b. 4, 12, 1835: d. 4. 14. 1893: m. Reuben Marker, (ix) Sarah Jane, b.
3. 9, 1837: m. John Fields, (x) Joseph, b. 5, 12, 1839. (^') John, b.
7, 9, 1841. (xii) Ruth Ann, b. 6, 25, 1843: m. William Fields, (xiii)
Orpha. b. 1845: d. in infancy. 3544. Samuel, b. 3, 18, 1805, same;
d. there, i. 12, 1861 ; bur. there, at Sandy Bank Burning Grounds,
near ^ledia. The name of his wife and her record were not obtained.
He was a paper maker and member with Friends. Children: (i)
Joseph, b. 10, 6. 1826: d. 2, 7, 1848: unm. (ii) Wilham L., b. 7. 20,
1828 : d. 4. 3. 1836. (iii) Sarah .nn, b. 3, 2, 1830: d. 5, 10, 1844.
(iv) Elvina. b. 2. 21. 1832; d. 5. 8. 1863: unm. ( v) Samuel, b. 10,
10, 1833 ; d. 3, 20. 1835. (vi) Samuel H., b. 8, 19, 1835: d. 4. 3,
1836. (vii) Eli^a Jane, 11. 30. 1837; d. 12, 21, 1879: ni. William H.
Ruth, who d. 9. 16. 1884. leaving a son, 'illiam R., b. 3. I. i860;
residence. 1890. Philadelphia, (viii) Hannah H., b. 3. 6. 1840 : d. 5.
8, 1844. (ix) Philena. b. 7, 16, 1842 ; d. 10, 25, 1868: unm. (x)
William R., b. 11. 11. 1844; d. 12, 22. 1847. .All b. in Upper
Providence Twp., Delaware Co., Pa. 3545. Hiram, b. i, 22. 1808.
3546. Caleb, b. II. 3, 1810. 3347. Jacob, b. II. 3. 1810: twin to
Caleb. 1127. George Martin' (Caleb," Hannah'), farmer, b. 11, 12,
1770. in West Bradford Twp,, Chester Co., Pa.; d. 3, 11, 1835. near
Wellsville, Columbiana Co., Ohio: bur. there. He m. about 1800. in
Chester Co.. Pa.. Naomi Taylor, b. there 8, 26, 1779 ; ci. 4. 26. 1847.
in Columbiana Co.. Ohio : bur. there : a dau. of ^^'illiam and
Elizabeth (Harvey) Taylor, both natives of Chester Co., Pa. George

and Naomi em. to Ohio in 1816. Issue : 3548. Olive, b. 3, 30, 1801 ;
d. 8, 26, 1884 ; m. James S. Stewart, 1820. 3549. Mahlon, b. 11, 18,
i8o6;d. 4, 22, 1876 ;m. .melia Stewart; (2) Farinda Jackson. Xo
further record. 3550. Caleb, b. in Chester Co.. Pa. 3551. Warwick, b.
11. 8, 1810; d. 12, 16, 1884; m. Rachel Harbaugh, 5, i. 1838. Xo
further record. 3552. Sarah. 3553. Alpheus, b. 9, 6, 1813, in
Delaware Co., Pa.; d. 9, 30, 1851 ; m. Mary ^^'illoughby ; (2) Ann
Eliza Robinson. Xo further record. 3354. Hannah. 3555. George, b. 4,
23, 1820: d. 4. 5. 1832 : m. .lbina Brown, 5. 25, 1843.

SEVENTH GENERATION. 321 3556. William F. (minister in
Christian Church), b. 7, 30, 1823. in Jefferson Co., ( )hio : m.
Limaville. Stark Co., Ohio, I'riulence Schilling, b. there, living, 1890.
willi her son at I.os Gatos, Cal. ; a dau. of Jacob Schilling
(merchant), b. 1 Mo., 1800, in Wiirttemberg, Germany, and Sarah
Ann Lewis, b. 9, 30, 1801, in Delaware. William F. and wife and
family removed in 1868 to Wisconsin, where he d. 3, 12, 1882, at
Mill Creek, in Richland Co.; bur. there. Children: (i) Leonard J., b. 2,
10, 1854, in Wellsville, ( )hio ; m. 10, 22. 1879, I'lora A. Barnes:
residence. 1890. Los Gatos, Cal. (ii) Clarence T., b. 3, 13. 1861 : (1.
5, 6, 1862, at Limaville. ( Hiio. 1128. Warwick Martin" (Caleb,"
Hannah^), farnu-r, b, 2. 24, 1773. in Chester Co., I'a. ; d. there, 7,
2, 1853, in West Bradford Twp. ; bur, there, in Friends' Burying
Grounds at Bradford IMeeting House. He m. 10, 6. 1796, by
E])iscopal ceremony at Holy Trinity (Old Swedes) Church.
Wilmington. Del.. Ruth Miller ( Xo. 1140). b. r. 29, 1772. in Chester
Co. : d. there, i, 17, 1850. in 'est Bradford Twp.; bur. at Bradford
Meeting House: a dau. of Benjamin Miller and Hannah Martin ( Xo.
300) : both b. in Chester Co., Pa. Issue : 3557. Rufus Taylor, b. 1,
18, 1797, in Chester Co.; d. there, 10, 2, 1826. He m. 9, 29, 1816.
Hannah Osborne, b. 7, 27, 1800. in that county: d. II, 24. 1872. at
Salem. Columbiana Co.. Ohio : bur. there ; a dau. of Samuel.
Children: ( i ) Elizabeth, b. I. I. 1818 : d. 9. 28, 1820. (ii) 'arwick
W.. b. 11. 17, 1819; m. Sarah McFarland, 9. i. 1842. (iii) Samuel (
)sljorne. b. 1. 17. 1822: d. 2. 18, 1886, in Columbiana Co., Ohio: m.
Caroline B. Gheen : (2) Hannah Gheen. (iv) Caleb Rufus. b. 6, 24.
1824 : m. McFarland. 3558. Maria, b. 8, 2y. 1798. in Chester Co. ; d.
there. 9. 8. 1802. 3559. Abner. b. 7, 9. 1802. in Chester Co. ; d.
there. 8. 24, 1802. Warwick Martin at one time was in charge of the
Chester County Infirmary. 1 134. George Woodward' ( Lydia,"
Hannah'), farmer, miller, b. 9 Mo., 1774, in 'est Bradford Twp.,
Chester Co., Pa.; d. 10, 8, 1842, in Champaign Co., Ohio; bur. there,
in Treacles Creek Burying Grounds. He m. 6, 30, 1796, at the home
of her uncle, Jonathan BufTington, and before Moses Marshall, Esq.,
Alice Buffington, b. 9, 11, 1777, in Chester Co., Pa. : d. 8. 11. 1851.

in Champaign Co.. ( )hio : bur. there, beside the husband ; a dau. of
Ephraim Buffington and Ann Clayton, the former a miller. In 1833
George and Alice, accompanied by their children, em. to and settled
on a farm near Urbana. Champaign Co.. Ohio. The children were all
b. in Chester Co.. Pa. Issue : 3560. Gideon, b. 8. 4. 1797; d. 9. 11.
1833. at Columl)Us. (.)hio. He was by trade a carpenter : d. unm.
3S6i. Lydia. b. 4. 18. 1799; d. 7. 18. 1889: m. John Harlan ( Xo.
1071 ). 9, 9, 1818. 3362. lonathan. 1). 6, ifi. 1801 ; rl. 8. 24, 1S88:
m. Sarah ( Yarnall) Robinson, 1836. 3363. John. b. 3. 17, 1803: m.
Elizabeth Hollingsworth, 3, ^3. 1828. 3364. .nn. b. 12. 23, 1803: d.
10. 17, 1898: m. Joshua Harlan (Xo. 1075). 2, 22. 1837.'

322 THE HARLAX FAMILY. 3565. Eliza, b. 2, 29. 1808 ; d. in
Champaign Co., Ohio : unm. 3566. 'ashington, b. 4. I, 1813: d. 9.
13, 1885; m. Jeanette Thompson, 1844. 3567. Ephraim. b. 8. 17,
1815 ; m. Margaret Lamborn (Xo. 6602) and Hannah Elsworth ;
residence, 1890, Woodstock, Champaign Co., Ohio. 3568. illiam, b.
6, 22, 1818: m. Susanna R. Crawford, 2. 18, 1845. 3569. George, b.
7, 14, 1821 : went to CaHfornia, 1849: unm., 1890: residence.
Johnsville, Cal. 1135. Joel Woodward' (Lydia," Hannah'), farmer, b. 3,
17, 1775, in West Bradford Twp., Chester Co., Pa. ; d. there, 12, 11,
1815 ; bur. in Friends' Burying Grounds at Bradford Meeting House.
He m. about 1797. in Chester Co., Pa., Elizabeth Nutt (a Friend), b.
there. 9. 21, 1781 ; d. there, 8, 11, 1837, near -Marshallton. in West
Bradford Twp.: bur. there, in Friends' Burying Grounds: a dau. of
John Xutt, cabinet maker in Chester Co. Children b. in ^^'est
Bradford Twp., Chester Co. Issue: 3570. Evan. b. II, 25, 1798: d.
there, 9, 9, 1800. 3571. Jehu, b. 4, 6, 1802 : m. Keziah Harry. No
further record. 3572. Sarah, b. 9, 5, 1803 : d. 3, 17, 1870: m.
Charles E. Moore, 4, 14, 1827. 3573. Imlah, b. 11,6, 1805 : d. there,
9 Mo., 1869: unm. 3574. Xelson, b. 8, 2"/, 1807 : m. Adeline Sparks.
Xo further record. 3575. Pamelia, b. 9, 6, 1810 : d. 2, 20, 1857 : m.
David C. Woodward. 3, 16. 1848. 3576. Mary. b. 7, 25, 1812 : m.
Cloud Marshall, i, 27. 1841. 3577. Joel, b. 7. 23, 1814: m. Phebe
Trimble. 10. 22. 1846. 1136. William Woodward' (Lydia.'' Hannah''),
wheelwright, farmer, a Friend, b. 2, 2S. 1779. in West I'radford Twp.,
Chester Co., Pa.: d. there. 4, 28, 1827: bur. there, in Friends"
Burying Grounds at Bradford Meeting House. He m. there, 6, 13.
1805. by ceremony of Friends, Lydia Seeds (a Friend), b. 10, 31,
1781, in East Bradford Twp.. Chester Co. : d. 12. 30. 1852. in West
Bradford Twp. ; bur. there, at Bradford Friends' Meeting House : a
dau. of George Seeds (farmer, weaver), b. 1750, in East Bradford
Twp., and Margaret Hoopes.b. 3. i, 1744/5. .At the time of their
marriage, 'illiam resided in West Bradford, and Lydia in East
Bradford Twp. Issue: 3578. Job, b. 1806, in Chester Co. : d. there,
1806. 3579. George, b. 1,8, 1807. same : d. there. 9. 14, 181 3.
3580. Emmor, b. 1808. same: d. there. 7. 25, 1808. ' 3581. Cheyney,

b. 1809, same ; d. there, 4, 23, 1811. '3582. Sidney M., b. 6, 29.
1814; m. Joel Hagerty, 2. 21, 1839. Xo further record. 3583. Hannah
S., b. 7, 31, 1817, same: d. 10, 14, 1868: m. William Loller ( Xo.
3400), II. I, 1838. 3584. Margaret H.. b. 7. 16, 1825 : d. 2, 4, 1869 :
m. Wilson Cheyney, 2, 2, 1843. Lydia m. (2) James Woodward,
whose first wife was Rachel Woodward (No. 1 139). 1138. John
Woodward' (Lydia," Hannah"), hotel-keeper, b. 1783. in West
Bradford Twp., Chester Co., Pa.: d. 6. 10. 1873. in Cain Twp., that
county; bur.

Slil-EMll GENERATION. 323 at Marslialltoii (formerly
liradford) I'.uryini; ( irounds. lie ni. y, i6, 1807. at her father's liome,
by Moses ^hlrshall, J£sq., Elizabeth Woodward, b. 2, 8, 1787, in
Chester Co., Pa. ; d. there, 12,22, 1854 ; bur. there, at Bradford
.Meeting House ; a dau. of James Woodward (a wheelwright), b. 2,
6, 1746, and Jane IjuUock, h. 10, 8, 1748: both b. in Chester Co.,
I'a. Children all b. in that county. Issue: 3585. Eliza A., b. 10, 24,
1807 : ni. Lewis Windle. No further record. 3586. Albina, b. 9, 14,
i8o

324 THE HARLAN FAMILY. Co.; d. there. 8, lO, 1853; bur.
there, in Coatesville Cemetery; a dau. of Joel Baily, Jr., b. 12, 16,
1732, and Elizabeth Marshall, b. 3, 6, 1741 ; both in Chester Co., Pa.
Their children were all b. in Chester Co. Issue ; 3606. Eliza, b. 8, 15.
1798; d. 5, 16, 1880, at Coatesville; bur. there. She m. 3, 21, 1833,
in East Cain Twp., Chester Co., Lewis Hoopes. She m. (2) J. Milton
Davis; (3) 9, 30, 1846, Robert Hope (farmer), b. in Chester Co. ; d.
there. 11. 21, 1877, at Coatesville; bur. there. Xo children. 3607.
Benjamin I., b. 7, 13, 1800: d. 5, 2, 1882; m. Catherine 'astine, 9,
19, 1822. 3608. Joel Baily (farmer), b. 11, 28, 1802; d. 4. 4. 1874,
at Coatesville; bur. there. He ni. 3, 13, 1854, at Coatesville, Martha
('astine) Miller, who d. there, and was bur. in Brandywine Manor
Cemetery ; widow of Joseph J. Miller; a dau. of Benjamin 'astine.
Joel m. (2) 3, 4, 1874, at Coatesville, Susan (Denny) Marshall, of
that boroug:h. Xo children. 3609. Hannah, b. 5, 8, 1805; d. there,
12. 6, 1886. at Coatesville: bur. there. She ni. William Scott, a
farmer, who d. 12, 9, 1891, at Coatesville; bur. there; a son of
Thomas Scott. Children: (i) Mary, who d. young, (ii) James L., who d.
1876. ( iii) Joseph M. 3610. Joseph, b. 9, 7, 1808; d. there, 9, 7,
1812. 361 1. Mary C. b. 7, 23, 181 1 ; d. 10, 28, 1867, at Trenton, N.
J. : bur. in Coatesville Cemetery. She m. 2, 10, 1842, Peter E. Vastine
(physician), who d. at Trenton and was bur. beside the wife. Child:
(i) Rachel Ella, who m. Abner T. Griffith, and in 1894 was residing at
Towanda, Pa. 3612. Pamelia, b. i, 25, 1813 ; d. there, 8. 20. 1872 ;
l)ur. there, at Coatesville ; unm. 3613. Ruthanna, b. 5, 2. i8iy; d.
there, 11, 11, 1861 ; bur. there, at Coatesville. She m. 5, 26, 1857,
Thomas Hope, and had; (i) Robert Heslop, whose residence, 1894,
was Philadelphia. 1 143. Mary Miller " (Hannah," Hannah '), b. 10,
22, 1778, in West Bradford Twp., Chester Co.. I'a. : d. there, 7, 22,
1852, at West Chester. She m. about 1804, James Chamberlain, b.
1772: d. 7, 14, 1822. at West Chester; both bur. probably near that
place. Issue : 3614. Miller, b. I, 6. 1805 ; d. in infancy. 3615. Maria
.1.. b. i, 29, 1814, in Chester Co.. Pa. ; m. 10, 29, 1843. =»t West
Chester, William Parker ( formerly in the hardware business ; then
secretary of a coal company), b. 10. 16. 1816, at Philadelphia ;

living, 1890; a son of Joseph Parker, who d. in Philadelphia, 1853.
and Deborah Smith, who d. there, 1846, aged 55 years ; residence,
1890, Xo. 1253 Mount 'ernon St., Philadelphia. Children: (i) Joseph
M., b. 3. 14, 1847. ■ ('') ^f'lry C, b. 12, 14, 1850. (iii) Emma, b. 6,
20, 1855. .Ml b. in Philadelphia, and all residing there, 1890. unm.
3616. .Xbner Miller, b. 9. 2. 1817, in Chester Co.; d. 7, 12, 1865, in
Philadel])hia ; bur. there. He m. 1 1, 30. 1842, at West Chester,
Harriet W. Jones, b. 10, 25, 1821, in Chester Co. : d. there, at est
Chester, 4. 26, 1891 : bur. there; a dau. of Joseph Jones (iron
merchant) and j Hainiali .Solomon. Children: (i) iicnjamin Miller, b. 9,
5, 1843;

SEVENTH GENERATION. 325 (i. 5, 25, 1885, at Chicas^o,
111.; m. luiiiiia Saunders, (ii) Mary, b. 7, 21, 1845: d. 6, I, 1851. (iii)
Joseph James, b. 3, 13, 1847: d. 7, II, 1848. (iv) Martha James, b.
6, 29, 1849; unm. (v) Anna, 1). 6, 16, 1855; unm., 1894; residence.
West Chester, Pa. (vi) P'rank b. 3, 10, i85(): attorney-at-lavv ;
residence, 1894, Portland, Ore. 1147. Elizabeth Martin' (Jusepli,"
Hannah '), b. 8, 14. 1782, in West I'.radford Twp., Chester Co., Pa.:
d. there, 5, 7, 1851, in that tnvnshi|.: hur. at I'.r.adford Friends'
Meetini; House (now Marshallton) Puryini; (irounds. She m. 2, 26,
1801, at the home of Thomas Worth, and before Moses Marshall,
Esq., Benjamin Ford, b. 8, 9. 1771, in Chester Co.; d. there, 4, 9,
1832; bur. at Bradford MeetinjT House ; a son of John b'ord and
Margaret .Mortjan. the latter 1). in Chester Co., Pa. Issue : 3617.
Lydia, b. 2, 5, 1802 ; d. 9, 8, 1888 : m. William Lewis. 2, 28, 1828.
3618. Joel Harlan, b. i, 30, 1804, in Chester Co.; d. there, 1, 30,
1873, in Juniata Co.. Pa. : m. Keech. No issue livinjj. 3619. Thomas
Worth, b. 12, 22. 1806; d. v 4. 1882; m. Rachel (iriffith. 2, 19, 1829.
3620. Benjamin, b. 7, 21. i8ck;; m. Catherine Zeigler, 1833. 3621.
Margaret, b. 2, 4, 1812 ; d. 4, 3, 1892 ; m. Charles Piaily. 3622.
Rufus M.. b. T, 17, 181 5 ; d. 12, 25. 1891 : m. Harriet Lewis ; (2)
Sidney Lewis. 3623. John. b. 8, 29, 1817; (1. 11, 7. 1897; '"■ Phebc
b'ling; (2) Mary L. Windle ; (3) Alary E. Johnson. 3624. Hannah, b. 7,
zt,. 1819; d. i, 26, 1890; m. Humphrey Ingram. 3625. Elizabeth, b.
7, 2^. 1822; d. 10, y, 1845; '"■ loshua Sharjiless, 10, i^, 1840.
1150. Joel Martin' (Joseph." Hannah'), farmer, b. 3. 22, 1786, in
West Piradford Twp.. Chester Co., Pa.; d. there, 2. 13, 1857; l)ur.
there, in Friends' Burying Grounds at Bradford Meeting House (now
Marshallton). He m. i, 13, 1825, in that county, Margery Townsend,
b. there, g, 22, 1803 ; d. there, 9, 7, 1877, in West Bradford Tw]). ;
linr. there, in Friends' Burying Grounds at Bradford Meeting House ;
a dau. of Jonathan Townsend : b. 8, 3, 1781, and Susanna Eachus,
b. 3, 18, 1783: both in Chester Co., Pa. Their children were all b. in
West Bradford Twp., Chester Co. Issue : 3626. Jonathan, b. 5. 26,
1825 ; d. there, 9, 5, 1832. 3627. Joseph, b. 7, 10, 1826 :'d. there,
7, 3, 1827. 3628. Abraham, b. 10, 9, 1827; d. there, i, 19, 1837.

3629. Susan A., b. 3, 3, 1829 ; d. there, 9, 19, 1832. 3630. Emmor,
b. 7, 23, 1830; d. there, 8. 21, 1832. 3631. Cheyney (farmer), b. 5,
14. 1832 ; m. there, 3 Mo., 1861, Sidney Osborn, who d. 3, 16, 1863
; a dau. of Joseph ; residence, 1890, Marshallton, Chester Co., Pa.
Child: (i) Pyle Osborn. b. 10, 17, 1862; m. Mattie Quillen, 1887.
3632. Lewis, b. 10. 30, 1833 : d. i, 20, 1884; m. Sarah E. Thornburg.
3633. Hannah, b. 6, 30, 1835 '• unm., 1890; residence. Marshallton,
Pa. 3634. Mary, b. r, 17, 1837; unm., 1890; residence, Marshallton,
Pa.

326 THE HARLAN FAMILY. 3635. George, b. 8, 4, 1838 ; d.
3, 30, 1879 : m. Mary T. Way, 5, 23, i860. 3636. Lydia M., b. 3, 14,
1840: m. 10, 25, 1866, William L. Johnson ; a son of George and
Susan (Lainhoff ) Johnson : residence, 1893, Philadelphia. Child: (i)
Carrie M.. b. 10, 22, 1867: unm., 1897; residence. Philadelphia.
3637. Alargaret, b. 6, 7, 1843 : d. there, 12. 5, 1844. 3638. Jacob, b.
4, 20, 1847. He m. 2, 15, 1871. Anna M. White, b. 7, 10, 1846, in
Chester Co. ; a dau. of Ezra and Lavina (Harvey) White ; both b. in
Chester Co. He is a farmer, and in 1899 was residing near
Marshallton. Pa. Children: (i) George H.. b. 12, 7, 1871. (ii) Helen L..
b. 4, 7, 1873. (iii) Joel, b. 7, 27, 1875. (iv) Emma, b. 7. 30. 1878. (v)
Lewis, b. 10, 26, 1880; d. 3, 14, 1882. (vi) Mabel, b. I. 13. 1883: d.
11,27. 1884. (vii) Ralph E..b. 1,30, 1885. (viii) Maude Alice, b. 8, 10,
1887 : d. 7. 29, 1888. ( ix) Erancis Earle, b. 7, 26, 1890. 1 166. Lydia
Richardson" (Dinah,'' Joel"), a Friend, b. 3. 16. 1785. in Lancaster
Co., Pa. : d. there. 6, 8, 1845 • 1"^""- there, in " Penn Hill "
Cemetery, Little Britain Twp. She m. there, 5, 16, 1817. David Parry
(a Eriend). b. there, 10, 20, 1778: d. there. 2. 28, 1875: bur. there,
in " Penn Hill" Cemetery: a son of John Parry, b. 9, 10, 1743, and
Rachel Eell, b. 8, 9. 1751 : both in Lancaster Co., Pa. Their children
were b. in Lancaster Co., Pa. Issue : 3639. Samuel, b. 9, 17, 1818;
d. there, 3, 4, 1879: m. Hannah Stubbs. 10, 31, 18733640.
Elizabeth, b. i, 19, 1820: d. 2. 5, 1855, in Harford Co., Md. ; m.
Charles C. Hairland, 4, 5, 1854. 3641. Sarah, b. 10, 24, 1821 : d.
there, i, 25, 1840: unm. 3642. David, b. i, 14, 1823 ; d. there. 5, 30,
1852 : unm. 3643. Joseph, b. 3. 29, 1825; unm., 1890. 3644.
Ruthanna. b. 4, 18, 1827: d. 6, 28, 1881, in Harford Co.. Md. : m.
Garrett Amoss, 3, 14, 1855. 1170, Isaac Pyle ' (Ruth,''' JoeH').
farmer. Friend, b. 5. 3, 1777, in Chester Co.. Pa. : d. there. 1 1, 2j.
1842, in London Grove Twp. ; bur. there, in Friends' Burying Grounds
at West Grove Meeting House. He m. about 1827, by magistrate,
Sarah E. Chalfant (a Friend), who d. in London Grove Twp. : bur.
there at West (irove Meeting. The compiler regrets that the records
of Sarah Chalfant are so incomplete. The children were b. in London
Grove. Issue: 3645. Eliza -Ann. b. 6. 4, 1828; d. there, I, 20. 1831.

3646. Isaac Xevvton, b. 8, 17. 1830; m. Hannah Baker. 6, 19, 1856;
(2) Elizabeth White (widow) ; (3) Sarah Baker, 12, 22. 1864. 3647. |.
Lindlev. b. 3. 9, 18^2: d. 6. 18. 1866: m. Tulia (Malone) Deiter. 3, 9,
■854' 3648. Cyrus, b. I, 26, 1834: m. Emily M. I'urtin. 5. 13. 1862.
3649. Franklin C, b. 3. 31, 1836: unm. 3650. Sarah Ann, b. i, 4. 1838
; m. David Martin. 12. 4. 1861. 1172. Joel Pyle ' (Ruth,'' Joel ■).
farmer, a Friend, b. 10, 2j, 1799, in London Grove Twp., Chester Co.,
Pa.: d. 12, 25. 1842. in Pennsbury Twp., that county;

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