![First International Coloquium on Numerical Analysis pp. 1-12 (1993)
D. Bainov and V. Covachev (Eds)
© 1993
O R T H O G O N A L P O L Y N O M I A L S A S S O C I A T E D W I T H
A N O N - S T A N D A R D M E A S U R E
Angel Arteaga* and Jaime Munoz Masque**
National Research Council of Spain (CSIC).
* IMAFF, Serrano 123, E-28006 Madrid, Spain, e-mail: ceeaiOO@cc.csic.es
**IETEL, Serrano 144, E-28006 Madrid, Spain, e-mail: vctqu01@cc.csic.es
Abstract
Let Z f be the group of infinite binary sequences, and let us denote by p. the Haar
probability measure on Zj". By means of the standard homeomorphism from TL^
onto the Cantor triadic subset K C [0,1], p induces a measure PK on K, which in
turn defines a measure v on the unit interval by simply setting i/(/) = PK(/K), for
every continuous function / 6 C ([0,1]).
According to the representation theorem, v is a Lebesgue-Stieltjes measure. Hence,
there exists a non-decreasing function a on [0,1] continuous on the right, such that
"(/) = fo /(*)<*«(*)•
It is proved that a is a "devil's staircase" in the sense of Mandelbrot, and the
associated orthogonal polynomials are studied.
Keywords: Orthogonal Polynomials, Cantor set, non-standard measures
1 Introduction
The purpose of this paper is to present some basic properties of the orthogonal polynomials
associated with a non-standard measure on the unit interval of the real line.
For the past ten years the interest of orthogonal polynomials has nothing but increased.
At the same time, the study of orthogonal polynomials associated to non-classical mea-
sures has become frequent (for instance, see the works of Nevai [1] and Ullman[2]).
Several works have recently appeared on the topic of what has been called semi-
classical orthogonal polynomials for some authors: Maroni[3], Ronveaux[4], Ronveaux
and Marcellan[5], etc.
Basically, these polynomials correspond to a weight function p which has been obtained
adding to a classical one p, a finite number of Dirac deltas located on some points x^ with
mass Ait > 0.
That is,
h'
p(x) = p(x) + Y , ~ x
k) •
k=I
From this point of view, the measure we consider is, clearly, a non-classical one, in the
sense that it does not correspond to any perturbation of a classical measure.
In a forthcoming paper we will try to extend the present results to more general
pro-finite abelian groups.](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-13-320.jpg)
![2 A. Arteaga and J.M. Masque
2 Homeomorphism between the group of sequences zeros and
ones and the Cantor triadic set
Let TL2 be the set {0, 1} and X ^ be the set of sequences of 0's and l's with the topology
product, considering in 'EI2 the discrete topology. Clearly TL2 is a compact topological
group.
Let K be the Cantor triadic set. As is well known, K is identified with the points of
the segment [0,1] whose irrational expansion in base 3 does not contain 1; i.e. x € K if,
and only if,
«1 . 0-2 , a3 an
and all the a;'s are zero or two. In this way, a homeomorphism between the sequences of
0's and l's and the Cantor triadic set is obtained: 9 : TL^ -> K , defined by ([6]):
2oi 2«2 2a,, , _ „ ,„.
= + + + + (a, £ TL2) . (1)
3 The Haar measure on TZ^
As is well known, any compact abelian group G has a distinguished measure fi: the unique
invariant measure for which the mass of G is equal to one; or, in other words, ¡i is the
standard Haar measure of the group G.
The Haar measure can be built in the following way:
Let / : ^ ^ —
> R be a continuous function only depending on a finite number of
components of the sequences of i. e., there exists n £ IN and a continuous function
g : ^ -> R such that:
, x 2 , . . . , xn, xn+,...) — , . . . , xn),
f o r all x = ( x j , x 2 , . . . , xn, xnJr,...) (E
Let T be the set of such functions. By the Weierstrass approximation theorem, T is
dense in the space C of continuous functions of the group so every continuous
function / : ¡Z^ —
> IR can be expressed as the limit of a sequence of functions /,, £ T
with respect to the uniform convergence: / = lim fn.
Again, let / £ T . We define ,«(/) to be:
= ^ £ / ( * ) ; (2)
i.e., M / ) = ^ ( 5 ( 0 , . . . , 0 ) + --- + «/(l,...,l)).
The measure defined above has the following properties:
i) n{f) < m a x | / ( x ) | = 11/11-
Proof:
= m a x
, - £ r ; lf(-T
)l = m a x
x - e ^ f ( * ) -](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-14-320.jpg)
![4 A. Arteaga and J.M. Masqué
Proof: By continuity, we only need to prove the above formula for a dense subset
of C (JZJ^ ; for example T• Each function / € T is a finite linear combination with real
coefficients of functions of separate variables; i.e., g ( x . . . ,xn) = g-i(xi) • • • gn(xn) .
By linearity, we only need to check the above formula for this last class of functions.
Then, according to the definition of /i (2), we have:
Moreover, for k > n we obtain:
^t Y , S(x) = ^ Y , ai(xi)---gn{xn)
l
2*-2
*" E 9i(xr) • • • gn(xn) = /*(<?),
(ll ,...,!„
thus proving the proposition.
4 The induced measure on [0,1]
Let us translate the homeomorphism V to the space of continuous functions on the unit
segment, (not only on the Cantor triadic set), defining a measure i/ that only pays attention
to the values of the continuous function on the triadic, in the following way:
" ( / ) = M / k ° Y > ) , all / e C([0,1]).
That is, we define the measure of a function in the interval [0,1] as the measure of the
function restricted to the points of the Cantor set.
R
5 Obtaining of the function <*(t)
It is well known that every measure on C([0,1]) is a Lebesgue-Stieltjes measure:
v
i . f )
=
/ ' / da , (4)
Jo
for a certain integrator a : [0,1] —
> R, monotone nondec.reasing, that we can assume to
be continuous on the right.
By definition:](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-16-320.jpg)
![Orthogonal polynomials 5
If in the formula (4) we could take / with discontinuities, taking / as the step function
f 1 0 < x < t
X,(x) = " "
0 t < x < 1'
u(Xt) = f Xtda= f da = a(t) - a(0).
Jo Jo
We can also assume a(0) = 0. Hence,
a(t) = w{Xt).
In order to obtain the a(t) function, we first consider the function X " ( : r ) , defined by:
f 1 0 < x <t
Xll
(i)= l - n ( i - i ) t<x<t+/n (5)
( 0 t+ l/n < x < 1
0 t t + l / n !
7 IN
Clearly, lim,,-.«, X ? = Xt. Hence, lim„^oo X " | k ° V = X ( | K °V on uu2 .
Now a(t) can be obtained as the mean of the values of X" in all the points of the
Cantor triadic set:
a ( i ) = / / ( X , ) = n(X,K °V ) = J i i n fi(X^K ° V)
(
= lim lim
n—oo k-^oo l / 2 f c
E W ( « ) ]
V /
( 6 )
as follows from the proposition in §3, been defined in §2 by (1).
Given the way the a function is obtained and the symmetry of the Cantor set about
the center of the segment [0,1], it is easy to see that a is symmetric about the point
(0.5,0.5). That is, a ( x ) = 1 - a(l - z); x 6 [0,1],
The a ( t ) function can also be seen as a normalized cumulative distribution function
with density distribution function
da{t) = lim 1/2* V] S(t-u)dt
k
—
>oo ^^'^
where <$(•) is the Dirac delta function. In this sense, the Cantor triadic set is the spectrum
of a(t).
With this notation,
a(r) = 1/2* £ f 6{t-u)dt = 1/2* £ 1.
U<T
That means: a ( r ) is the fraction of the number of points of the Cantor triadic set in
the domain [0,r] relative to the number in the whole domain [0,1].
Let us make two remarks concerning the Lebesgue-Stieltjes measure defined by the
integrator a function:](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-17-320.jpg)
![6 A. Arteaga and J.M. Masqué
I .ODO 0.000 0.002 0.004 0.006 0.008 0.010
Figure 1: a(t) in the domains [0,1] and [0,0.01]
1. The value JQ f{x) da(x) only depends on the values the function / takes on the
Cantor set. However, note that if / is a polynomial, then / is completely determined
by the values it takes on the triadic.
2. The picture of a looks like a devil's staircase in the sense of Mandelbrot [7, p. 62]
(see figure 1).
A similar measure has also been considered by Barnsley, Bessis and Moussa [8] in
dealing with a problem concerning the activity of the ferromagnetic Ising model, but
their work it is not related with ours.
In order to implement the algorithm to obtain a ( x ) , first let us consider the truncation
error that is obtained when only the first k digits are taken in defining the Cantor triadic
points:
k ° ° •
= Yi + e r r ' w h e r e e r r = i f ! ( a i 6
¿=1 6 !=/
f
c+l J
(7)
The maximum error will take place when in the point of the triadic considered all the
digits after the A;th are ones. In this case, the expresión of the error (7) is a geometric
progression, and hence:
E.g.: For k=10 max(err) ss 2 x 10~5
max(err) =
2/3fc+1 _ J_
1 - 1/3 ~ 3* '
k=15
k=25
7 x 10-8
1 x 10~12
To obtain a(t) we have to generate the sequences of O's and l's defining the triadic
points (truncated) a, obtain V(a) by (1), X"[V(a)] by (5), and, finally a(t) by (6) as
avereage of these values X1}tp(a)].
6 Orthogonal Polynomials
The function a(-) obtained can be employed as a nonstandard measure to build a sequence
of orthogonal polynomials with the product defined as:
< P{x),Q{x) >= f P(x).Q(x) da(x)
Jo](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-18-320.jpg)
![Orthogonal polynomials 1
B„
This product defines a sequence of orthogonal polynomials given by the recurrence
formula:
Pn(x) = (x - An)Pn-i(x) - BnPn-i{x), n> 1, (8)
with P-i{x) = 0 and P0(x) = 1.
According to the results by Nevai [1, §4.20], it is obvious that our orthogonal polyno-
mials can not satisfy a second-order linear homogeneous differential equation. This fact
is one of the most outstanding differences from semi-classical orthogonal polynomials.
It is easy to see that the recurrence coefficients are defined by:
_ < I P , I - I ( I ) , P„-i(a) >
< Pn-i(x),Pn-i{x) >
(9)
< xPn^(x),Pn^(x) >
< Pn_2[x),Pn.2{x) >
The coefficient B is arbitrary, but it is convenient, for sake of uniformity [9], to define
it as: Bx = fd da(t) = 1.
When the a function is symmetric about the origin, it is well known that all the
coefficients An are zero (see [10, THEOREM 4.3]). In this case, we are going to demonstrate
the following
T H E O R E M : When ail integrator function a is symmetric about the point (0.5,0.5) it
is verified that:
i) An = 0.5 , n = 1,2,...
ii) Pn(x) has even (odd) symmetry about x = 0.5 if n is even (odd).
That is, Pn(x) = ( —1 )"/'„( 1 - n = 0, 1,2,...
Proof: We have,
Po(x) = 1
< xP0(x)P0(x) > p
= „ , . ,. .—: = / xaa = Ml •
< P0(x)P0(x) > Jo ^
Given the symmetry of a, clearly A — 0.5 and, then, P(x) = x — 0.5.
Hence, i) and ii) are true for n = 0 and n = 1. By induction, supposing they both
are true for all k < n:
i)
_ < xPn-i(x)Pn-i(x) >
n~ <Pn-i(x)Pn-i(x)> '
that can be expressed as
, < (x-0.5)Pn.,(x)Pn_,(x) > , n r
—
r! 7—wi — r U.O .
< Pn^(x)Pn^(x) >
Here, the denominator of the fraction is always not zero, while in the numerator:
< (X - O^Pn-iMPn-^x) >= f x ~ 0 . 5 ) [Pn-^X,)]1
da(x) ;
Jo
if we do the change x = x — 0.5, results:
< (a - 0.5)/'„_i(x)/'„_i(a:) >= /°'5 (x [Pn.x{x + 0.5)]2
da(x + 0.5) = 0 ,
J-0.5](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-19-320.jpg)
![8 A. Arieaga and J.M. Masqué
given that [Pn _1 (x + 0.5)]2
is even and x odd. Hence, An = 0.5, for all n.
ii) In the recurrence formula:
Pn(x)(x - 0.S)Pn-x(x) - BnPn-2(x)
if n is even we have Pn-{x) = —Pn _i(l — x) and Pn _2 (a:) = f n _ 2 ( 1 — x), hence:
Pn(x)( 1 - z - 0.5)Pn _i(l - x ) - B„P„_2 (1 - x ) = Pn( 1 - x).
And similarly for n odd. •
T h e coefficients Bn have been obtained numerically, following the bootstrap Stieltjes
procedure [9]. An open question would be to find the general expression for Bn.
As far as we know, there is no method to do the quadrature in a simple form. Taking
into account that we only consider k digits in the expansion of f(x), the easier way
to obtain the quadratures involved in (9) is to solve them as Riemman-Stieltjes sums,
considering partitions such that in every interval there were at most one point of the
triadic set.
In this way:
< P(x),Q(x) >= j i ' P(x).Q(x)da(x) « /2k £ P[p{a)].Q[^a) (10)
afz'fflj2
where k is the number of digits taken to define the triadic points, and, thus, 2k is the
number of points of the triadic in [0,1]
T h e equations (9), taking into account the formula (10), let us obtain the coefficients
Bn. These coefficients are given in table 1 up to n = 26, obtained with k = 25.
n Bn Tl Bn n Bn n Bn
1 1.0000000000 8 0.0812180463 15 0.0189201066 22 0.0774248549
2 0.1250000000 9 0.0324016054 16 0.1410187727 23 0.0452801655
3 0.0500000000 10 0.1510749676 17 0.0386977786 24 0.1140167807
4 0.0914835164 11 0.0130941801 18 0.0801319585 25 0.0142362331
5 0.0276826008 12 0.0898122397 19 0.0086722736 26 0.1150376125
6 0.1267576108 13 0.0445443616 20 0.1577325433
7 0.0278390606 14 0.1026727454 21 0.0278943990
Table 1: Values of Bn for k = 25
In a similar way, the moments /i^. of the function a(x) can be obtained:
fin = fnda(x) « 1/2* £ M « ) ] " .
Jo k
The moments up to the order n = 26, obtained with k = 25, are given in table 2.
Those coefficients An, Bn allow us to put the polynomials in the usual form,
f ' n ( x ) — Xn + <ln,n-Xn 1 + (ln,n-2x" * + • • ' + 1ri,ll'' + an,0 ,
where the coefficients a;^ are obtained by iteration.
T h e coefficients an<J from n = 1 to n = 26 are given in table 3.](https://image.slidesharecdn.com/25523251-250512190734-3bc34645/85/Proceedings-Of-The-First-International-Colloquium-On-Numerical-Analysis-Reprint-2020-D-Bainov-Editor-V-Covachev-Editor-20-320.jpg)
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- 5. Proceedings of the First International Colloquium on Numerical Analysis
- 7. Proceedings of the First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 -17 August 1992 Editors: D. Bainov and V. Covachev ///VSP/// Utrecht, The Netherlands, 1993
- 8. VSPBV P.O. Box 346 3700 AH Zeist The Netherlands © VSP BV 1993 First published in 1993 ISBN 90-6764-152-9 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopy- ing, recording or otherwise, without the prior permission of the copyright owner. Printed, in Great Britain by Bookcraft (Bath) Ltd., Midsomer Norton.
- 9. CONTENTS Preface Orthogonal polynomials associated with a non-standard measure A. Arteaga and J. Muhoz Masqué 1 The numerical treatment of ordinary and partial Volterra integro-differential equations H. Brunner 13 Numerical methods for y" = f(x,y) J.P. Coleman 27 Numerical-analytical method for finding periodic solutions of nonlinear impulsive singularly perturbed differential systems V. Covachev 39 Special algorithms for the numerical integration of problems in orbital dynamics J.M. Ferrândiz and P. Martin 51 Polynomial approximation with side conditions: Recent results and open problems H.H. Gonska and X.-i. Zhou 61 Continuous numerical solutions and error bounds for matrix differential equations L. Jôdar and E. Ponsoda 73 Convergence rate estimates for the finite difference schemes compatible with the smoothness of data B.S. Jovanovic 89 Numerical solutions of differential-function problems Z. Kamont 97 An iterated method of computing the inverse mapping J. Muhoz Masqué and G. Rodriguez Sanchez 113 On using high orders finite elements for solving structural mechanics problems T. Scapolla 123
- 10. Numerical solution of differential equations for the analytic singular value decomposition K. Wright A new trust region algorithm for nonlinear optimization Y.-X. Yuan
- 11. Preface The First International Colloquium on Numerical Analysis was organized by UNESCO and Plovdiv Technical University with the help of the Austrian Mathematical Union, the Canadian Mathematical Society, Hamburg University, Institute of Mathematics of the Bulgarian Academy of Science, Kyushu University, the London Mathematical Society, Plovdiv University 'Paissii Hilendarski', Technical University-Berlin, Union of the Bulgarian Mathematics - Plovdiv Section, and sponsored by Bulgarian Mails and Telecommunications Ltd., and the firms 'Eureka' and 'Microtechnika Ltd.' It was held on August 13-17, 1992 in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 13-17 August in Plovdiv, Bulgaria. The Address of the committee is: Stoyan Zlatev, Mathematical Faculty of the Plovdiv University, Tsar Assen Str. 24, Plovdiv 4000, Bulgaria. The Editors
- 13. First International Coloquium on Numerical Analysis pp. 1-12 (1993) D. Bainov and V. Covachev (Eds) © 1993 O R T H O G O N A L P O L Y N O M I A L S A S S O C I A T E D W I T H A N O N - S T A N D A R D M E A S U R E Angel Arteaga* and Jaime Munoz Masque** National Research Council of Spain (CSIC). * IMAFF, Serrano 123, E-28006 Madrid, Spain, e-mail: ceeaiOO@cc.csic.es **IETEL, Serrano 144, E-28006 Madrid, Spain, e-mail: vctqu01@cc.csic.es Abstract Let Z f be the group of infinite binary sequences, and let us denote by p. the Haar probability measure on Zj". By means of the standard homeomorphism from TL^ onto the Cantor triadic subset K C [0,1], p induces a measure PK on K, which in turn defines a measure v on the unit interval by simply setting i/(/) = PK(/K), for every continuous function / 6 C ([0,1]). According to the representation theorem, v is a Lebesgue-Stieltjes measure. Hence, there exists a non-decreasing function a on [0,1] continuous on the right, such that "(/) = fo /(*)<*«(*)• It is proved that a is a "devil's staircase" in the sense of Mandelbrot, and the associated orthogonal polynomials are studied. Keywords: Orthogonal Polynomials, Cantor set, non-standard measures 1 Introduction The purpose of this paper is to present some basic properties of the orthogonal polynomials associated with a non-standard measure on the unit interval of the real line. For the past ten years the interest of orthogonal polynomials has nothing but increased. At the same time, the study of orthogonal polynomials associated to non-classical mea- sures has become frequent (for instance, see the works of Nevai [1] and Ullman[2]). Several works have recently appeared on the topic of what has been called semi- classical orthogonal polynomials for some authors: Maroni[3], Ronveaux[4], Ronveaux and Marcellan[5], etc. Basically, these polynomials correspond to a weight function p which has been obtained adding to a classical one p, a finite number of Dirac deltas located on some points x^ with mass Ait > 0. That is, h' p(x) = p(x) + Y , ~ x k) • k=I From this point of view, the measure we consider is, clearly, a non-classical one, in the sense that it does not correspond to any perturbation of a classical measure. In a forthcoming paper we will try to extend the present results to more general pro-finite abelian groups.
- 14. 2 A. Arteaga and J.M. Masque 2 Homeomorphism between the group of sequences zeros and ones and the Cantor triadic set Let TL2 be the set {0, 1} and X ^ be the set of sequences of 0's and l's with the topology product, considering in 'EI2 the discrete topology. Clearly TL2 is a compact topological group. Let K be the Cantor triadic set. As is well known, K is identified with the points of the segment [0,1] whose irrational expansion in base 3 does not contain 1; i.e. x € K if, and only if, «1 . 0-2 , a3 an and all the a;'s are zero or two. In this way, a homeomorphism between the sequences of 0's and l's and the Cantor triadic set is obtained: 9 : TL^ -> K , defined by ([6]): 2oi 2«2 2a,, , _ „ ,„. = + + + + (a, £ TL2) . (1) 3 The Haar measure on TZ^ As is well known, any compact abelian group G has a distinguished measure fi: the unique invariant measure for which the mass of G is equal to one; or, in other words, ¡i is the standard Haar measure of the group G. The Haar measure can be built in the following way: Let / : ^ ^ — > R be a continuous function only depending on a finite number of components of the sequences of i. e., there exists n £ IN and a continuous function g : ^ -> R such that: , x 2 , . . . , xn, xn+,...) — , . . . , xn), f o r all x = ( x j , x 2 , . . . , xn, xnJr,...) (E Let T be the set of such functions. By the Weierstrass approximation theorem, T is dense in the space C of continuous functions of the group so every continuous function / : ¡Z^ — > IR can be expressed as the limit of a sequence of functions /,, £ T with respect to the uniform convergence: / = lim fn. Again, let / £ T . We define ,«(/) to be: = ^ £ / ( * ) ; (2) i.e., M / ) = ^ ( 5 ( 0 , . . . , 0 ) + --- + «/(l,...,l)). The measure defined above has the following properties: i) n{f) < m a x | / ( x ) | = 11/11- Proof: = m a x , - £ r ; lf(-T )l = m a x x - e ^ f ( * ) -
- 15. Orthogonal polynomials 3 ii) Let fl6Sf, and let ta : -> ^ be the translation ta(x) ~a + x. l i f e J7, then / o ta G T and /i(/ o i„) = p { f ) . Proof: With the same hypotheses we get: ( f o t a ) ( x u . . . , x n , . . . ) = /(fl!+a;,,...,a = flf(ai + xi,.. . , a „ + i n ) . . . ) = (g ° t a ) ( x i , . . . ,xn,...). Besides, we get /¿(/0<a) = ^- Y, ( J o 1 . ) W = ^ E «/(a + a;)- Let us do the change of variable y = a + x. As far as is a group, x runs over if, and only if, y runs over W^. Thus /*(/ = 9(a + x) = -L </(t/) = //(/). Q ^ f T O T l ^ T 7 T I x^llu^ 3/6^2 The property i) allows us to extend /i to all the space C ( X ^ j of continuous functions by setting the condition: /j.(f) = lim f i ( f n ) , if / = l i m ( / „ ) , fn € .F. n—*oo n—»oo We will prove the definition is correct by showing that there exists the limit of the right side of the above equation and that it does not depend on the sequence chosen. Actually, K / » ) - M / m ) l = K/"n-/m )|<||/n-/m || < Win ~ f II + 11/ - U I -> 0, because fn /. Thus f i { f n ) is a Cauchy sequence and converges. If /„ is another Cauchy sequence which tends to /, it is verified: I M / n - K f n ) | = | H{fn ~ /n)| < ||/n ~ /n|| 0, b e c a u s e / „ - / „ - » / - / = 0. Then lim (//(/„) - /*(/„)) = 0, or, equivalently, lim/*(/„) = lim/z(/n). • By passing to the limit, the property ii) shows that /z, extended to C is also invariant under translations. Effectively: //(/ o ta) = lim f i ( f n o ta) = lim /t(/n) = l * { f ) - Moreover, the total measure is the unity. Actually, the function 1 belongs to T , and then fi(l) = jp • 1 = 1. Hence, the measure fi above constructed coincides with the Haar measure on ^ ^ • P r o p o s i t i o n : For every f eC ( x ^ ) , we have: M(/) = tlim ^ £ / ( * ) , (cf. (2)). (3)
- 16. 4 A. Arteaga and J.M. Masqué Proof: By continuity, we only need to prove the above formula for a dense subset of C (JZJ^ ; for example T• Each function / € T is a finite linear combination with real coefficients of functions of separate variables; i.e., g ( x . . . ,xn) = g-i(xi) • • • gn(xn) . By linearity, we only need to check the above formula for this last class of functions. Then, according to the definition of /i (2), we have: Moreover, for k > n we obtain: ^t Y , S(x) = ^ Y , ai(xi)---gn{xn) l 2*-2 *" E 9i(xr) • • • gn(xn) = /*(<?), (ll ,...,!„ thus proving the proposition. 4 The induced measure on [0,1] Let us translate the homeomorphism V to the space of continuous functions on the unit segment, (not only on the Cantor triadic set), defining a measure i/ that only pays attention to the values of the continuous function on the triadic, in the following way: " ( / ) = M / k ° Y > ) , all / e C([0,1]). That is, we define the measure of a function in the interval [0,1] as the measure of the function restricted to the points of the Cantor set. R 5 Obtaining of the function <*(t) It is well known that every measure on C([0,1]) is a Lebesgue-Stieltjes measure: v i . f ) = / ' / da , (4) Jo for a certain integrator a : [0,1] — > R, monotone nondec.reasing, that we can assume to be continuous on the right. By definition:
- 17. Orthogonal polynomials 5 If in the formula (4) we could take / with discontinuities, taking / as the step function f 1 0 < x < t X,(x) = " " 0 t < x < 1' u(Xt) = f Xtda= f da = a(t) - a(0). Jo Jo We can also assume a(0) = 0. Hence, a(t) = w{Xt). In order to obtain the a(t) function, we first consider the function X " ( : r ) , defined by: f 1 0 < x <t Xll (i)= l - n ( i - i ) t<x<t+/n (5) ( 0 t+ l/n < x < 1 0 t t + l / n ! 7 IN Clearly, lim,,-.«, X ? = Xt. Hence, lim„^oo X " | k ° V = X ( | K °V on uu2 . Now a(t) can be obtained as the mean of the values of X" in all the points of the Cantor triadic set: a ( i ) = / / ( X , ) = n(X,K °V ) = J i i n fi(X^K ° V) ( = lim lim n—oo k-^oo l / 2 f c E W ( « ) ] V / ( 6 ) as follows from the proposition in §3, been defined in §2 by (1). Given the way the a function is obtained and the symmetry of the Cantor set about the center of the segment [0,1], it is easy to see that a is symmetric about the point (0.5,0.5). That is, a ( x ) = 1 - a(l - z); x 6 [0,1], The a ( t ) function can also be seen as a normalized cumulative distribution function with density distribution function da{t) = lim 1/2* V] S(t-u)dt k — >oo ^^'^ where <$(•) is the Dirac delta function. In this sense, the Cantor triadic set is the spectrum of a(t). With this notation, a(r) = 1/2* £ f 6{t-u)dt = 1/2* £ 1. U<T That means: a ( r ) is the fraction of the number of points of the Cantor triadic set in the domain [0,r] relative to the number in the whole domain [0,1]. Let us make two remarks concerning the Lebesgue-Stieltjes measure defined by the integrator a function:
- 18. 6 A. Arteaga and J.M. Masqué I .ODO 0.000 0.002 0.004 0.006 0.008 0.010 Figure 1: a(t) in the domains [0,1] and [0,0.01] 1. The value JQ f{x) da(x) only depends on the values the function / takes on the Cantor set. However, note that if / is a polynomial, then / is completely determined by the values it takes on the triadic. 2. The picture of a looks like a devil's staircase in the sense of Mandelbrot [7, p. 62] (see figure 1). A similar measure has also been considered by Barnsley, Bessis and Moussa [8] in dealing with a problem concerning the activity of the ferromagnetic Ising model, but their work it is not related with ours. In order to implement the algorithm to obtain a ( x ) , first let us consider the truncation error that is obtained when only the first k digits are taken in defining the Cantor triadic points: k ° ° • = Yi + e r r ' w h e r e e r r = i f ! ( a i 6 ¿=1 6 !=/ f c+l J (7) The maximum error will take place when in the point of the triadic considered all the digits after the A;th are ones. In this case, the expresión of the error (7) is a geometric progression, and hence: E.g.: For k=10 max(err) ss 2 x 10~5 max(err) = 2/3fc+1 _ J_ 1 - 1/3 ~ 3* ' k=15 k=25 7 x 10-8 1 x 10~12 To obtain a(t) we have to generate the sequences of O's and l's defining the triadic points (truncated) a, obtain V(a) by (1), X"[V(a)] by (5), and, finally a(t) by (6) as avereage of these values X1}tp(a)]. 6 Orthogonal Polynomials The function a(-) obtained can be employed as a nonstandard measure to build a sequence of orthogonal polynomials with the product defined as: < P{x),Q{x) >= f P(x).Q(x) da(x) Jo
- 19. Orthogonal polynomials 1 B„ This product defines a sequence of orthogonal polynomials given by the recurrence formula: Pn(x) = (x - An)Pn-i(x) - BnPn-i{x), n> 1, (8) with P-i{x) = 0 and P0(x) = 1. According to the results by Nevai [1, §4.20], it is obvious that our orthogonal polyno- mials can not satisfy a second-order linear homogeneous differential equation. This fact is one of the most outstanding differences from semi-classical orthogonal polynomials. It is easy to see that the recurrence coefficients are defined by: _ < I P , I - I ( I ) , P„-i(a) > < Pn-i(x),Pn-i{x) > (9) < xPn^(x),Pn^(x) > < Pn_2[x),Pn.2{x) > The coefficient B is arbitrary, but it is convenient, for sake of uniformity [9], to define it as: Bx = fd da(t) = 1. When the a function is symmetric about the origin, it is well known that all the coefficients An are zero (see [10, THEOREM 4.3]). In this case, we are going to demonstrate the following T H E O R E M : When ail integrator function a is symmetric about the point (0.5,0.5) it is verified that: i) An = 0.5 , n = 1,2,... ii) Pn(x) has even (odd) symmetry about x = 0.5 if n is even (odd). That is, Pn(x) = ( —1 )"/'„( 1 - n = 0, 1,2,... Proof: We have, Po(x) = 1 < xP0(x)P0(x) > p = „ , . ,. .—: = / xaa = Ml • < P0(x)P0(x) > Jo ^ Given the symmetry of a, clearly A — 0.5 and, then, P(x) = x — 0.5. Hence, i) and ii) are true for n = 0 and n = 1. By induction, supposing they both are true for all k < n: i) _ < xPn-i(x)Pn-i(x) > n~ <Pn-i(x)Pn-i(x)> ' that can be expressed as , < (x-0.5)Pn.,(x)Pn_,(x) > , n r — r! 7—wi — r U.O . < Pn^(x)Pn^(x) > Here, the denominator of the fraction is always not zero, while in the numerator: < (X - O^Pn-iMPn-^x) >= f x ~ 0 . 5 ) [Pn-^X,)]1 da(x) ; Jo if we do the change x = x — 0.5, results: < (a - 0.5)/'„_i(x)/'„_i(a:) >= /°'5 (x [Pn.x{x + 0.5)]2 da(x + 0.5) = 0 , J-0.5
- 20. 8 A. Arieaga and J.M. Masqué given that [Pn _1 (x + 0.5)]2 is even and x odd. Hence, An = 0.5, for all n. ii) In the recurrence formula: Pn(x)(x - 0.S)Pn-x(x) - BnPn-2(x) if n is even we have Pn-{x) = —Pn _i(l — x) and Pn _2 (a:) = f n _ 2 ( 1 — x), hence: Pn(x)( 1 - z - 0.5)Pn _i(l - x ) - B„P„_2 (1 - x ) = Pn( 1 - x). And similarly for n odd. • T h e coefficients Bn have been obtained numerically, following the bootstrap Stieltjes procedure [9]. An open question would be to find the general expression for Bn. As far as we know, there is no method to do the quadrature in a simple form. Taking into account that we only consider k digits in the expansion of f(x), the easier way to obtain the quadratures involved in (9) is to solve them as Riemman-Stieltjes sums, considering partitions such that in every interval there were at most one point of the triadic set. In this way: < P(x),Q(x) >= j i ' P(x).Q(x)da(x) « /2k £ P[p{a)].Q[^a) (10) afz'fflj2 where k is the number of digits taken to define the triadic points, and, thus, 2k is the number of points of the triadic in [0,1] T h e equations (9), taking into account the formula (10), let us obtain the coefficients Bn. These coefficients are given in table 1 up to n = 26, obtained with k = 25. n Bn Tl Bn n Bn n Bn 1 1.0000000000 8 0.0812180463 15 0.0189201066 22 0.0774248549 2 0.1250000000 9 0.0324016054 16 0.1410187727 23 0.0452801655 3 0.0500000000 10 0.1510749676 17 0.0386977786 24 0.1140167807 4 0.0914835164 11 0.0130941801 18 0.0801319585 25 0.0142362331 5 0.0276826008 12 0.0898122397 19 0.0086722736 26 0.1150376125 6 0.1267576108 13 0.0445443616 20 0.1577325433 7 0.0278390606 14 0.1026727454 21 0.0278943990 Table 1: Values of Bn for k = 25 In a similar way, the moments /i^. of the function a(x) can be obtained: fin = fnda(x) « 1/2* £ M « ) ] " . Jo k The moments up to the order n = 26, obtained with k = 25, are given in table 2. Those coefficients An, Bn allow us to put the polynomials in the usual form, f ' n ( x ) — Xn + <ln,n-Xn 1 + (ln,n-2x" * + • • ' + 1ri,ll'' + an,0 , where the coefficients a;^ are obtained by iteration. T h e coefficients an<J from n = 1 to n = 26 are given in table 3.
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- 22. Towards six, Célestin, still unconscious, gave a little shiver, as if at the coldness of her lover, and Dodor in the cage fluttered his wings as if in fear. The priest, who had been standing patiently, fell upon his knees, and prayed with fervor for the passing soul. Struve told me most of this story as we sat one day before a café on the boulevards. “That is the man,” he said, indicating a good-looking young fellow on a coal-black horse, who was riding by, accompanied by a girl with auburn hair, mounted on a magnificent gray; “that is Toto.” “But the girl?” “His wife, the Princesse; she was Helen Powers.” “But surely—is she married to him?” “Very much so. He confessed all his sins, and she gave him absolution. No woman, you see, can withstand a confession of folly; you see, it is a far more genuine thing than a confession of love— with ordinary men.” “You do not think Toto an ordinary man?” “I have never thought of him as a man. Come, it is five o’clock; I am tired of sitting still.” “A moment. Where has old De Nani gone to?” “He is living at Monte Carlo. He lost a hundred thousand francs there, and they have pensioned him; they give him sixty francs a week, I believe.” “Pelisson did not prosecute him?” “Oh, no! all that did Pantin a lot of good.” “If I had been Toto, I would have made him fight.” “Thank goodness we were saved from that! A duel between Toto and De Nani was the only thing wanted to cap the business and kill
- 23. everyone outright.” “Kill them?” “With laughter.” “And about Garnier?” “Ah, Garnier—he only wanted one thing before he met Célestin.” “What was that?” “Célestin—she has made him. Célestin is not dead; she will never die so long as men have eyes and Garnier’s pictures exist. She might have lived with Toto and produced little Totos; she lives instead with Garnier, and through him will live forever.” “A moment. What of Gaillard?” “He has grown very fat. You know, Toto shook him off when he married, Pelisson forsook him, De Brie gave him the cold shoulder; and what did he do? He sat down and wrote ‘Poum-Poum,’ and turned all the minor poets into ridicule, and sold a hundred thousand copies in a month, and ‘slew art,’ to use his own expression, because it tried to slay him. He is making eighty thousand francs a year, if he is making a sou. I am glad of it; he is not a bad sort—Gaillard.”
- 24. Transcriber’s Notes: The original spelling, hyphenation, accentuation and punctuation has been retained, with the exception of apparent typographical errors which have been corrected.
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