Reconstruction of partial sum without gibbs phenomenon

1
Reconstruction of Partial Fourier Sum
without Gibbs Phenomenon
Deahyun Kim
May 17, 2019
Abstract
Fourier series allow us to represent a perhaps complicated periodic function
as simply a linear combination of projections onto a basis. Such a compact
representation has proven exceedingly useful in the analysis of many real-
world systems. However a disadvantage of any Fourier-based methods arises
when dealing with highly discontinuous fields. It is well known that finite
Fourier series expansion of discontinuous functions suffer from oscillations
around the discontinuities, it is known as Gibbs phenomenon, which in
practice significantly reduced the performance of system. In terms of this, we
introduce new idea to reduce Gibbs phenomenon and reconstruct partial
Fourier sum using formulization of truncation error.
1. Introduction
Fourier series is an expansion of a periodic function by an infinite summation,
̃ ∑ (1)
where ̃ is a periodic function with period T, and is the Fourier coefficients. There are
old-established questions that every function can be represented by Fourier series? If a
finite data set of is applied, which is called partial Fourier sum, then could it provide a
good representation? Fourier series is usually not an exact replica of its original function.
Thus, a natural question is exactly how does the series approximate the function? It is well
known if the function is not reasonably smooth, such as jumping discontinuity, then partial
Fourier sum would not be a good approximation due to Gibbs phenomenon.
We introduce new idea to represent the partial sum error by formularization, where it is
the error between the original function and partial Fourier sum. It is called traditionally as
truncation error1. This is the new approach for truncation error, which is described by
universal form and works for every truncation error. Until now, the study about truncation
error itself is under-researched. Many researches have been focused on the minimization of
the truncation error such as mean square error.
1
Due to the restriction to a finite bandwidth of frequencies in its Fourier field, then the Fourier series is cut short
or truncated, hence this cause Gibbs phenomenon.

2
This traditional approach has no room for more development of reducing error.
The formularization of the truncation error makes it possible to remove the Gibbs
phenomenon.
We will deal with the process that reconstruct the partial Fourier sum with given spectrum
. One of engineering application for this situation is the magnetic resonance imaging
(MRI)2, where the samples of the Fourier transform of the image function are directly
measured. It means that of the image function is given, and reconstruct the image
function. This process is mentioned as ‘Fourier image reconstruction’.
As was the case with the reconstruction of partial Fourier sum from given , if original
function has discontinuities, then recovered partial Fourier sum suffer from the Gibbs
phenomenon. In real applications, this phenomenon will reduce the quality of performance.
In terms of this, we developed the novel method to remove Gibbs phenomenon, which is
called as ‘composite error method(CEM)’, utilizing the formularization of the truncation
error.
Reconstruction of partial Fourier sum, all research and effort was performed under the
definition that the period T is equal to the length L of original function. This idea has got rid
of very important results. Many information was buried under the case of only . Some
article3 defined about T such as
(2)
However we do not constrain about T and consider only for all process in analyzing
the function and extract meaningful results. This is the first step for new approach for the
reconstruction of original function from the sampled spectrum .
2. Definitions
2.1 Partial function
Suppose is a partial function of , defined on finite interval , and a time-
limited that has zero value outside the interval, where is positive real number. is
real, integrable on .
{ (3)
2.2 Periodic function
Denote the period by T. Periodic function ̃ of is defined by
̃ ∑ { (4)
2
Magnetic Resonance Imaging, 2nd Edition – Robert W.Brown, Y.C. Norman Cheng, E. Mark Haacke, Michael
R. Thompson, Ramesh Venkatesan. 2014
3
Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method,
Ruonan Shi, Jae-Hun Jung ,Ferdinand Schweser, January, 2019

3
2.3 Partial Sum
Denote Fourier coefficients of ̃ by and is Fourier transform of .
Suppose and exist. Partial sum of ̃ is defined by
∑ ∑ (5)
Denote the bandwidth of partial sum by , where M is non-zero positive integer.
(6)
is a truncated function of by rectangular spectral filter
{ (7)
is an inverse Fourier transform of , which is a convolution of and
function,
↔ (8)
then the partial sum is a periodic function of with period T.
∑ ∑ (9)
The eq(9) is derived using Poisson Summation Formula.
Statement 1. Suppose is defined on , real, integrable, and its Fourier coefficients of
̃ exist. is not band-limited, and is truncated by , then
is given for the further process, where is known and are selectable variables.
2.4 Partial Sum Error
The is known as ‘truncation error’. Partial sum error is defined by
̃ ∑
| |
(10)
Fourier coefficients is to minimize the mean square error(mse) of ,
(‖̃ ‖ )
(11)
where,
‖̃ ‖ √∑|̃ | (12)

4
The study for the of is well developed in the error analysis for Fourier series
We introduce the envelope of | | for easy analysis of . The envelope detection
is connecting all the peaks in the signal, | | . One of the envelope detection
methods is using Hilbert transform4,
| | { }
where is the envelope of | | for error function .
Now we introduce new measurement of error, ‘error value’, is denoted by
| ∫ | | ∑ | (13)
Statement 2. is an index to determine approximation quality. The smaller error value
is, the better the recovery performance of the partial sum is.
We will use two values, and for approximation quality.
Example 1. is given as,
(14)
Set . The periodic function ̃ has discontinuous points at , where
. The partial sum is obtained by using eq(5),
Figure 1. , | |, and
Using the eq(12),eq(13), set and compare the recovery performance of
the partial sum for fixed and different .
Figure 2. mse and error value for different T
4
https://www.mathworks.com/help/dsp/examples/envelope-detection.html

5
Facts we found,
1. Lower error value means good approximation, recovery performance
2. As T increase for fixed bandwidth , error value converge to up or down bound
according to the shape of function, depends on magnitude of discontinuities.
3. As bandwidth increase with fixed , error value converge into down bound
and provides the better recovery performance.
Further study,
1. More study about relationship among error value , approximation, and resolution
2. Mathematical proof for statement 2
3. Fourier uncertainty theorem in terms of error value
3. Reconstruction of Partial Sum
Recall partial sum error eq(10). We introduce the new approach for formularization of the
partial sum error , and reconstruct the partial sum so that reduce the Gibbs
phenomenon. It is convenient way for removal of Gibbs phenomenon.
Recall eq(14), calculate the partial sum error using eq(10), where
.
Figure 3. by eq(10), where
As shown in the figures, we can presume that the partial sum error is consist of the
pulse shape at discontinuous points. This pulse shape is caused by the convolution of
and as mentioned in eq(8) and its magnitude is the biggest at the points of
discontinuity. The magnitude of pulses in the figures is the half of difference between the
left and right of one-sided limit at discontinuous point.
3.1 Composite Error
Statement 3. The convolution of a rectangular function and a sinc function, , is
described by the sine integral, and corresponds to truncating the Fourier series, which is
the cause of the Gibbs phenomenon.5
∫
( )
[ ( ) ( )] (15)
5
Reference 1: Signal Analysis – Athanasios Papoulis, page 58
Reference 2: Partial Differential Equations with Fourier Series and Boundary Value - Nakhle H. Asmar, page 394

6
| | ∫
is sign integral function and converges as , its asymptote is for .
Let’s compose the partial sum error using at .
Denote that is ‘composite error’ for the partial sum error .
[( ) ( ) ] (16)
for , and where is the max frequency(bandwidth) and at discontinuous
point . Constant ‘a’ is the half of difference between the left and right of one-sided
limit at discontinuous point .
| |
(17)
For the eq(14), the composite error at discontinuous point is
( ) (18)
{[ ] [ ]
(19)
At discontinuous point , we have , where constant can be
calculated by
∑ ∑ (20)
Figure 4. composite error with
3.2 Composite Recovery Function
Using the composite error denoted by the eq(19), reconstruct the partial sum
by simply adding the composite error .
Denote that is the ‘composite recovery function’, defined by
(21)
The above equation will show that simple adding the composite error to the partial
sum can attenuate the overshoot caused by Gibbs phenomenon near discontinuous
point of ̃ .

7
Figure 5. composite recovery function,
As shown in the above figures, the composite error attenuates the overshoot in the
partial sum . We call this approach as ‘composite error method(CEM)’.
Statement 4. Suppose is real, continuous on , , however its periodic
function ̃ be discontinuous, where , or is piecewise
continuous. Then the reconstruction of the partial sum by simple adding the composite
error reduce the Gibbs phenomenon.
The composite error method is based on the following assumption.
(22)
The partial sum error from eq(10) has aliasing due to the periodicity. However the
composite error is not periodic function and no aliasing happen. Therefore two
functions are not identical. But with enough large T, the effect of aliasing will be trivial.
3.3 Recovery Error
Denote is ‘recovery error’ defined as the error between the partial sum error
and the composite error .
̃ (23)
[Example 2] Recall the eq(14) again, . Compare the partial sum
error of | | and the recovery error of | |. We also utilize the envelope
of | | and | | with log sacle.
Figure 6. envelope of | | and | |

8
As shown in the above figure, the effect of Gibbs phenomenon at is
attenuated by composite error . Suppose that max frequency of is fixed as
(bandwidth). Using envelope , compare recovery error in | | for
the fixed and different .
Recall eq(13), is envelope of | |, The ‘recovery error value’ is denoted by
|
∑
| (24)
Set , . mse is measurement between .
Figure 7. the error value for fixed and different
In the case of fixed , as T is increased, the error value of | | is decrease. This
statement is due the fact that the aliasing effect is trivial as T is increased.
It also implies that the recovery performance does not depend on only bandwidth . For
example, the error value ( of ( ) is less than the
error value ( of ( ).
Comparing the case of two bandwidth .
Figure 8. error value for
As shown in the figure, for equal T, the case of provide less error value .
However for different T(one is small and one is large), the above statement is not always
true. For example, consider a pink line( ), for the case of its
error value is greater than one of the case of .
This imply that the enough large T will provide the low error value . It means that the
recovery performance of does not depend on only the bandwidth of , but also
period T.

9
Statement 5. This is against the fact that is described as the ‘Fourier uncertainty principle’. In
terms of the recovery function using the composite error , if the period T is fixed
such as , then Fourier uncertainty principle would be true. However if T is variable,
then Fourier uncertainty principle is not always true for certain conditions.
In real application, the bandwidth , period T, number of sample M will be determined so
as to maximize the performance of application, which depends on number of calculations,
cost, time, etc.
3.4 General Form of the Composite Error
Condition 1: is real, defined on finite length L, integrable and ̃ is periodic defined
by eq(4) with .
Condition 2: is piecewise continuous, having finite number of discontinuities, or
is continuous, but ̃ is discontinuous on .
Condition 3: The locations of discontinuity in ̃ are known.
Condition 4: is the Fourier transform (CTFT) of , and of eq(7) is given,
and is known, are selectable.
Suppose satisfy the above conditions, and make the general form of the composite
error . Let ‘ ’ be the sign index telling us ‘ascending’ or ‘descending’ of the curve of
at discontinuous points,
(25)
where , are respectively right and left of one-sided limit at discontinuous point
.
 If is positive, then the jump at discontinuous point is ascending and use
to compose the error .
 If is negative, then the jump at is descending and use .
Statement 6. Suppose that ̃ has P number of discontinuity that its is positive,
and S number of discontinuity with negative . There are total P+S discontinuous points at
in ̃ , then the general form of the composite error is
∑ ∑ (26)
(27)
where is the eq(19), , are the coefficients of composite error and the half of
difference between the left and right of one-sided limit at discontinuous points as
mentioned in the eq(17).
For the case of , the method to make compose the error is a little different from
the eq(26). It is due to the direct effect of periodicity T, even though has only one
discontinuous point at , we need to add one more pulse shape at .
The composite error of eq(26) is consisted of , where depends only three
variables (a) magnitude of the coefficients , , (b) bandwidth , (c) locations of

10
discontinuities . Composite error method is more convenient and efficient in the
process comparing with traditional methods. Its merits are followings.
 Simple mathematics with three variables ( , ), ,
 Works for piecewise continuous function
 Reasonable recovery performance to remove Gibbs phenomenon.
However the composite error is still not good enough for completely
removal of Gibbs phenomenon, this is due to the assumption of and non-
periodicity of . Next article will introduce another method to overcome this
disadvantage, so that reduce the more precisely.
[Example 3] is defined on and piceswise continous. ̃ is periodic and
has discontinuity at .
{ (28)
Make the composite error and reconstruct the partial sum with .
First solve the integral for .
∫ ∫ (29)
Set , using the eq(5), the partial sum is
∑ (30)
Gibbs phenomenon happen at the discontinuous points on .
Figure 9. partial sum of eq(28)
Next make the composite error with known location of discontinuities.
At discontinuous point , the sign of is positive, but negative at
. Using the eq the (26), the composite error is
(31)
where is the eq(19). We have for the eq(26).
The coefficients of the composite error , are obtained as

11
| |
(32)
Using the eq(31), the composite error is
Figure 10. composite error
Simply add the composite error to the partial sum , then the recovery function
can be obtained as
Figure 11. for
3.5 Key functions
Key functions and their definitions
function definition
Original function, time limited, real, its Fourier transform exist
̃ Periodic function, real, ̃ ∑
Partial sum, periodic, real, ∑| |
Partial sum error, periodic, real, ̃
Composite error, real,
∑ ∑
Composite recovery function, real,
Recovery error, real,
Table 1. key functions and its definitions

12
4. Analysis of Composite Error
A square wave(rectangular pulse) is given,
{ ̃ ∑ (33)
̃ has discontinuous point at .
Set , , the Fourier transform, is
∫ (34)
Using eq(5), the partial sum is described by trigonometric polynomial form as
∑ ( )
̂
(35)
where ̂ ⌈ ⌉, ceiling of M/2.
For instance, if , then ̂ , and  is .
Figure 12. for
4.1 Analysis of Partial Sum Error
The composite error method is based on the assumption that the partial sum error
can be composed by reasonable combination of , .
Set for eq(33), and from the eq(10), is
̃ ∑ ( )
⌈ ⌉
(36)
Differentiate the eq(36) to locate the global and local extrema of .
{ (37)

13
Figure 13. derivative of
The eq(37) imply that the partial sum error has the extrema at , where is
non-zero integer. In the eq(37), the first local minimum at is about 9 percent of the
jump near discontinuous point.
Figure 14. extrema of for T=2, M=20
The behavior of the partial sum error is identical to the partial sum of the
square wave as shown in the eq(35) and figure 12.
4.2 Analysis of Composite Error
Recall again eq(33). Using the eq(26), the composite error for the rectangular pulse is
(38)
where and is defined by eq(19).
Differentiate6 the above eq(38) to find the location of extrema of .
{
( )
(39)
Unlike of eq(37), the eq(39), is not periodic function. Solve , the
k that makes first term of eq(39) to be zero also make second term zero.
Set ,
6
Hint: derivative of is function,

14
Figure 15. derivative of
The extrema of the eq(39) is located at
(40)
Figure 16. composite error extrema points
Like the partial sum error , the composite error has same extrema points at
. Compare the first extrema value of the composite error and the partial
sum error .
-0.08990700 -0.08959407 -0.08949054 -0.08949004
-0.08948987 -0.08948987 -0.08948987 -0.08948987
-0.09481958 -0.09208751 -0.08969292 -0.08959129
Table 2. first extrema value of
of eq(19) represent Gibbs phenomenon and identical for any M.
( ) ∫ (41)
It is well known that the general form for Gibbs phenomenon is described as
∫ (42)

15
The eq(42) shows the height of the left-most peak(first maximum of overshoot) doesn’t
exceed about 9% of the difference between the left and right derivative at discontinuous
point. The eq(42) imply the important fact that “Gibbs phenomenon is the step response of a
low-pass filter”. The eq(41) imply the relationship between error and .
( ) ( ) (43)
It should be known that the eq(43) works for only square wave like the eq(33).
Derive the general relationship between the partial sum error and the composite
error . Start from the eq(8), denote the error by
(44)
It is obvious the error is not periodic function.
Using the eq(9), the partial sum is represented by
∑ ∑ ∑ (45)
Since ̃ , the eq(45) imply is a periodic function of .
∑ (46)
The principal basis of the composite error method is based on the following assumption.
(47)
If is the rectangular function like the eq(33), then error would be identical to
the composite error such as
(48)
In this case, the error is determined by the eq(15) and the composite error is
obtained by the eq(26). And eq(49) and eq(50) are identical.
[ ( )] (49)
[ ] [ ] (50)
Next consider general case of , which is not the rectangular function.
Start from eq(44). Denote the Fourier transform of by ,
↔ [ ] (51)
where is defined by eq(5).
Suppose that is defined, continuous on , and -time integrable, derivatives,

16
∫ ∫
(52)
where is the Fourier transform of , which is -time derivative of .
Note that . Using the eq(51), is
[ ] (53)
Iteratively calculating of the eq(53), we have
( ) [ ] (54)
(55)
Statement 7. For , if , then ̃ is continuous everywhere and the
eq(54) shows that goes to zero faster than 1/ , since decay with very fast rate.
And if , then decays exactly as 1/ , since the rate of
decay depends on .
Statement 8. For , if , then ̃ is continuous everywhere, and the
decay faster than 1/ due to . And if , then decays
exactly as 1/ .
From above statements, we conclude
 if ̃ has discontinuities, then decays with rate as 1/ .
 is reasonable to be ignored, since goes to zero faster than 1/ .
Denote by
[ ] (56)
The second term of eq(54) is
(57)
According to above statements and eq(57), we can assume
(58)
Next derive the general form for .
Statement 9. Suppose ̃ satisfy the condition 1~4 and has P number of
positive discontinuity and S number of negative discontinuity. There are total
discontinuous points at . The general form of is

17
∑ [ ] (59)
where ,where , are respectively left and light limit at
discontinuous point .
Denote that is the Fourier transform of of eq(26), the general form of is
{∑ ∑ } [ ] (60)
where , are the coefficients of composite error .
Obviously eq(59) and eq(60) are identical,
(61)
From above statements and eq(58), eq(61), we may conclude
(62)
To complete the discussion for the assumption , we need to
consider an additional fact, which is aliasing due to periodicity of eq(46).
Since is a pulse shape that fast goes to zero , if we take large enough T, then
the effect of aliasing will be trivial, and this is the reason why for the fixed , as T is
increased, the error value is decrease and the recovery performance of is
improved. The assumption for is quite reasonable to apply into composite
error method.
Let’s take a simple example, suppose the has one discontinuity at ,
and composite error has only at ,
(63)
∫ (64)
is odd function with one side function such as
(65)
where [ ] .
Denote the Fourier transform of by ,
∫ (66)
then is the imaginary part of ,
(67)

18
Solving eq(66)7, we have
[ ] (68)
The eq(68) shows .
[Example 4] A function and its Fourier transform are given as,
{ (69)
∫ (70)
̃ has discontinuities at
Using eq(51), is
( ) [ ] (71)
From eq(59), is
{ } [ ] (72)
where
By eq(60), is
{ } [ ] (73)
where , . The above equations imply
(74)
From eq(57), is
( ) [ ] (75)
From eq(75), it is obvious that and only first derivartive, exist, but
higher derivative than , i.g. are zeros.
7
Use Fourier transform of ,where ∫ , substitute and
{ } {∫ } ∫ { }
∫

19
Set ,
Figure 17.
As mentioned in statements and shown in above figure, decay very fast than
and can be ignored.
[Example 5] Recall eq(14), .
{ } (76)
where
(77)
Using eq(51),eq(59),eq(60), are
(
( )
) [ ] (78)
{ } [ ] (79)
where .
{ } [ ] (80)
where , .
It is obvious that,
(81)

20
Set , ,
Figure 19.
As shown in the figure, goes to zero very fast. Therefore is
well approximately equal to .
4.3 Convergence of Composite Error
One method for the convergence is described by convergence8,
| | (82)
From eq(57), denote inverse Fourier transform of by ,
(83)
Statement 10. If is square-integrable, where ∫ | | , then the recovered partial
sum converges to ̃ .
However the above convergence does not guarantees zero deviation between ̃ and
for every .
Statement 11. If has -time integrable, derivatives on then its Fourier transform
have upper bounds9
| |
(84)
For to be integrable, ̃ ̃ ̃ should be continuous.
The continuity of ̃ ̃ ̃ requires the following conditions.
8
The symbol of converges is certainly different from of the length of
9
On the application of spectral filters in a Fourier option pricing technique-M. J. Ruijter,M. Versteegh, C. W.
Oosterlee

21
(85)
For instance, for , the eq(84),(85) imply
 For , if ,
- then ̃ is not continuous,
- and decays at a rate of .
 For , if , and
- then ̃ is continuous, and first derivative exist, but ̃ is not continuous,
 For , if , , and/or ,
- then ̃ is continuous, the first derivative ̃ is continuous, and the second
derivative exist, but ̃ is not continuous,
Using statement 11, we conclude as,
Statement 12. Recall the eq(55), has no term of in eq(55). It implies that and
is continuous on , and converges uniformly to .
| | (86)
5. Applications
Composite error method is applicable to all industries where Fourier transforms are used
and suffer from Gibbs phenomenon.
5.1 Wireless Telecommunication
In wireless telecommunication, almost all modern networks are built on OFDM, because
OFDM improved data rates and network reliability significantly by taking advantage of
multi-path in wireless transmissions. 4G(LTE) and 5G(NR) networks also employ OFDM as
the fundamental element. OFDM is Fourier-based modulation/demodulation and cannot be
free from Gibbs phenomenon. Composite error method remove Gibbs phenomenon and
improve the performance of wireless telecommunication.
5.1.1. Channel Estimation in OFDM
Channel estimation indicates attenuation and phase of the transmit signal and is very
important in order to get the promised theoretical capacity and to achieve maximum gain
of diversity in OFDM systems. For high accuracy estimation even during high speed
movement, in general channel estimation makes use of the pilot symbol. However for

22
packet transmission by OFDM, there are definitely discontinuities in the frequency and
time domain of signal for packet transmission by OFDM. If DFT interpolation is used for
channel estimation, then Fourier transform of discontinuous signal suffer from the Gibb
phenomenon and it cause the accuracy of channel estimation to drop.
5.1.2 High OOBE in 5G Network
One of critical disadvantage of CP-OFDM(including any multicarrier system) is high out of
band emissions(OOBE) that may interfere with other users.
OOBE is reduced by various windowing/filtering approaches along with the guard band
allocation. LTE uses 10% of total bandwidth as guard bands and decreases the spectral
efficiency. OOBE is due to the fact that OFDM signal is well localized with a rectangular
pulse shape in the time domain, that results in sidelobes at edge carriers causing significant
interference, ACI. Without proper guard band and any windowing/filtering process, if to
reduce the guard band, only main bandwidth using sharp rectangular shape filter in
frequency domain are applied, then what happen in time domain OFDM signal? Answer is
the Gibbs phenomenon occurs on discontinuous point in time domain. It will critically
reduce the performance of 5G network. (of cause no one do like this)
5.2 Image Processing
The Gibbs phenomenon is usually experienced during attempts to remove noise from an
image. Filtering of medical images to remove noise is very important, because of the risk of
artifacts that can lead to misdiagnosis. The most common image artifact happens from low-
pass filtering to remove high frequency noise and the Gibbs phenomenon is typically
viewed as a numerical artifact in the numerical representation of a function due to
truncation.
5.2.1 MRI (Magnetic Resonance Imaging)
Gibbs artifacts occur as a consequence of using Fourier transforms to reconstruct MR
signals into images. In theory, any signal can be represented as an infinite summation of
sine waves of different amplitudes, phases, and frequencies.
In MR imaging10, however, we are restricted to sampling a finite number of frequencies and
must therefore approximate the image by using only a relatively few harmonics in its
Fourier representation. The Fourier series, then, is cut short or truncated, hence the name
for this artifact.
6. Conclusion
As discussed, the assumption of give us meaningful results to remove Gibbs
phenomenon. However it still doesn’t remove Gibbs phenomenon completely due to
periodicity of .
10
It is well elaborated in ‘Magnetic Resonance Imaging’, 2nd Edition, 2014

23
We developed enhanced version of composite error method, which is called ‘Partial Sum
Error Method’. This method is based on the assumption of .
is the ‘fundamental partial sum error’ and is to reconstruct the partial sum ,
instead of using the composite error . Then it will enhance the reduction of the Gibbs
phenomenon. The detail will be discussed in next article.
[Example 6] Recall eq(14) again,
Figure 20. comparison of errors,
The errors are represented by the envelope of | | with log scale.
In next article, also we will introduce current developed various methods such as
reprojection method, i.g. Gegenbauer reprojection, of which performance of reduction for
Gibbs phenomenon, is similar with our discussed methods.
The benefit of composite error method and composite error method is
 simple mathematics, low calculation, more economic.
 works for any functions, piecewise continuous function, no restriction.
- Gegenbauer reprojection method require function should be continuous and analytic.
 no restriction for interval for .
- it should be in Gegenbauer reprojection method.

Reconstruction of partial sum without gibbs phenomenon

More Related Content

What's hot (17)

Similar to Reconstruction of partial sum without gibbs phenomenon (20)

Recently uploaded (20)

Reconstruction of partial sum without gibbs phenomenon