30 nov 2016 · Fourier series in 2-D (convergence) Proof of convergence of double Fourier series 2 Fourier series examples Laplace's Equation in a Cube
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[PDF] Chapter 1 - Fourier Series
Ed Alexander D Poularikas 1 5 Two-Dimensional Fourier Series Appendix 1 1 5 Complex form of the series: f t dt a b ( ) < ∞ ∫ f t f t f t f t k k k 1 2 1 ()
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Fourier transforms and spatial frequencies in 2D • Definition and the 1D Fourier analysis with which you are familiar 2D Fourier transform Definition
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Contemporary applications of the Fourier transform are just as likely to come from problems in two, three, and even higher dimensions as they are in one —
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2 Fourier Series • J B Joseph Fourier, 1807 – Any periodic function can be expressed as a weighted sum of sines and/or cosines of different frequencies
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Just as the Fourier transform of a rectangle function in one dimension is a sinc function, so the two- dimensional transform of rect r is a jinc function Naturally, a
Multiple Fourier Series
30 nov 2016 · Fourier series in 2-D (convergence) Proof of convergence of double Fourier series 2 Fourier series examples Laplace's Equation in a Cube
TWO-DIMENSIONAL FOURIER TRANSFORMS - ScienceDirectcom
In this chapter we will discuss functions of two variables, f(x,t), and we will consider the properties of their two-dimensional Fourier transform Spatial aliasing and
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Multiple Fourier Series
Abdul Raheem, Anees Abrol
University of New Mexico
araheem@unm.edu, aabrol@unm.eduNovember 30, 2016
Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 1 / 46Agenda
1Fourier series
Fourier series in 1-D
Fourier series in higher dimensions (vector notation)Fourier series in 2-D (convergence)
Proof of convergence of double Fourier series
2Fourier series examples
Laplace's Equation in a Cube
3D Wave Equation in a Cube
Symmetrical Patterns from Dynamics
Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 2 / 46Fourier seriesFourier series in 1-D
Fourier series in one dimension
A periodic functionf(x) with a period of 2and for whichR20f(x)2dxis nite has a Fourier series expansion
f(x)12 a0+1X n=1[ancos nx+bnsin nx] and, this fourier series converges tof(x) in the mean [Weinberger, 1965].Iff(x) is continuously dierentiable, its Fourier series converges uniformly. Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 3 / 46 Fourier seriesFourier series in higher dimensions (vector notation)Periodic Functions
Consider a functionf(x1;x2) (p1;p2)-periodic in variablesx1andx2 [Osgood, 2007] f(x1+n1p1;x2+n2p2) =f(x1;x2)8x1;x22 R;n1;n22 Z:Assumingp1andp2to be 1, the new condition isf(x1+n1;x2+n2) =f(x1;x2)8x1;x22[0;1]2:If we use vector notation, and writexfor (x1;x2), andnfor pairs
(n1;n2) of integers, then we can write the condition as f(x+n) =f(x)8x2[0;1]2;n2 N:Inddimensions, we havex= (x1;x2;:::xd) andn= (n1;n2;:::nd): and so the vector notation becomesf(x+n) =f(x)8x2[0;1]d;n2 N:Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 4 / 46
Fourier seriesFourier series in higher dimensions (vector notation)Complex Exponentials
In 2-D, the building blocks for periodic functionf(x1;x2) are the product of complex exponentials in one variable. The general higher harmonic is of the form e2in1x1e2in2x2;
and we can imagine writing the Fourier series expansion asX n 1;n2c n1;n2e2in1x1e2in2x2; with an equivalent vector notation usingn= (n1;n2).X n2Z2c ne2in1x1e2in2x2:So the Fourier series expansion in2-Dlooks likeX n2Z2c ne2in.x:Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 5 / 46 Fourier seriesFourier series in higher dimensions (vector notation)Complex Exponentials (contd.)
Similarly, ind-D, the corresponding complex exponential is e2in1x1e2in2x2::::e2indxd;
and we can imagine writing the Fourier series expansion as X n1;n2;::;ndc
n1;n2;::nde2in1x1e2in2x2:::e2indxd: with an equivalent vector notation usingn= (n1;n2;:::;nd). X n2Zdc ne2in1x1e2in2x2e2indxd:So the Fourier series expansion ind-Dlooks like X n2Zdc ne2in.x:Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 6 / 46 Fourier seriesFourier series in higher dimensions (vector notation)Vector Notation Summarized
The Fourier series expansion ind-Dis approximated as f(x) =X n2Zdc ne2in.x; wherex= [x1;x2;::: ;xd]2[0;1]d, andn= [n1;n2;::: ;nd]2 Zd.The Fourier co-ecients ( ^f=cn) can be dened by the integral f(n) =Z [0;1]:::Z Z [0;1]:::Z Z [0;1]de2in.xf(x)dx:Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 7 / 46Fourier seriesFourier series in 2-D (convergence)
Fourier series in two dimensions
Letf(x;y) be a continuously dierentiable periodic function with a period of 2in both of the variables: f(x+ 2;y) =f(x;y+ 2) =f(x;y):For each value of y, we can expandf(x;y) in a uniformly convergentFourier series
f(x;y) =12 a0(y) +1X n=1[an(y)cos nx+bn(y)sin nx]:The co-ecients a n(y) =1 Z f(x;y)cos nx dx b n(y) =1 Z f(x;y)sin nx dx are continuously dierentiable in y. Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 8 / 46Fourier seriesFourier series in 2-D (convergence)
Fourier Series in two dimensions (contd.)
Co-ecients can be expanded in uniformly convergent Fourier series a n(y) =12 an0+1X m=1(anmcos my+bnmsin my) b n(y) =12 cn0+1X m=1(cnmcos my+dnmsin my) where a nm=1 2Z Z f(x;y)cos nx cos my dx dy b nm=1 2Z Z f(x;y)cos nx sin my dx dy c nm=1 2Z Z f(x;y)sin nx cos my dx dy d nm=12Z Zf(x;y)sin nx sin my dx dy:Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 9 / 46
Fourier seriesFourier series in 2-D (convergence)
Fourier Series in two dimensions (contd.)
Putting the series for the coecients into the series for f(x,y), we have f(x;y)14 a00+12 1 X m=1[a0mcos my+b0msin my] 12 1 X n=1[an0cos nx+cn0sin nx] 1X n=11 X m=1[anmcos nx cos my+bnmcos nx sin my+cnmsin nx cos my+dnmsin nx sin my]Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 10 / 46
Fourier seriesProof of convergence of double Fourier seriesProof of convergence of double Fourier series
The Parseval equation gives
Z f(x;y)2dx=2 a0(y)2+1X n=1[an(y)2+bn(y)2]:The series on right converges uniformly in y. Hence we may integrate with respect to y term by term: Z Z f(x;y)2dx dy=22 Z a20dy+21X n=1Z [a2n+b2n]dy:We now apply the Parseval equation to the functionsan(y) andbn(y): Z a n(y)2dy=2 a2n0+1X m=1(a2nm+b2nm) Z b n(y)2dy=2 c2n0+1X m=1(c2nm+d2nm)Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 11 / 46 Fourier seriesProof of convergence of double Fourier series Proof of convergence of double Fourier series (contd.) Thus, we get the Parseval's equation for double Fourier series derived under the hypothesis that f(x,y) is continuously dierentiable. Z Z f(x;y)2dx dy=24 a200+22 1 X m=1(a20m+b20m) 221 X n=1(a2n0+c2n0) +21X
n=11 X
m=1(a2nm+b2nm+c2nm+d2nm)Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 12 / 46
Fourier seriesProof of convergence of double Fourier series Proof of convergence of double Fourier series (contd.)