[PDF] 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



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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



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Multiple Fourier Series

Abdul Raheem, Anees Abrol

University of New Mexico

araheem@unm.edu, aabrol@unm.edu

November 30, 2016

Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 1 / 46

Agenda

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 / 46

Fourier seriesFourier series in 1-D

Fourier series in one dimension

A periodic functionf(x) with a period of 2and for whichR2

0f(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 is

f(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 becomes

f(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 e

2in1x1e2in2x2;

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 e

2in1x1e2in2x2::::e2indxd;

and we can imagine writing the Fourier series expansion as X n

1;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 / 46

Fourier 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 convergent

Fourier 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 / 46

Fourier 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 Z

f(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 series

Proof 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) 22
1 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.)

Assuming f(x,y) is such that

R R f(x;y)2dxdyis nite implies that f(x,y) can be approximated in the mean by continuously dierentiable functions. As such, Parseval equation remains valid for such functions.Additionally, we know that the functionscos(nx)cos(my), cos(nx)sin(my),sin(nx)cos(my), andsin(nx)sin(my) are orthogonal in the sense that Z Z cos(nx)cos(my)cos(kx)cos(ly)dxdy= 0unless n=k;m=l; Z Z cos(nx)cos(my)cos(kx)sin(ly)dxdy= 0 and so forth. Abdul Raheem, Anees Abrol (UNM)Multiple Fourier SeriesNovember 30, 2016 13 / 46 Fourier seriesProof of convergence of double Fourier series Proof of convergence of double Fourier series (contd.)

Therefore, we nd that:

Z Z f(x;y)14 a00+12 M X m=1[a0mcos(my) +b0msin(my)] +12 N X n=1[an0cos(nx) +cn0sin(nx)] NX n=1M X m=1[anmcos(nx)cos(my)] + [bnmcos(nx)sin(my)] + [cnmsin(nx)cos(my)] + [dnmsin(nx)sin(my)] 2 dxdy Z Z f(x;y)2dxdy24 a2 00+22 M X m=1[a2 0m+b2quotesdbs_dbs21.pdfusesText_27