One and Two Dimensional Fourier Analysis
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.
Multiple Fourier Series
Nov 30 2016 Fourier series in higher dimensions (vector notation). Fourier series in 2-D (convergence). Proof of convergence of double Fourier series.
Chapter 1 - Fourier Series
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Lecture 2: 2D Fourier transforms and applications
Fourier transforms and spatial frequencies in 2D the 1D Fourier analysis with which you are familiar. ... Fourier series: just a change of basis.
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Two-dimensional Fourier cosine series expansion method for pricing financial options. M. J. Ruijter?. C. W. Oosterlee†. October 26 2012. Abstract.
Introduction to two-dimensional Fourier analysis
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Jun 8 2012 ISSN 1555-578X. All rights of reproduction in any form reserved. Page 2. Ferenc Weisz: Summation of multi-dimensional Fourier series. 2.
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The first two-dimensional Fourier transform NMR experiment of Jeener (1) has proved to be of considerable historical importance in the development of two-.
CHAPTER 4 FOURIER SERIES AND INTEGRALS - MIT Mathematics
This section explains three Fourier series: sines cosines and exponentialseikx Square waves (1 or 0 or?1) are great examples with delta functions in the derivative We look at a spike a step function and a ramp—and smoother functions too Start with sinx Ithasperiod2?since sin(x+2?)=sinx
CHAPTER 4 FOURIER SERIES AND INTEGRALS - Massachusetts Institute of
Two-Dimensional Fourier Transform So far we have focused pretty much exclusively on the application of Fourier analysis to time-series which by definition are one-dimensional However Fourier techniques are equally applicable to spatial data and here they can be applied in more than one dimension
Fourier analysis - Scholars at Harvard
2 CHAPTER 3 FOURIER ANALYSIS physics are invariably well-enough behaved to prevent any issues with convergence Finally in Section 3 8 we look at the relation between Fourier series and Fourier transforms Using the tools we develop in the chapter we end up being able to derive Fourier’s theorem (which
Chapter 1 Fourier Series - Chinese University of Hong Kong
First the Fourier series of a function involves the integration ofthe function over an interval hence any modication of the values of the function overa subinterval not matter how small it is may change the Fourier coecientsanandbn This is unlike power series which only depend on the local properties (derivatives of allorder at a designated
Searches related to two dimensional fourier series PDF
1 FOURIER ANALYSIS 1 2 Discrete Fourier Transforms The c j are now generally complex numbers They are functions of the a j and b j and thus comprise the frequency spectrum 1 1 2 Two-Dimensional Fourier Series It is the same principle with images Here g() is a function of two variables g(uv) where u and v are the
What are the three Fourier series?
This section explains three Fourier series: sines, cosines, and exponentialseikx.Square waves (1 or 0 or?1) are great examples, with delta functions in the derivative.We look at a spike, a step function, and a ramp—and smoother functions too. Start with sinx.Ithasperiod2?since sin(x+2?)=sinx.
How do you get a two dimensional Fourier transform?
where x is the spatial coordinate and ? is the wave number. If a(x,y) is a function of two spatial variables then the two-dimensional Fourier transform is simply obtained by repeating the one dimensional Fourier transform in both dimensions ( (
How many types of Fourier expansions are there?
There are two types of Fourier expansions: † Fourier series: If a (reasonably well-behaved) function is periodic, then it can be written as adiscrete sumof trigonometric or exponential functions with speci?c fre- quencies.
Does a non-periodic function need a Fourier transform?
But when dealing with the Fourier transform of a non-periodic function, there is no natural length scale of the function, so it doesn’t make sense to count the number of oscillations, so in turn there is no need for the… ’s. 3.4 Special functions The are a number of functions whose Fourier transforms come up often in the study of waves.
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.)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[PDF] two sample anderson darling test r
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