1
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2answers
16 views

Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
0
votes
2answers
17 views

Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?
1
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2answers
76 views

Fourier series with half range

Question What are $a_0,a_n,b_n$ equal to with range $-L\leq x \lt0$, rather than the standard $-L\leq x \leq L$? For example: $$f(x)=2x^2,\quad-1\leq x\leq0$$ Instead of $f(x)=2x^2,\quad-1\leq ...
0
votes
0answers
8 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
1
vote
2answers
36 views

Why does the point spread function not violate the linearity of the Fourier transform?

In radio astronomy the point spread function is the Fourier inverse of the $uv$-sampling function of a telescope. The $uv$-sampling function is a sum of sampling functions (one for each baseline). So ...
2
votes
2answers
41 views

A simple Fourier Transformation

I am a bit stuck with this small basic signal. I have this $$y(t)=\frac{\sin(200\pi\,t)}{\pi\,t}$$ and I want to take its Fourier Transformation. Obviously it looks like the sinc function. But that ...
0
votes
1answer
14 views

Solving the wave equation bounded by one free end and one fixed end

Given that $\{\sin\left[\frac{(2n-1)\pi}{2L}x\right] : n\in\mathbb N\}$ is the complete set of eigenfunctions of a regular Sturm-Liouville with boundary points $0$ and $L$ and weight function $1$, and ...
1
vote
1answer
48 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
0
votes
1answer
16 views

Fourier transform and Z transform question?

Lets suppose we have an exercise where I have to find the Z transform and its region of convergence.I find the Z transform and the region.How do I determine if the Fourier transform exists from this ? ...
0
votes
1answer
103 views

How to get fourier series of 8-bit character to be transmitted?

I have been reading this in a book, but can't understand how he used the 8-bit in fourier series equation to get the result below. The transmission of the ASCII character ‘‘b’’ encoded in an 8-bit ...
0
votes
0answers
19 views

Need a closed form for fourier coefficients (if it exists)

i have a set of 53 fourier coefficients. the dc term is 0. the 26 positive frequency amplitudes (coefficients) are given below. the 26 negative frequency amplitudes are the same. {0.014451, ...
-1
votes
1answer
51 views

Fourier series of rescaled cosine function [closed]

How would I find the Fourier series of $\cos\left(\, 5x/2\,\right) $ on $\left[-\pi,\pi\right]$? Progress $$A_0={1\over 2\pi}\int_{-\pi}^\pi \cos(5x/2)dx={2\over 5\pi}$$ $$A_n = {1\over \pi} ...
1
vote
1answer
31 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
1
vote
1answer
43 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
3
votes
0answers
42 views

Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
1
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0answers
29 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
4
votes
0answers
43 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
0
votes
1answer
20 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
1
vote
0answers
31 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
8
votes
0answers
71 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
1
vote
3answers
38 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
0
votes
1answer
60 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
1
vote
1answer
40 views

What is the Fourier series of $\frac1T\sum^{\infty}_{m=-\infty}\delta(f-\frac mT)$?

As the title mentioned, I've not known exactly about Fourier series and when I was reading an digital communication textbook, I wondered about below equation derivation of Fourier series like ...
4
votes
4answers
86 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
2
votes
1answer
39 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
1
vote
1answer
36 views

A problem concerning finite number of Fourier coefficients

Is there a smooth, non-zero $2\pi$-periodic function $f,$ with support of $f$ contained in an interval $[a,b]\subset[0,2\pi],$ such that $b-a<2\pi$ and only finitely many Fourier coefficients of ...
1
vote
1answer
31 views

Prove the uniform convergence of a Fourier series

Suppose that $f$ is a $2\pi$-periodic function that satisfies the estimate $$|f(x)-f(y)|\leq M|x-y|^\alpha$$ for an $0<\alpha<1,$ and let ...
4
votes
1answer
69 views

Show $\lim_{n\to\infty} n^p f(nx) = 0$ exists in the distributional sense

Let $f\in C^\infty(\mathbb R)$ be periodic, with period $2\pi$ and have mean zero ($\int^{2\pi}_0 f(x)dx =0$). Show that for any positive integer $p$ the following limit is valid in the ...
0
votes
1answer
30 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
0
votes
1answer
42 views

Interpreting Fourier transform frequency graph

I've been trying to understand Fourier transform for some time now and I think I've perhaps finally got the idea now. What I would like to do now is to make an example of Fourier transform for ...
0
votes
0answers
14 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
1
vote
1answer
35 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
1
vote
0answers
50 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
0
votes
2answers
31 views

Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
2
votes
1answer
56 views

Fourier series of oscillation in form $\cos(2 \pi \frac{k}{T}+\phi)$

I would like to calculate the fourier coefficients of $\cos(2 \pi \frac{k}{T}+\phi)$ where $T \in \mathbb{N}$ is the period and is arbitrary but fixed, $k \in [1, N-1]$ is the number of oscillations ...
3
votes
1answer
80 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
1
vote
1answer
43 views

How can we represent an image using basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But i don't understand terms like "orthogonal " and "basis ...
0
votes
0answers
8 views

Are the dominant frequencies preserved under fractional inversion

Let $f(t)$ be a signal that is a function of time. Let $F(f)=\mathcal{F}\{f(t)\}$ be the Fourier transform of $f(t)$. If $F(f)$ is dominated by a sparse set of frequencies $(f_1,f_2,\cdots,f_n)$ (only ...
1
vote
1answer
57 views

Proof of Fourier series Theorem (k-continuous derivatives)

Here's the theorem: Theorem: If $f$ is periodic with Fourier coefficients $a_n,b_n$ and if the series $$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$ converges for some integer $k \geq 1$, then f ...
0
votes
1answer
23 views

N-point FFT and 2-radix FFT

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$) For example a is a sequence: ...
3
votes
0answers
79 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
1
vote
1answer
55 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
1
vote
1answer
57 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
2
votes
2answers
36 views

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
0
votes
1answer
83 views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
1
vote
0answers
36 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
0
votes
1answer
55 views

Questions about the Fourier series

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$ The symbol $\sim$ has the following meaning: We know that the right ...
4
votes
1answer
67 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
2
votes
0answers
39 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
1
vote
2answers
66 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...