Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

learn more… | top users | synonyms

0
votes
2answers
188 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
2
votes
0answers
90 views

Understanding JPEG compression.

I have some problems in understanding a passage of the JPEG compression algorithm: Consider an $8\times8$ matrix $M$ that in our case is a "piece'' of a channel (for example the red channel $R$) of ...
3
votes
0answers
76 views

Special functions, Fourier series

Well known are the Fourier expansions (presented, e.g., in Abramovitz and Stegun): $$ \cos ( A \sin x) = J_0(A) + 2 \sum_{k=1}^{\infty} J_{2k}(A)~\cos(2kx)~~, $$ $$ \sin ( A \sin x) = 2 ...
0
votes
1answer
25 views

Fourier transformation of h(-t)

Ask a simple question: we know $F[h(t)] = H(f)$, where $h(t)$ is the impulse response. How to show $F[h(t)] = H^*(f)$? My answer is just $H(-f)$.
3
votes
0answers
173 views

Fourier Transform of Heaviside Function

I'm trying to find the Fourier transform of $H(k - |x|)$, where $H$ is the Heaviside step function. I've solved a few Fourier transforms recently, but this one is giving me a bit of trouble. I'd ...
8
votes
1answer
2k views

Fourier transform in $L^p$

Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$? I'd be grateful for any kind of help including providing a literature or ...
2
votes
2answers
2k views

Example of a function whose Fourier Series fails to converge at One point

Can one think of an example of a continuous $2\pi$ periodic function whose Fourier series fails to converge on $\mathbb{R}$. I referred this in the wikipedia page but no avail: It might be ...
0
votes
1answer
36 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
2
votes
0answers
283 views
0
votes
1answer
246 views

Fourier Transform of $1/(\pi\cdot t)$ by Duality

I'm asked to prove using "duality property" the Fourier transform of $$\frac{1}{\pi t} = -j sgn(f)$$ I have the proof steps but I'm quit not understanding it: multiply by $j = \frac{j}{j(\pi t)}$ ...
2
votes
0answers
132 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
1
vote
0answers
162 views

Using Fourier Analysis to determine Green's Function of Laplace's equation

I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have ...
1
vote
1answer
34 views

Fourier transform Excercise

I am stuck on an excerise which says that prove the fourier transform $f(k)$ of a real function satisfied the condition $f(-k)=f*(-k)$. Where the astericks denotes the complex congugate. I am ...
23
votes
4answers
810 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
-1
votes
1answer
36 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
2
votes
1answer
40 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...
3
votes
2answers
141 views

Approximation of a $L^1$ function by a dominated sequence of continuous functions

Consider $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$ and the Lebesgue measure on it. Denote by $L^1(\mathbb{T})$ the set of integrable functions on $\mathbb{T}$ and by $C(\mathbb{T})$ the set of ...
1
vote
1answer
36 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
1
vote
1answer
220 views

Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform $$ f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$ where $F$ is an arbitrary function and ...
3
votes
1answer
74 views

$L^2$ and uniform norm of $\text{sinc}\, x$ and its derivatives

Looking at the graphs of the derivatives of $\mathrm{sinc}\,x$, it appears that they all are bounded by $1/x$, with $[\mathrm{sinc}\,(x)]'$ the sole exception: A few questions: 1) With the ...
0
votes
1answer
29 views

What happens to fourier transform of the sampled output of pure sinusoidal input of 26kHz if sampled with 44.1kHz sample frequency?

Because pure sinusoidal signal only contains impulses, I was wondering what happens to the fourier transform of the sample output from the sinusoidal input of $26$kHz if the sampling is done with ...
1
vote
0answers
62 views

bessel function with Fourier transform

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
0
votes
1answer
64 views

Odd or Even for Fourier Series?

I have the function $f(x) = -x^2 + x\pi$ and $0\le x\le \pi$ and without seeing the graph I want to show if it is odd or even, but of course $f(x) = f(-x)$ doesn't show that it is even because I can't ...
3
votes
3answers
49 views

$f\in M(\mathbb{R})$ but $\hat{f}$ is not

I am studying Fourier analysis. I noted some problems state $f,\hat{f}\in M(\mathbb{R})$ as assumption, where $M(\mathbb{R})$ denote the collection of all continuous and of moderate decrease functions ...
3
votes
0answers
69 views

Riemann-Lebesgue Lemma for Spherical Harmonics expansion

Here is my question: A basic result of classical Fourier analysis is that the fourier coefficients of an $L^1$ function must tend to zero (Riemann-Lebesgue Lemma). Is there analogous result to the ...
2
votes
2answers
89 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
0
votes
1answer
42 views

Sharpening a curve

I have a frequency domain graph as shown. I need to "sharpen" the curve to get a better response, and computing large butterworth orders is not possible on my machine. Hence, I would like to know if ...
0
votes
1answer
9 views

Support of Auto-correlation

Suppose $f\in C_0^{\infty}(\mathbb{R}^n ),$ then clearly we have supp$(f\ast f)\subseteq$ supp$(f)+$ supp$(f)$. The question is whether supp$(f\ast f)\subseteq 2$ supp$(f)$ holds? Any counterexample?
0
votes
1answer
33 views

Support of polynomial distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
0
votes
1answer
54 views

Solving convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside step function

How does one solve convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside (unit) step function? I tried using Fourier transform of both functions to ...
1
vote
0answers
64 views

Power law in power spectrum and memory.

If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law ...
0
votes
1answer
77 views

Find the Fourier transform of $\frac1{1+t^2}$

Find the Fourier transform of $$f(t)=\frac1{1+t^2}$$ using contour integration that $$F\{f(t)\}=\int^\infty_{-\infty}\frac1{1+t^2}e^{2\pi ft}dt$$ How can I do this?
0
votes
1answer
97 views

Fourier transform of compactly supported differentiable function

Let $K$ be the space of infinitely differentiable functions $\mathbb{R}\to\mathbb{C}$ with compact support. I read the unproved statement in Kolmogorov-Fomin's Элементы теории функций и ...
2
votes
0answers
22 views

Strategies for approximating fourier transform of $k$-th power of the $n$-th derivative of a function

For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I ...
7
votes
2answers
934 views

Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
0
votes
1answer
147 views

Young's inequality for convolutions

Let's assume that the convolution $f * g$ is continuous with $\lim_{|x| \to \infty}(f*g)(x) = 0$ and that $f, g \in L^2$. Then the following inequality holds $$ \| f * g \|_{\infty} \leq \| f \|_2 ...
2
votes
0answers
33 views

Fast Fourier Transform and its example

I read the wikipedia and my textbook, but I can't understand the whole process of Fast Fourier Transform. Especially the book uses the Cooley-Tukey algorithm and it gives an example of 4X4 matrix like ...
0
votes
1answer
31 views

Support of tempered distribution under exponetiation and differentiation

Suppose $u$ is a tempered distibution in $\mathbb{R}^n$. How are supp$(\widehat{u})$ and support of $\sum_{|\alpha|\leq k}\frac{\partial^{\alpha}\widehat{u^n}}{\partial x^{\alpha}}$ compared , where ...
0
votes
1answer
45 views

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$.

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$. Normally, I would do is find a matrix representation and ...
0
votes
1answer
49 views

using fourier method to compute this integral

Use the method of Fourier analysis to calculate the following integral: $$ \int_{0}^{\infty} \frac{\cos x}{1+4x^2} \operatorname{d} x .$$ Could someone help about this question? what ...
2
votes
1answer
68 views

The inverse Fourier transform of $\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}$

I need help to compute the following integral $$\int_{\mathbb{R}^n}\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}e^{2\pi i\xi \cdot x} \mathrm{d}\xi $$ where $\widehat{\varphi}$ is the Fourier ...
2
votes
0answers
42 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...
4
votes
2answers
131 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
2
votes
1answer
62 views

Prove a trigonometric series is positive

Let $f(x)= \sum_{n=-\infty}^\infty \frac {e^{inx}}{1+n^2}$ on $[-\pi,\pi]$. Prove $f(x)>0$ for $x\in[-\pi,\pi]$. This is an review question for my Fourier course. I am not sure how to approach ...
1
vote
0answers
26 views

Product of Fourier transform

I'm supposed to calculate : $ F[\delta(x-a)](\nu)F[e^{i2\pi \nu x}](\nu)$ Since $F[\delta(x-a)] = e^{i 2\pi \nu a}F[\delta(x)] = e^{i 2\pi \nu a}$ it leaves : $$e^{i 2\pi \nu a}F[e^{i 2\pi \nu x}] ...
2
votes
2answers
80 views

Why am I allowed to set a fixed point in a fourier series?

I'm working with $f(t)=\cos(at)$, for $a\in (0,1)$, on the interval $(-\pi,\pi)$. I've calculated the fourier series on this interval. what I would want to do next is to fix $t=\pi$ and get a nice ...
1
vote
1answer
62 views

Evaluate integral using Fourier analysis

$\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$ $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$ There is no hint for these two questions. I think for Q2, since it's a square, I can use Plancherel ...
1
vote
0answers
23 views

Evaluate the limit of function presented as a series

This is an additional exercise given in my Fourier analysis course. Define $F(t)=\sum_{n=-\infty}^\infty (-1)^n e^{-2n^2t^2}, \,t>0$, Prove that $\lim_{t\to \infty}F(t)=1$. ...
3
votes
0answers
54 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...