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).

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What is the Fourier transform of $\frac{x}{\sin(x)}$?

What is the Fourier transform of $\frac{x}{\sin(x)}$? (Not $\frac{\sin(x)}{x}$!)
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Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
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Support of polynomial distributions

Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let $p_1,\cdots,p_m$ be ...
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Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
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14 views

Fourier series qn determine the fourier series coefficients

Can someone please help me with this Fourier series $q_n$: determine the fourier series coefficients of $x(t)$ given as $x(t) = \cos4t + \sin8t+3$?
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136 views

Multidimensional Fourier Transform

I'm having difficulty with multidimensional Fourier Transforms. I have the following problem for $u=u(t,x) \in \mathbb{R}$ $$ \frac{\partial u}{\partial t} = \sum_{m,n=1}^d ...
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52 views

Construction bump function with positive Fourier transform

I am looking for the construction of a smooth bump function, $f$, mapping the real line to itself which has two special properties: (1) $f$ is constant on some interval in its support (for instance ...
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25 views

Clarifying the Fourier Transform of $f_c(x)=\exp(-cx^2)$.

I believe I have found the Fourier transform of $f_c(x)=\exp(-cx^2)$ (where $0<c<\infty$) by noting first that $f_c'(x)=-2cx\exp(-cx^2)$. Taking the Fourier transform of both sides of this ...
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297 views

Is there software that interpolates/extrapolates data using a discrete Fourier?

I've read various methods of Fourier interpolation and extrapolation detailed in articles such as Interpolation and Extrapolation Using a High-Resolution Discrete Fourier Transform—so what I'm ...
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33 views

Fourier transform on trig wave

Find the fourier transform for signal in this picture (sorry for the bad quality) Could it be done like this? The signal is a sum of two triangular waves that are each delayed. ...
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13 views

Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...
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45 views

Continuity in the complex plane

I was reading a book where it is claimed that a sufficient condition for \begin{equation} f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2 \end{equation} to be continuous and is ...
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27 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...
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36 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 ...
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29 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 ...
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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 ...
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23 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 $H(-f)$.
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53 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 ...
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856 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 ...
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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 ...
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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 ...
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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 $j = \frac{j}{j(\pi t)}$ ...
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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 ...
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51 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 ...
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28 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 ...
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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 ...
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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 ...
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1answer
37 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 ...
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32 views

A highly oscillatory integral

I am considering the following integral $$ \int_{-\infty}^{\infty} \text{d} z' e^{-i\alpha(z-z')}e^{iV(z')(z-z')}\text{sign}(z-z'), $$ where $\alpha\in\mathbb{R}$ is a (large constant) and ...
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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 ...
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how to show that Kirchhoff's formular solves wave equation.

There is one exercise in text book, Fourier Analysis: An Introduction, Stein p.211 Ex# 11. I have no idea how to handle that formula. please help me!! this is Theorem 3.6 The definition of ...
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61 views

Proving the Riemann-Lebesgue Lemma in $L^1(\mathbb{R}^n)$

$\mathbf{Riemann-Lebesgue \ Lemma \ in \ L^1(\mathbb{R}^n)}$. Suppose that $f \in L^1(\mathbb{R}^n)$. Then $\hat{f}(k) \rightarrow 0$ as $|k| \rightarrow \infty$. I cannot understand any of the ...
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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?
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169 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 ...
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43 views

Should I start with complex variables before fourier series?

My math professor has said that it would be useful to start with complex analysis before learning fourier series in the signals and systems course(I'm an undergrad EE). Do you agree with that and why ...
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Does $|f\sin (x)|$ integrable on $\mathbb{R}$ imply that $|f|$ integrable on $\mathbb{R}$?

I guess not. Because we usually require $|f|$ to be integrable on ℝ so that it has the fourier transform. Can anyone give me an counterexample for the statement in the title? I have searched for ...
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1answer
46 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 ...
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1answer
21 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 ...
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30 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 ...
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1answer
37 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 ...
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$f\in M(\mathbb{R})$ 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 ...
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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 ...
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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 $ ...
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36 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 ...
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estimate on a convolution

Let $\psi$ be a non-negative Schwartz function on $\mathbb{R}$ such that supp$\hat{\psi}$ is contained in $[-0.1, 0.1]$ and $\hat{\psi}(0)=1$. Define $\psi_k(x)=2^k\psi(2^kx)$ for any integer $k$. Let ...
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8 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?
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31 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?
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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 ...