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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Fourier Transform existence

Let $\varphi:\mathbb{R}^2\to\mathbb{R}$ be a continuous function. Moreover, consider that $f:\mathbb{R}\to\mathbb{R}$ is a schwarzian function, i.e. $f\in C^{\infty}$ and $\lim\limits_{x\to\pm\infty} ...
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coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The ...
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Inverse of a toeplitz matrix with fft based methods

I have a covariance matrix, Q and I need to find out Q^-1. Here, Q is a Toeplitz matrix. Now, I want to calculate the inverse of the matrix with fft based methods rather than the conventional ones ...
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Bounding a sum on lattice points in an annulus

How do we bound $\Sigma_{\beta<|\vec{k}|<\alpha}|\vec{k}|^{-2}$ by using an integral comparison type method ? How about $\Sigma_{\beta<|\vec{k}|<\alpha}|k|^{-4}$ Here ...
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How could I continue to show the inequality?

Let $g: [0, \pi]\rightarrow \mathbb{R}$ a $C^{\infty}$ function for which the following stands: $$g(0)=0 \ \ , \ \ g(\pi)=0$$ I have to show that $$\int_0^{\pi}g^2(x)dx \leq ...
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Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
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why the convolution of two functions of moderate decrease is again function of moderate decrease?

I need to prove that a convolution of 2 functions of moderate decrease is a function of moderate decrease. I tried to split the integral into two integrals but I couldn't manage to bound any one of ...
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Problem using the Fourier transform and convolution to compute an integral

I'm trying to write a subroutine (in Fortran) to compute integrals of the form $$I=\int_{-L}^{L} f(x)g(y-x) \:\mathrm{d}x, $$ using the convolution theorem and fast Fourier transforms. In my routine, ...
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Check if the orthogonal system is complete

According to my lecture notes: The orthogonal system $\sin{kx}, k=1,2, \dots $ in $[0, \pi]$ is a complete system in $L^2[0, \pi]$. $1, \cos{kx}, \sin{kx}$ in $[0, 2 \pi] $is a complete system in ...
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Condition for existence of Fourier transform?

We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits. So I want to ask ...
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What is Fourier Transform of $\phi(x,y) = 2x $

How to calculate Fourier transform of this 2D function? $\phi(x,y)=2x$ for $-1<=x <= 1 ; and -1<=y<=1$ and $\phi(x,y)=0$ ; otherwise I tried like this: $\psi(u,v) = ...
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Discrete Fourier Transform of a function

Is it possible to find the discrete Fourier Transform of a non-discrete function? I've been asked to find out the discrete Fourier Transform of the function $g(x)=\mathcal{X}_{[-2,2]}f(x)$, where ...
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Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$ where $ \displaystyle ...
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how do you do this integral from fourier transform.

I am trying to find the fourier transform of $$\frac{\sin(ax)}{x}$$ for $a >0$. This is clearly an even function so we only need to do the real part, but I could not evaluate ...
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What is the Fourier transform of the product of two functions?

Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer. The motivation behind the question is homework, but this is a basic principle in ...
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Signum function and Fourier transform

I'm extracting a portion of my notes which I believe I might have copied wrongly. Given this equation: $$\frac{G(\omega)}{2ic\omega} [e^{ic\omega t}-e^{-icwt}]$$ I want to find the Fouerir ...
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Features of phase and magnitude spectrum?

I have read in many books that whether the signal is 1D or multidimensional , The magnitude spectrum tells you how strong are the harmonics in the signal and The phase spectrum tells where this ...
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different formulas for the fourier series.

Quick question, I see both of these $$f(w) = a_0 + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw) \quad f(w) = \frac{a_0}{2} + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw)$$ Why the difference ( ...
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What's the difference? Absolute convergence of trigonometric series

Consider $$\sum_{n=-\infty}^\infty c_n e^{ixn}$$ What's the difference between this series being absolutely convergent for $x = 0$ and just being absolutely convergent? I mean, for $x = 0$, the ...
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definition of the Fourier transform of function on the sphere

Let $f: S^{n-1}\longrightarrow R^n$ be even continuous function. What is the Fourier transform of $f$?
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Fourier coefficients of $\cos(x/2)$

Is there a straightforward way to calculate the fourier coefficients of $\cos(x/2)$ in closed form on the interval $[0,2\pi]$? (I mean in terms of the generic $n$) From a calculation of the first ...
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Derivation of the momentum operator

\begin{align*} \langle p \rangle &= \int_{-\infty}^\infty \frac{d p}{2\pi \hbar}\, \phi(p, t)^\ast \, p \, \phi(p, t) = \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \int_{-\infty}^\infty dx' ...
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Please recommend good text on complex Fourier series/analysis

I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular ...
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How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
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Fourier transform of a Green's function

I was studying for an exam and I found this question which has caused me a bit of trouble: Given the Green's function that satisfies the equation $$\Box ...
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$L^1$ Estimates involving bi-Laplacian

The following inequality can be shown to be true in the cases $p>1$: If $n\ge 5$, $\frac{1}{q} = \frac{1}{p} - \frac{4}{n}$, then there exists $C_{p,q,n}>0$ such that, for every $f \in ...
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Recovering Time Shift Using DFT of Translated Square Pulse?

As an exercise, I attempted to manually translate a pulse $n_0$ steps to the right and recover the translation using the time-shift property. The problem I'm encountering is that the phase unwrapping ...
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Fourier Transform of Gaussian-like function

I need to find the Fourier transform of the following Gaussian-like function: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2(x)}}\,e^{-\frac{x^2}{2\sigma^2(x)}}$$ where $\sigma(x) = \sigma_0 ...
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Example of tempered distribution

I am learning about tempered distributions. Today we learned that on $\mathbb{R}^n$, $\frac{1}{|x|^p}$ is a tempered distribution as long as $0 < p < n$. My question is, what goes wrong when $p ...
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Fourier transform accumulation property

It is known that Fourier Transform has an "accumulation property". If we define: $$X(\omega) = F[x(n)]$$ and $$y(k) = \sum\limits_{n=-\infty}^k x(n)$$ i can write: ...
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What do real and imaginary parts of phase spectrum represent?

In frequency domain, we can compute phase spectrum of a signal. Usually phase spectrum is complex valued. So my question is what do real and imaginary parts of phases of phase spectrum represent ? ...
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Check smoothness at point looking at Fourier transform

Let $u \in \mathscr E'(\mathbb R^n)$ we a distribution with compact support. Then $u \in C^\infty(\mathbb R^n)$ if and only if for any $N \in \mathbb N$ there exists $C_N > 0$ such that $$ ...
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Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...
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Identity using the Fourier transform.

Use the representation as Fourier integral to prove that $$\int_0^\infty \displaystyle\frac{cos(xw)}{1+x^2}dw=\frac{\pi}{2}e^{-x}$$ I am really confuse with this kind of problem, I do not know ...
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What does it mean that a sine wave is unchanged when added to another sine wave?

From the wikipedia article on sine waves: The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and ...
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Smoothness of Fourier transform of $\frac{1}{|x|^p}$

Consider the "function" (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the ...
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Fourier Series: Heat Bar Problem [closed]

Need help understanding how to do this problem.
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Fourier transform of $x\left ( t \right )= t\cdot e^{-t^{2}}$

Knowing that the Fourier transform of the function $y\left ( t \right )= e^{-t^{2}}$ is equal to $\sqrt{\pi}e^{-\frac{\omega ^{2}}{4}}$ I then proceeded using the the derivative rule $$\mathcal ...
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Approximation of the coefficients of the Fourier Series via the FFT

Is there literature on the approximation of the coefficients of the Fourier Series via the FFT? The approach I'm interested is merely numerical, consisting of computing the integrals with the ...
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Fourier transform of a rational function with spike at origin

Consider a rational function $f(x) = \frac{p(x)}{q(x)}$, where both $p(x)$ and $q(x)$ are polynomial functions of the multivariate $x = (x_1, x_2,..., x_n) \in \mathbb{R}^n$. Also, let us say that the ...
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Why the spatial/mathematician's Fourier Transform?

I was wondering why the sign-change in the exponential of the spatial/mathematician's Fourier Transform and why is it called mathematician's spatial in either case?
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derivative of a function using cosine transform

If we have a periodic function y(x) the derivative of function can be found using Fourier transform as $derivative = y'(x)$ taking Fourier transform $F(derivative) = -kiy(x)$ i is the complex ...
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Fourier transform with non sine functions?

Fourier says that any periodic function can be represented like a infinite sum of sine functions with their appropriate periods,amplitudes and phases. My question is: is it possible to represent the ...
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Question about Fourier transforms of gradient, curl and divergence

Consider a vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$. Denote by $F_u$ the Fourier transform of a scalar or vector field $u$. Can one finds an equality relation between ...
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Relation between discrete and continuous inner product of a function

Let $g_1^d$ be descrete (sampled) version of continuous function $g_1$ , same for $g_2$. So we have $$\left<g_1^d,g_2^d\right>=\sum_{n=-\infty}^\infty {g_1^d[n] ...
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Fourier transform of $f'(t)$

The Fourier transform of f′(t) is : $$\hat{f'}(\omega) = \int_{-\infty}^{+\infty}f'(t)e^{-i\omega t}dt = f(t)e^{-i\omega t}\bigg\vert_{-\infty}^{+\infty} ...
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Increasing the points in a time scale changes the shape of the fft-function

This question is derived by this question and the corresponding answer. The problem was that I had a $sech(x)$-function in a specific time interval, and I applied a ...
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Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
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How can I show the approximate version of the fourier inversion formula?

Let f be $L^1(R) \cap C_0(R)$ and satisfies $|\hat{f}(\alpha)|\leq A\frac{1}{|\alpha|}$, for all non zero real $\alpha$, for some positive A. Then, show that for any $x \in R$, $f(x) = ...
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How can I show that there is M>0 for all positive a<A s.t $|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}| <= M $? [closed]

Let f be $L^1(R)$ and odd function. Then, for any positive $a < A$, there is $M>0$ such that $$ \left|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}\right| \leq M $$ ($\hat{f}$ is the ...