Questions tagged [fourier-analysis]

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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Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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Toeplitz matrices question with Fourier coefficients

Denote: $f(e^{i\theta})$ is continuous and strictly positive on the interval $ 0 \le \theta \le 2\pi$ with Fourier coefficients $$ t_j = \frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta} \quad ...
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202 views

Lower Bound on Oscillatory Integral

Let $p,y\in\mathbb{R}^d\setminus\{0\},\beta>0$ be given and fixed and define for all $\alpha>0$, $$I(\alpha) := \int_{x\in\mathbb{R}^d}\exp(\mathrm{i}\alpha p\cdot x-\alpha\beta \|x-y\|^2)f(x)\...
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188 views

The number $\pi$ in an unexpected context

[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square ...
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202 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
10
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1answer
153 views

Why is this integral is super-exponentially small?

Consider the integral $$I_n^{(a,b)} = \int_{-1}^1 (1-x)^a\,(1+x)^b\, P_n(x)\, dx,$$ where $P_n(x)$ is the $n$-th Legendre polynomial. Here's a plot of $|I_n^{(50,20)}|$ for $n=0,\dots,70$: (I just ...
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2k views

Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
9
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543 views

Discrete Fourier Transform: Shift by Fraction of a Sample

I use spectral Fourier methods to numerically solve PDEs and these methods make heavy use of the discrete Fourier transform (DFT). With respect to the DFT I have some issues understanding the discrete ...
9
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1answer
257 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
8
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96 views

A simple Fourier-analytic proof of $\zeta(s)\neq 0$ for $\text{Re}(s)=0$: is it possible/already known?

This is a naive attempt to show that the Riemann $\zeta $ function is non-vanishing over line $\text{Re}(s)=0$: Let us devise an integral representation for the $\zeta$ function which works over a ...
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332 views

Is the Fourier transform a special case of this version of the Yoneda lemma?

The (co)Yoneda lemma tells us that for a presheaf $F\in \hat{\mathbb{C}}$, the following formula holds: $$ F=\int^{c\in\mathbb{C}} Fc\ \times\ h_{c}\ , $$ where $h_c$ is a representable presheaf for $...
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282 views

Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
8
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236 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
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345 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall x \...
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466 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in S^{n-1}\subseteq\...
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373 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{...
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Fourier transform of vector-valued functions (e.g. differential forms)

Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined. The application ...
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Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Theorem 1.25: Suppose $ \phi \in L^1(\mathbb{R^n}) $ and $ \int_{\mathbb{R^n}} \phi =1 $ . Also, let $\phi_{\epsilon}(x)=\frac{\phi\left(\frac{x}{\epsilon}\right)}{\epsilon^n}$.Moreover , suppose ...
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Under what conditions does the “third-order version of Plancherel's theorem” hold?

I have read in a few places that the formula $$ \int_\mathbb{R} x(t)^3 \, dt \ = \ \int_{\mathbb{R}^2} \hat{x}(f_1)\hat{x}(f_2)\hat{x}(-f_1-f_2) \, d(f_1,f_2) $$ holds (where $\hat{x}$ denotes the ...
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Deconvolution question

Suppose $a,b,x:\mathbb{R}\mapsto\mathbb{R}$ are three functions of which $a$ and $b$ are known and $x$ is unknown. Suppose they are related by the integral equation. $$\int_{-\infty}^\infty a(t-s)\,x(...
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What is the Fourier transform of $ f=\chi_{[-1,1]} $?

I am totally new to Fourier transforms, so I would be very thankful for a feedback on my calculation. I want to calculate the Fourier transform of $ f=\chi_{[-1,1]} $, that is, $f(x)=1$ for $-1\le x\...
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231 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $n$ be a natural number and $B$ be the $n\times n$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \...
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221 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
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183 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
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343 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the Fourier-...
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58 views

Reed and Simon definition of product of distributions

Let $\mathcal{D}$ denote the space of $C^{\infty}$, compactly supported functions on $\mathbb{R}^{d}$, and let $\mathcal{D}'$ denote its dual (i.e. the space of distributions). In volume II of Reed ...
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Heisenberg's inequality

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The ...
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Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a bounded $L^1$ function that is piecewise-smooth (with the boundaries of the pieces having no accumulation points), but not necessarily continuous. Define ...
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657 views

Generalising Parseval's Identity using the Convolution Theorem

Suppose that we have a $2\pi$-periodic, integrable function $f: \mathbb{R} \rightarrow \mathbb{R}$, whose Fourier coefficients are known. Parseval's identity tells us that: $$\displaystyle \frac{1}{2\...
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Fourier transform of integral related to zeta function

In this MO question here, I asked about the Fourier transform of the zeta function. The second answer lists the following as a representation for $\zeta(s)$, with $E(x)$ as the floor function: \begin{...
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344 views

Computing the Fourier transform of the distribution $\|x\|^{-\alpha}$.

Question: Suppose we are given the tempered distribution $\|x\|^{-\alpha}$. We want to compute the Fourier transform $\mathcal{F}[\|x\|^{-\alpha}](\xi)$. What techniques are available for ...
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88 views

For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$.

As the title states: For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$ where $C_c^\infty(\mathbb{R})$ is the ...
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How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
6
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149 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in \...
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Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
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596 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\...
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Density of a class of function in $L^2(\mathbb{R}, e^x\,dx)$

Consider the class of function defined by $$\mathcal{G}=\operatorname{Span}\left\{e^{-\frac{(x+a)^2}{2}}-e^{-x}e^{-\frac{(x+a)^2}{2}}\mid a\in\mathbb{R}\right\}.$$ Is $\mathcal{G}$ dense in $L^2(\...
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Solution of advanced functional differential equation

Statement Consider an advanced functional differential equation $$ Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1} $$ Let's construct a solution of Eq. $(1)$ with finite ...
5
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1answer
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Solution of $u_t=\mathcal{F}u$

I tried to solve the equation $u_t=\mathcal{F}u$, where $\mathcal{F}$ denotes the Fourier transform, with initial data $u(x,0)=u_0(x)$. The solution should be given by $$ u(x,t)=e^{\mathcal{F}t}u_0=\...
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If a function is extended to make it periodic then must the integration limits also be extended?

The following extract is taken from "Riley, Hobson and Bence - Mathematical methods for physics and engineering", Section 12.5 - "Non-periodic functions", page 422 and 423: Find the Fourier series ...
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124 views

Fourier transform and fourth root

Given a well-behaved convex function $f(t):\mathbb{R}\to \mathbb{R}$, its Fourier transform (FT) $\hat{f}(\omega)=\mathcal{F}[f(t)](\omega)$ is positive (and decreasing) proof here. It follows that ...
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111 views

Fourier series: $\hat f(n)=O(1/n)$ and $f$ continuous implies uniform convergence?

Littlewood's Tauberian theorem: Let $a_n=O(1/n)$. (In particular, given any $0<r<1$, the power series $\sum a_nr^n$ converges.) If the function defined by the power series $$f(r)=\sum a_nr^n$$ ...
5
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1answer
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Triebel-Lizorkin space $F_{p,q}^s(\mathbb{R}^d)$, $p=\infty$

One way to define the Triebel-Lizorkin space is using dyadic resolution of unity. Let $\psi$ be a Schwartz function which satisfies $\hat{\psi}(\xi)=1$ when $|\xi|\leq 1$ and $\hat{\psi}(\xi)=0$ when $...
5
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285 views

Regularity of a function from asymptotics of its Fourier transform?

I know that if $f(t) \in L^1(\mathbb{R}) \cap C^k(\mathbb{R})$ then we must have $\hat{f}(s)=o(s^{-k})$ for the Fourier transform. Is there some sort of converse to this statement? Let $f \in L^1(\...
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97 views

Are the canonical representatives of the Hilbert space $L^2$ basis-dependent?

The space $\mathcal{L}^p(\mathbb{R}^n)$ of functions $f$ such that $\int |f(x)|^p\, d^nx$ converges is only a seminormed rather than a normed vector space, because any function $f$ whose support has ...
5
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1answer
295 views

Proof of Fejer's lemma

How does ont prove Fejer's lemma: If $f \in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$, then $\lim_{n \rightarrow \infty} \int f(t) g(nt) \, dt = \hat{f}(0)\hat{g}(0).$
5
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2answers
649 views

Laplace and Fourier transform

I have a doubt about the equivalence between Fourier Transform and Laplace Transform. It was told me that if I have a function such that: $f(t)=0$ if t<0 $f\in L^1(R) \bigcap L^2(R)$ I can ...
5
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103 views

Decay of positive definite functions in Lp

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means $$ \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0 $$ for all $...