# 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).

10,136 questions
Filter by
Sorted by
Tagged with
5 views

### Intersection of subspace of cyclical rotations with orthant

In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector $\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
14 views

### Trigonometric polynomial derivative upper bound

Let $P$ be a trig poly of degree $N$ on the torus $\mathbb T$. Prove that $$\Vert P' \Vert_\infty \lesssim N\Vert P \Vert_\infty.$$ I'm not sure how to approach this problem, though I feel like some ...
26 views

30 views

### How does this definition of Fourier transform in Fulton and Harris 3.32 relate to the usual notion of Fourier transform?

This is exercise 3.32 in Fulton and Harris' Representation Theory: A First Course. It defines Fourier transform in a form unfamiliar to me, and I could not find any definition of Fourier transform ...
45 views

### All DFT of binary numbers subsets of prime length are nonzero

Let $p$ be a prime. Consider a sequence $S$ of $p$ binary numbers $x_n \in \{ 0, 1 \}$, i.e. $S = \{x_1, x_2, \cdots, x_p\}$, where the number of zeroes in $S$ is neither $0$ nor $p$. Then the ...
1 vote
32 views

32 views

### persavel identity

i want to prove parseval identity $||f||_2^2 = S_N(f)^2$ ($S_N(f)$ partial sum of Fourier series) in my proof i know bessel identity: $||f||_2^2 \geq S_N(f)^2$ and i know that giving f function ...
17 views

### Fourier Transform Method for Solving Fredholm Translation-Invariant Covariance-Kernel Integral Equations

I'm examining a problem involving the Fredholm integral equations of the second kind and trying to apply Fourier transform techniques. Particularly, the challenge arises when seeking to express the ...
32 views

### Semigroup property between pseudodifferential operators and differential operators

Given a positive integer $n$ and a>0. Let consider the operators $\nabla^n (\cdot)= \sum_{i=1}^d \partial_i^{n}(\cdot)$, and $(1- \Delta)^{\frac a 2}$ defined at the Fourier level as (modulus ...
84 views

75 views

### Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?

One nice property of the Fourier transform is it's famous convolution theorem : $$f*g = \mathcal{F}^{-1} \left\{ \mathcal{F}\left\{ f \right\} \cdot \mathcal{F}\left\{g\right\} \right\}$$ If we want ...
1 vote
34 views

### Asymptotics of a multi-dimensional Fourier series

Let $\alpha\in(0,2]$ and $$f_t(u)=\sum_{m\neq 0}\frac{1}{m^\alpha}\exp(-\|m\|^2 t^2)\exp(im\cdot u),u\in \mathbb R^d.$$ Would anyone know a closed form expression for $f_t(u)$, in all dimension $d$? I ...
1 vote
41 views

### Is the convolution of a tempered distribution and a Schwartz function also a function?

Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as $$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$ where $\tilde{f}(x) = f(-x)$. The above ...
1 vote
50 views

### double doubt working with fourier inverse theorem and a calculation of a sup

the first one is this : I am working with Fourier inverse theorem and I have $f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega$ with $f \in L^1(\mathbb{R})$ ...
31 views

1 vote
36 views

### Question on Rudin's proof of Theorem 9.5 in Real and Complex Analysis

I'm working through Theorem 9.5 in Rudin's Real and Complex Analysis and I'm having trouble understanding one of the steps. The statement of the proof is as follows: Rudin uses Theorem 3.14, which ...
30 views

### Taylor series for radial function

Denote by $\hat{J}$ the Fourier transform of $J\in C(\mathbb{R}^n,\mathbb{R})$, a nonnegative, radial function with \begin{equation} \displaystyle\int\limits_{\mathbb{R}^n}J(x)~\mathrm{d}x=1. \end{...
1 vote
198 views

### How do I compute the Fourier Coefficients of a Riesz Product?

Let $(\alpha_n)_{n\in \mathbb{N}}$ be a sequence of non-zero real numbers such that $\sum_{n=1}^\infty \lvert \alpha_n \rvert^2 < \infty$. We consider the Riesz Product \begin{align*} \prod_{n=1}^\...
273 views

### Fourier transform of $\frac{1-e^x}{1+e^x}$

I was trying to compute $\int_{-\infty}^{\infty}e^{ikx}\frac{1-e^x}{1+e^x}\, dx$ from a Mathematical trivium. I tried first with contour integration but finding the right shape was hard. I gave it a ...
43 views

### $A \subset \ell^2(\Bbb Z)$ as a collection of Fourier coefficients

Let $A$ be a closed subspace of $\ell^2(\Bbb Z)$. Suppose for every $\{a_n\}_n \in A$, we have $\{a_{n+m}\}_n \in A$ for each $m \in \Bbb Z$. Show that there exists a measurable set $E\subset \Bbb T$ ...
Let $F\subseteq \mathbf{Z}^2$, ($\mathbf{Z}^2$ is the integer lattice), such that $|F| = \alpha < \infty$. For any $(m_1,m_2)\in \mathbf{Z}^2\setminus F$, we can define a double trigonometric ...