# Questions tagged [fourier-analysis]

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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### Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the Torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
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### Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the ...
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### Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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### (Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
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(UPDATED for rapid decay considerations + new question) In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
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### Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
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### eigenvectors of a graph Laplacian VS Fourier basis

Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
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### Lower bound on exponential sums

Let $k\geq 2$. Consider the following norm of exponenetial sum: $$I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.$$ Bourgain mentioned on Page 118 of https://...
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### Reference request: Band limited interpolation of data

I have come up with an interpolation method for irregularly placed data points on a square domain. The method assumes the data points are discrete, that is they coincide with nodes of a uniform ...
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I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski. During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
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### Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) [duplicate]

I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens. ...
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### What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
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### Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that: $$\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0$$ where $\Vert \cdot \Vert_1$ is the $L_1$ ...
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One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$\sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(... 0answers 81 views ### Wigner distribution The Wigner distribution of u\in L^2(\mathbb R) is defined as a function W(u) on \mathbb R^2 given by$$ W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} ...
In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...