# 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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questions

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55 views

### 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 ...

**0**

votes

**1**answer

64 views

### 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 ...

**2**

votes

**0**answers

131 views

### 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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61 views

### (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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52 views

### Radon transform range theorem and radial functions

(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 ...

**8**

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180 views

### 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 $a\in (1,2)$.

**2**

votes

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33 views

### 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.

**3**

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**0**answers

61 views

### Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$

It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...

**5**

votes

**2**answers

184 views

### Analog of the Birkhoff's ergodic theorem for the sequence of squares

Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation.
Then by the classical ...

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21 views

### Unable to do question 3 in 7.3 from Folland's Fourier Analysis and its Application [migrated]

I'm unable to answer this question, where we were given $f(x)$:
$$f(x)=\begin{cases}
1, & \text{if }-1<x<1 \\
0, & \text{otherwise}\end{cases}$$
The questions asks me to compute $f*f$ ...

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votes

**2**answers

602 views

### Fourier series of $\log(a +b\cos(x))$?

By numerical computation it seems like, if $a_0 < a_1$:
$$
\begin{multline}
\log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\
+ \frac{a_0}{a_1}\cos(\omega t)
- \frac{...

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vote

**0**answers

184 views

### Problem with completeness of an orthogonal system

For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by
\begin{equation}...

**3**

votes

**1**answer

99 views

### Example of a bounded function whose mean-zero mollification diverges at a point

For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\...

**2**

votes

**0**answers

81 views

### Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity

The classical heat semigroup on $\mathbb{R}$ is given by
$$
W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0.
$$
Then the Littlewood-Paley vertical square ...

**6**

votes

**1**answer

161 views

### the fractional integration method of the proof of Stein-Tomas theorem?

In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...

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68 views

### Strichartz estimate for the Schrödinger equation

Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t ...

**3**

votes

**1**answer

100 views

### Strict inequality in decoupling inequality

I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=...

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votes

**2**answers

320 views

### 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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votes

**0**answers

136 views

### 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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votes

**0**answers

84 views

### Inequality about exponential integrals

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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votes

**1**answer

184 views

### 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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92 views

### A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define
$$
I(\alpha, x)=\int_\mathbb R \cos(xt) e^...

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**1**answer

81 views

### Weyl symbol of product

Are there explicit formulas for the Weyl symbol of $-f(x)D_x^2 $ where $D_x:=-i\partial_x $ and $\partial_x$ is the derivative and $f$ some sufficiently smooth function?
In the standard quantization ...

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votes

**1**answer

230 views

### Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...

**4**

votes

**1**answer

172 views

### Idea behind Carleson's theorem modern proof “intitial reductions”

I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.
For any $f \in L^2(\mathbb{R})$, let $\...

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45 views

### Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.
...

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**1**answer

77 views

### Evolution equation generated by Fourier multiplier

I am on the hunt for techniques regarding a field which I am not familiar with.
More precisely, I am considering equation of the form
$$ i \partial_{t} u(t,x) + p(D)u(t,x) = 0, \ \ u_{|_{t=0}}=u_0(x)...

**7**

votes

**1**answer

273 views

### Compactly supported probability measure in high dimensions with fast Fourier decay?

For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...

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vote

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31 views

### Higher-order inner products of an orthonormal basis

Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...

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202 views

### 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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votes

**1**answer

102 views

### 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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104 views

### On Pitt's inequality (weighted Fourier inequality)

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(...

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votes

**0**answers

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)} ...

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68 views

### Fourier dimension of radial set

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 ...

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380 views

### Earliest use of deconvolution by Fourier transforms

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...

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72 views

### Sobolev convergence of Fourier series

Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...

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74 views

### Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...

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33 views

### Example of periodic semidifferentiable function without absolutely convergent Fourier series

Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...

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129 views

### Logarithm of the Fourier transform?

I've found this paper on the logarithm of the discrete fourier transform which proves that
$$
log F = 1/4 i \pi (I - (1 +i)F + F^2 - (1 - i)F^3)
$$
where $F$ is the unitary discrete Fourier ...

**9**

votes

**1**answer

245 views

### Bounding a Fourier coefficient of a non-negative periodic function in terms of its $L^2$-norm

This question is motivated by the earlier MO question: Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$ .
It is a cleaned up ...

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71 views

### Reconstructing a sine wave using square waves and Möbius inversion: L² convergence?

Let $s$ be the (“square wave”) $1$-periodic real function such that $s(x) = 1$ if $0<x<\frac{1}{2}$ and $s(x) = -1$ if $\frac{1}{2}<x<1$ (and maybe $s(0)=s(\frac{1}{2})=0$ for the sake of ...

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47 views

### Fourier coefficients of a variation in Teichmuller theory

Prove that for $\dot w[\mu](\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{...

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66 views

### Point wise convergence of Laplace transform and convergence of functions

Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},
$$
...

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votes

**1**answer

112 views

### Stationary phase method for $\varphi''(x_0)= 0$

Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...

**26**

votes

**1**answer

691 views

### Linear combination of sine and cosine

I was explaining to my students the other day why $\cos(2x)$ is not a linear combination of $\sin(x)$ and $\cos(x)$ over $\mathbb{R}$. Besides the canonical method of using special values of sine and ...

**4**

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196 views

### Simultaneous Hahn-Banach theorem

Let $C(\mathbb{T})$ be the Banach algebra of continuous functions on the unit circle. Let $n \in \mathbb{N}$ and let $P_n(\mathbb{T})$ be the subspace of trigonometric polynomials of degree at most $n$...

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**1**answer

121 views

### Explicit construction of Kakeya sets using Perron tree

I have found many excellent notes online that illustrate how to construct a Kakeya needle set (with measure $<\varepsilon$.) Yet none of them gives full argument about the construction of a Kakeya ...

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votes

**1**answer

199 views

### Earliest reference on the calculation of derivatives by Fourier transform

I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?
The Fourier transform of the derivative is (...

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vote

**1**answer

87 views

### Duality relation of Lorentz space $L^{p,1}$

want to prove the duality relation:
$$||f||_{L^{p,1}} =C_{p} \cdot \sup\{\int_X fg d\mu: \text{ for any } ||g||_{L^{p',\infty}}\le 1 \} $$
where $\frac{1}{p}+\frac{1}{p'}=1, p>1, \mu$ is $\sigma$-...

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100 views

### Fourier Inversion Theorem

I have a reference request concerning the Fourier Inversion Theorem.
Let $G$ be a compact hausdorff abelian group. $L_2(G)$ has a unique Haar integral $\int_G - d \mu$ arising from the completion of ...