**3**

votes

**0**answers

50 views

### Convolution of decaying polynomials [closed]

I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying
$$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some
$M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$...

**1**

vote

**0**answers

24 views

### Extension of $\ell^{2}$ decoupling to $C^{2}$ hypersurfaces

Let $\phi : [0,1]^{n-1} \rightarrow \mathbb{R}$ be a $C^{2}$ function such that $\phi(0)=\nabla\phi(0)=0$ and $0<\frac{1}{C}<\nabla^{2}\phi<C$. Consider the compact $C^{2}$ hypersurface $S \...

**1**

vote

**0**answers

50 views

### A Global Restriction Estimate from Local Estimate

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose ...

**6**

votes

**1**answer

129 views

### Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...

**5**

votes

**1**answer

68 views

### Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert_s = \...

**2**

votes

**0**answers

100 views

### Decoupling Implies Local Decoupling

My question comes from Bourgain and Demeter's The proof of the $l^2$ Decoupling Conjecture. A related question was asked about a year ago go on MO, but the author was interested in the reverse ...

**1**

vote

**1**answer

150 views

### trigonometric series with nonnegative coefficients and prescribed values

Given real numbers $x_1$, $\dots$, $x_n$ and $r_1$, $\dots$, $r_n$ (with reasonable restrictions), is there a trigonometric series $T(x)=\sum_k a_k \cos(kx)$ with $a_k\ge 0$ such that
$$
|T(x_i)-r_i|&...

**0**

votes

**0**answers

55 views

### Is there an analogue of Fourier series using infinite series of Jacobi elliptic functions to model doubly-periodic functions?

We can use Fourier series to model general singly-periodic functions. Infinite sums of orthogonal trigonometric polynomials can approximate any singly-periodic function that is continuous along its ...

**0**

votes

**0**answers

48 views

### Optimality of Exponent in $\ell^{2}$ Decoupling Theorem

For $\delta>0$, let $\mathcal{N}_{\delta}$ denote the $\delta$-neighborhood of the truncated paraboloid $P^{n-1}$ and let $\theta$ denote a $\delta^{1/2}\times\cdots\times\delta^{1/2}\times\delta$ ...

**3**

votes

**0**answers

86 views

### Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...

**0**

votes

**0**answers

82 views

### An integral form of sum $\sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2}$ for two real analytic at 0 functions? Fourier|Taylor series parallels

Thinking of parallels between Fourier series and Taylor series, you might find out that the integral $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...

**1**

vote

**0**answers

41 views

### A question about Fourier transform of function of the type $(1+P(x))^{z}$

Let $$f= (1+P(x))^{z},$$
where $P(x)\ge 0$ is a real polynomial in $\mathbb{R}^n$ of degree $2m>0$, and $z=a+ib$ with $a<0$. I want to study the behavior near the origin of the Fourier ...

**0**

votes

**0**answers

55 views

### Fourier analytic estimate

The following question arises naturally from applications to the image processing. Let $\alpha\in [0,1]$ and assume that for infinitely many $n\ge 1$ we have
$$\sum_{k=1}^n\frac{1-|\cos(2\pi k\alpha)|...

**1**

vote

**0**answers

48 views

### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...

**2**

votes

**1**answer

152 views

### Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first.
Let $\mathbb{Q}^{\times}$ be the multiplicative ...

**1**

vote

**1**answer

250 views

### Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...

**1**

vote

**2**answers

68 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

**5**

votes

**1**answer

211 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

**1**

vote

**1**answer

77 views

### Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...

**11**

votes

**3**answers

1k views

### Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$
by
$
\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.
$
It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...

**3**

votes

**1**answer

177 views

### On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series:
$$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$
Using ...

**1**

vote

**0**answers

70 views

### Hilbert Transform and multiplier in $\mathbb{C}(X)$

I found myself trying to solve an equation of that kind :
$$
H f= R f,
$$
where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...

**3**

votes

**2**answers

237 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**0**

votes

**0**answers

62 views

### Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on math....

**14**

votes

**2**answers

519 views

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

**2**

votes

**1**answer

210 views

### Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...

**2**

votes

**0**answers

43 views

### Can we say translation/dilation of the $L^p-$multiplier is again a $L^{p}-$multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ such that
$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}$ (where $C$ is some constant, $f\in L^{p}$). (That is, $m$ is an $L^{p}-$ multiplier) ...

**1**

vote

**0**answers

68 views

### A complicated integral inequality

How can we bound this integral:
$${\displaystyle \int_{-1}^{1}2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\left(\hat{f}\left(\xi\right)\right)^{2}...

**3**

votes

**1**answer

155 views

### How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.
Say we have a stochastic process described by a ...

**2**

votes

**1**answer

143 views

### Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story
I want to prove Euler's reflection formula by showing that
\begin{equation*}
f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s)
\end{equation*}
is constant, where $s = \sigma + it$. It's easy to see ...

**1**

vote

**1**answer

156 views

### Fourier transform of tanh [closed]

tanh is not absolutely integrable, so a direct fourier transform does not exist. But even for the signum function, which is not absolutely intrgrable, we can get the fourier transform by applying some ...

**5**

votes

**1**answer

620 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...

**1**

vote

**1**answer

98 views

### Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...

**7**

votes

**1**answer

97 views

### Fast Fourier Transforms for non-trigonometric bases

The fast Fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other bases, e.g. orthogonal polynomial bases ...

**50**

votes

**1**answer

2k views

### Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.

**7**

votes

**0**answers

105 views

### Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...

**2**

votes

**1**answer

51 views

### How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...

**9**

votes

**1**answer

261 views

### Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains
$$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$
The standard ...

**5**

votes

**1**answer

296 views

### Application of Factorization Theory to Oscillatory Integral Estimates

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
$$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...

**8**

votes

**1**answer

282 views

### Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...

**6**

votes

**1**answer

126 views

### Positivity of power of positive PSD matrices

Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries.
Suppose now that, on top of having nonnegative entries, ...

**0**

votes

**0**answers

97 views

### How to estimate the Fourier transform in high dimension?

Let $\epsilon>0$. Can we compute explicitly or estimate the following Fourier transform in $n$ dimension?
$I_\epsilon(x)=\int_{\mathbb{R}^n} e^{i<x,y>}\frac1{1+i\epsilon-|y|^2}dy.$
Here $i=\...

**3**

votes

**1**answer

97 views

### Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...

**1**

vote

**1**answer

51 views

### extremal kernels for functions on the torus

Let $\delta, \epsilon>0$ and $f: \mathbb{T} \rightarrow \mathbb{C}$ such that (i) $f(0)=1$ and (ii) $|f(x)| \le \epsilon$ for all $|x| > \delta$. What is the smallest $L = L(\delta, \epsilon)$ (...

**1**

vote

**0**answers

71 views

### Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous
function, $ \omega $ be a modulus of continuity and assume
$$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$
whenever $ x,h \in \...

**1**

vote

**1**answer

109 views

### What are the spaces for which the Fourier transform is an automorphism? [closed]

this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...

**2**

votes

**0**answers

77 views

### Bilinear Approach to Bochner-Riesz Conjecture in Two Dimensions

In some old lecture notes on the Restriction and Kakeya conjectures (Notes 5, specifically), Terence Tao gives a proof of the restriction conjecture (for the sphere) in two dimensions via a bilinear ...

**7**

votes

**1**answer

321 views

### The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...

**1**

vote

**0**answers

58 views

### Perturbation in Besov space

$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...

**2**

votes

**2**answers

311 views

### Why decompose a function with eigenvectors of Laplace operator? [closed]

On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...