# Tagged Questions

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### Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
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### How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0)$$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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### $L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
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### How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
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### Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space ...
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### How Fourier-Lebsgue spaces operates functions?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
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### Ultrafilter-based Fourier-Walsh-like Functions

Here is a (little wild) question about Boolean functions with countably many variables and a wild analog for Fourier-Walsh functions and analysis based on them. Let $x_1,x_2,\dots,x_n,\dots$ be ...
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### Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
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### Sets $E$ in $\mathbb{Z}$ such that any $l^2$ function with support on $E$ comes from Fourier of a continuous function

Are there infinite sets $E\subset\mathbb{Z}$ such that any $f\in l^2(\mathbb{Z})$ with support on $E$ comes from the Fourier transform of a continuous function on $\mathbb{T}$ ? If yes, is there a ...
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### Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f

If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
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### $L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$

It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
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### decay rate or bound of short-time Fourier transform of nonlinear waves

Suppose we have a nonlinear wave $f(x)=e^{2\pi i N(x+\epsilon \sin(x))}$ with positive $\epsilon$ small enough. Let $w(x)$ be a smooth window function supported in a unit ball. Define the short-time ...
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### Obtaining a pointwise bound on the convolution of two singular measures

I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids. We are in Section 7, near equation (34) (pag.16 of the arxiv). Notations and ...
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### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
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### local moments of measures whose Fourier transform vanish in an interval

Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form \begin{equation*} \int_{-\delta}^{+\delta}|h|(dt)\le ...
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### On lower bounds of exponential frames in l1 norm

Let $\{t_k\}_{k=-\infty}^\infty$ be a sequence of real numbers. I'm interested in finding the largest number A such that \begin{equation*} \int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i ...
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### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k ...
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### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$ \begin{equation*} f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})} ...
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### Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is ...
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### Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
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### About the boundedness of a multiplication operator.

Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator: Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}. ...
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### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
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### Sufficient condition for $L^p$ multiplier on the torus

Hello, I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from ...
Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...