# Tagged Questions

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### What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
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### Modulus of of continuity of a convolution operator with respect to Wasserstein metric

For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as $$f_G := f*G(dx) := \int f(dx|\theta) G(d\theta)$$ for some nice kernel function ...
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### Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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### How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO) (For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
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### Fourier Analysis in Kahane and Zelasko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Zelasko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
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### When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
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### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
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### Fourier transform and support of a distribution

Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on ...
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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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### Let $f \in M^{1,1} (\mathbb R)$ (Feichtinger's algebra /Modulation Space). Can we say $Fof\in M^{1,1}(\mathbb R)$; $F$ is an entire function?

The Modulation space ( Feichtinger's algebra), $$S_{0} (\mathbb R) = M^{1, 1}(\mathbb R): = \{ f\in L^{2}(\mathbb R) : V_{g}(f) \in L^{1}(\mathbb R^{2}) \};$$ where $V_{g}f (x, w)$ is the short- ...
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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 ...
This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either. Fourier series define an isometry $L^2(\mathbb{Z}) ... 2answers 258 views ### A sufficient condition for a probability measure to have compact support Consider a probability measure$\mu$on, let's say,$\mathbb R$. Is there a necessary and sufficient condition so that$\mu$has compact support$Supp(\mu)$? I agree this question is too vague, ... 1answer 235 views ### Is this set a Riesz Basis of$L^2(0,\pi)$Let$A=\{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\} _{n=1}^\infty$, where$a$is a positive real number. Is$A$a Riesz Basis of$L^2(0,1)$? 1answer 212 views ### About the boundedness of a multiplication operator. Let be$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}. ... 1answer 235 views ### 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 ... 3answers 760 views ### Integral kernel for the resolvent of the laplace operator Consider the Laplace operator defined in the biggest possible subset of$L^2(\mathbb{R}^2)$and let$z \in \mathbb{C}\backslash\mathbb{R}$. Therefore$z \notin \sigma (\Delta)$the spectrum of ... 2answers 371 views ### Does the Fourier series of an$L^1$function converge to the function *weakly* in$L^1$? Let$f$be a periodic$L^1$function, and$S_n[f]$the$n$-th partial sum of its Fourier series. I am aware that$S_n[f]$might not converge toward$f$in$L^1$(i.e., in norm). However, does it at ... 2answers 575 views ### What is the simplest oscillatory integral for which sharp bounds are unknown? I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $are unknown when the critical ... 1answer 755 views ### In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ... 1answer 1k views ### A Fourier-analytic inequality used by Jean Bourgain I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in$R^k$and would appreciate some help in understanding a Fourier-analytic estimate used in ... 1answer 346 views ### Ask for theory about the weighted L^2(R^d) space. Dear MOs, I am now considering the following norm: $$||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.$$ where the integral is over the whole space$R^{2d}$and$H(x,y)$is some non-negative ... 2answers 275 views ### Is it possible that the intersection of two nest algebras contains only scalars? Dear all, I really want to know the answer of the following question. I would appreciate any help. Assume H is a separable Hilbert space, is it possible to find two nests N1, N2 such that the ... 1answer 436 views ### how to prove the range of a closed linear operator is closed ? The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence$R(T)$is closed in$Y \iff R(T^{*})$is ... 4answers 1k views ### General Procedure for Inverse of an Integral Transform Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform)$\int^{a}_{b} f(x) g(x,\xi) dx $? ... 1answer 496 views ### Fastest decay of Fourier Transform for Generalized Functions of compact support What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ... 0answers 197 views ### Continuous function sort If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ... 0answers 99 views ### Differential equation with switched parameters and boundary conditions in integral form Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem. ... 2answers 274 views ### Wiener Tauberian Theorem for nonunimodular group Is there a nonunimodular group for which Wiener's Tauberian theorem is true? Is a locally compact topological group whose volume grows polynomially with radius always unimodular? 0answers 280 views ### Fourier series/transform of an amplitude-limited sinusoid I am trying to estimate the amplitude of an original unlimited sine wave from a measurement of the power spectral density (PSD) of an amplitude-limited version. I expect that I may be able to do so ... 1answer 573 views ### A problem concerning$L^2([0,1]\times[0,1])$Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it. Let ... 0answers 369 views ### The log kernel and Bochner Theorem I was wondering if it possible to find a measure$\eta$on$\mathbb{R}$such that $$L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)$$ for every$x\in [0,1/2]$. On a structural ground, this ... 3answers 703 views ### How to isolate$f(x)$in$f(x+a)=f(x)+a\times g(x)$?$a \in \mathbb{R}f:\mathbb{R} \rightarrow \mathbb{R}g:\mathbb{R} \rightarrow \mathbb{R}$For generic functions$f$and$g$, how isolate$f(x)$in the equation below?$f(x+a)=f(x)+a\times ...
Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...