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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.

3
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0answers
61 views

On smoothness of a function and decay of its Fourier transform

I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here. I am interested in a quantitative version of the principle that smoothness of ...
1
vote
1answer
173 views

Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples? This seems to be a well-known result, but I can ...
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vote
0answers
59 views

Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?

Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$. Is it true ...
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votes
1answer
116 views

Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support. What is the relationship between $$ \widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
3
votes
0answers
28 views

Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ ...
5
votes
0answers
70 views

Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
3
votes
1answer
254 views

Detecting if a series represents a rational function

A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to ...
0
votes
1answer
81 views

Uniqueness of solution to system of integral equations

Given the following system of integral equations for an integrable function $f(x)$: For all integers $k \ge 1$ holds $\int_{0}^{2\pi} [f(x)]^k e^{(ikx)} dx = 0$. If $f(x)$ is real-valued and non-...
0
votes
0answers
35 views

Initial value theorem for Fourier transform

Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
1
vote
2answers
167 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
5
votes
0answers
88 views

Convolution theorem on a non-abelian Lie group

Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
11
votes
1answer
372 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
1
vote
0answers
108 views

Calculus of variation with discontinuous solutions?

I'm thinking of the following question: Consider a function $f: [0,L]\rightarrow\mathbb{R}$ and an energy functional $$F=\int_{0}^{L}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\mathrm{d}x.$$ The ...
7
votes
2answers
164 views

Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
0
votes
1answer
66 views

The minimum of the maximum of a sequence of sinc functions

I apologise if this is trivial or well known to be impossible: Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$ such that for the function defined as $$ f_{a_1,\...
2
votes
0answers
46 views

Star-convex curve and Fourier series

Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
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vote
0answers
43 views

Conditions for Poisson summation (for discontinuous functions)

Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
13
votes
1answer
407 views

A question on the sine function

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ Here I ask the ...
4
votes
1answer
150 views

$L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
10
votes
0answers
102 views

A combinatorial proof of the Harrow--Kolla--Schulman theorem

Let $Q^n := \{0,1\}^n$ be the Hamming cube with the Hamming metric. (Recall that the Hamming is defined by the distance $d(x,y) := \# \{ i : x_i \neq y_i \}$. For integers $0 \leq k \leq n$, define a ...
11
votes
1answer
247 views

Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?

One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...
2
votes
0answers
114 views

Theory of distributions on various domains

The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at ...
6
votes
1answer
420 views

graph signal processing

I have read this article https://arxiv.org/abs/1307.5708 about vertix-frequency analysis on graph. David IShuman in this article claims that,"we generalize one of the most important signal ...
2
votes
1answer
71 views

Are the Prolate Spheroidal Wave Functions absolutely integrable?

I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability. ...
2
votes
0answers
115 views

Average of irrational flow on the torus

Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$ defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that $$ \lim_{T\...
2
votes
1answer
290 views

Can we say that the Riemann zeta zero spectrum backbone asymptotic is $\frac{H_{\text{n}}+\frac{\partial \vartheta (t)}{\partial t}}{t}$?

Let the Riemann zeta zero spectrum (blue) be: $$f(N,t)=\Re\left(\zeta(1/2+it)\sum\limits_{\substack{n=1 \\ d|n}}^{N} \frac{\mu(d)}{n \cdot t \cdot d^{1/2+it-1}}\right)$$ where $N$ approaches ...
3
votes
0answers
133 views

A uniform Riemann sum approximation of the integral of the Fejer kernels

Let $F_N(t)$ denote the Fejer kernel $$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$ Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
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0answers
37 views

Discrete Wavelet Transform and Gaussian decay

I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
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0answers
38 views

Composing functions whose derivatives have $L^1$ Fourier transforms

Let $f, g \in C^n(\mathbb{R})$. Given bounds $\|f^{(i)}\|_\infty \le a_i, \|g^{(i)}\|_\infty \le b_i \forall i \in \{1, ..., n\}$, we can derive bounds for $\|(g \circ f)^{(i)}\|_\infty$ using the ...
2
votes
1answer
163 views

About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
2
votes
0answers
79 views

A specific Schwartz function $f$ on $\mathbb C^2$

Choose a Schwartz function on $\mathbb C$ of the form $f(z)=f(r e^{i\theta})= f_0(r) e^{in\theta}$. Then $$(*) \quad f(e^{i\alpha} z)= e^{in\alpha} f(z), \quad \forall z\in \mathbb C.$$ Now, let $f$ ...
6
votes
1answer
105 views

$L^p$ estimates and functions with positive Fourier transform

For $f\in\mathcal{S}$ a Schwartz function on $\mathbb{R}^n$ and $m$ a bounded function, define $Tf$ by $\widehat{T f}=m\cdot \widehat{f}$. Fix $1<p<\infty$, $p\not=2$. Suppose we have proved ...
1
vote
1answer
311 views

Which Fourier transform is the correct one?

Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it: $$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$ while from Sokhotski–Plemelj theorem it follows ...
0
votes
0answers
40 views

Positivity of an integral with product of functions and their Fourier transforms

What condition two functions $f$ and $g$ defined on $\mathbb{R^+}$ must fulfilled to have: $$\int\limits_{0}^\infty \ln(x) (fg+ \hat{f} \hat{g} ) dx >0$$ Where we note $\hat{f}(x)= 2\int\...
1
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0answers
131 views

On the bound of the Stein-Wainger oscillatory integral

Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by $$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$ Stein-Wainger [1] showed ...
4
votes
0answers
96 views

Is there a Lie group Fourier transform for D-modules?

Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,...
1
vote
0answers
91 views

Functional equation with Fourier transform

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$: $$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$ Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
3
votes
2answers
85 views

Decay of bandlimited function

Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have. I read on "G. ...
0
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0answers
65 views

Positive Convolution Root

I try to compute the convolution root of a symmetric, positive definite, nonnegative, one dimensional function $f: \mathbb R\to \mathbb R^+_0$. Furthermore I assume $f$ is bounded and $\int_{\mathbb R}...
1
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0answers
40 views

Comparison of (square) of a function and its Fourier transform in an integral

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
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votes
1answer
82 views

A question on the zeros involving the equation containing exponential factor [closed]

I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation $$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$ has at most only one ...
2
votes
0answers
46 views

Existence of unbounded $M \subset \Bbb{R}$ of finite measure s.t. $1_M$ is $L^p$-Fourier multiplier

I would like to know if there is a measurable set $M \subset \Bbb{R}$ such that $M$ has finite Lebesgue measure $0 < \lambda(M) < \infty$, $M$ is unbounded in the sense that $\lambda(M \...
0
votes
0answers
88 views

Can we deduce the sign of this integral which includes cosine transform?

Let's $v(t)$ be a negative real function and consider the following integral with $f(t)$ complex (rapidly decreasing at infinity): $$A= \int\limits_{0}^\infty \frac{1}{x} \int\limits_{x}^\infty \Big(...
15
votes
3answers
2k views

Is there an “analytical” version of Tao's uncertainty principle?

Let $p$ be a prime. For $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ let its Fourier transform be: $$\hat f(n) = \frac{1}{\sqrt{p}}\sum_{x \in \mathbb{Z}/p \mathbb{Z}} f(x)\, e\left(\frac{-xn}{...
0
votes
0answers
52 views

Solution of a functional equation with cosine transform

What are the functions verifying: $$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$ With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
2
votes
1answer
110 views

Solving classical parabolic equation by using Littlewood-Paley theory

Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the ...
3
votes
0answers
104 views

Reverse Markov-Bernstein inequality for trigonometric polynomials

Let $r(t)$ be a real trigonometric polynomial of degree $n>1$. Assume it has zero at $t=0$ of multiplicity $k>0$. What can be said about the lower bound of the constant $c(k,n)$ such that $$ \...
4
votes
0answers
94 views

Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?

As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
0
votes
0answers
72 views

Is the product of two Sobolev functions in L^p?

Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$. Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$. Under what conditions on $\beta$ can we get ...
3
votes
0answers
91 views

Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a ...