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.

**-1**

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39 views

### Short-time Fourier transform of $f\ast g (y)- f\ast g(x)$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$
It is also known that the ...

**-2**

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**0**answers

48 views

### Fourier series of a heat equation [closed]

I am solving this PDE $$2u_{xx}=u_t$$ $$u(x,0)=x$$ $$u(0,t)=u_x(4,t)=0$$.
I found this solution $$u(x,t)=\sum_0^{\infty}\left(D_n\sin\left(\frac{n\pi+\frac{\pi}{2}}{4}x\right)\exp\left(\left(\frac{4}{...

**7**

votes

**0**answers

113 views

### Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...

**10**

votes

**2**answers

814 views

### Differentiability of Fourier series

Consider the function defined by the Fourier series
$$ f(x;\alpha) = \sum_{n=1}^\infty \frac{1}{n^\alpha} \exp(i n^2 x ) , $$
where $\alpha >1 $.
For what values of $\alpha $ is $f$ ...

**0**

votes

**0**answers

64 views

### Speed of divergence series

Given a positive sequence $\{a_k\}$ and an $s>0$ such that $\sum_{k=1}^\infty k^sa_k = \infty$, but $\sum_{k=1}^\infty k^s a_k/(\log k)^2 = 1$. For any $t\in [0,b]$, where $b$ is a fixed number, ...

**8**

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**0**answers

115 views

### A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...

**9**

votes

**1**answer

189 views

### The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...

**1**

vote

**1**answer

79 views

### Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation

This is a restated version of my original very broad question.
Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...

**2**

votes

**1**answer

129 views

### Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?

I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.
As we ...

**2**

votes

**1**answer

101 views

### $L^2(X) \cong L^2(X',\xi)$

Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example:
Let X=A\G, X'=N\G, where G=PGL(2), A={
$\left[\begin{array}{...

**1**

vote

**1**answer

66 views

### Maximum Magnitude Deviation between DFT and DTFT

This is a cross-post from signal processing forum as it was not conclusive.
Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum_{n = 0}^...

**12**

votes

**3**answers

500 views

### Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle?

Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by
$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$
Clearly, $(...

**0**

votes

**1**answer

89 views

### A question about the convolution theorem

I have the following "argument" about Fourier series, which I know is wrong because it yields a ridiculous conclusion. However, I don't know where the mistake is, and need to know which step is the ...

**3**

votes

**1**answer

117 views

### Does zero Fourier dimension imply there is no Rajchman measure?

Let $K\subset R^d$ be a compact set. It is well known that its Fourier dimension is defined by
$$\dim_F K=\sup\{s\ge 0: \exists \mu \in M_1(K) s.t. \hat{\mu}(x)=O(|x|^{-s/2})\}(|x|\to\infty),$$
where $...

**9**

votes

**1**answer

306 views

### Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...

**0**

votes

**0**answers

19 views

### Bound for nonhomogeneous low-frequency cut-off operator in littlewood-paley decomposition

In Remark 2.11 of Fourier analysis and nonlinear PDEs written by Bahouri and chemin, it suggests nonhomogeneous low-frequency cut-off operator $S_j$ is bound operator from $L_p$ space to $L_p$ space, ...

**6**

votes

**3**answers

324 views

### Connections between martingales and Fourier analysis

I have had this strange feeling recently that somehow, the theory of martingales we study in probability, and the theory of Fourier analysis are very alike. But I am not able to formalize my thoughts.
...

**4**

votes

**0**answers

70 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 ...

**2**

votes

**1**answer

360 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 ...

**1**

vote

**0**answers

68 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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**1**answer

118 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

**0**answers

32 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 $\lambda_i\to \infty$ ...

**6**

votes

**0**answers

134 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

**1**answer

256 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 ...

**1**

vote

**1**answer

105 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

**0**answers

53 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

**2**answers

170 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

**0**answers

96 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

**1**answer

383 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

**0**answers

111 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

**2**answers

168 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

**1**answer

68 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

**0**answers

50 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(...

**1**

vote

**0**answers

61 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

**1**answer

420 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

**1**answer

163 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

**0**answers

105 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

**1**answer

262 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

**0**answers

118 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 ...

**7**

votes

**1**answer

565 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

**1**answer

73 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

**0**answers

117 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\...

**3**

votes

**0**answers

146 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 $\int_{-\pi}^\pi F_N(t) dt$ ...

**1**

vote

**0**answers

40 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 ...

**0**

votes

**0**answers

39 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

**1**answer

188 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

**0**answers

81 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

**1**answer

110 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

**1**answer

315 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

**0**answers

42 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\...