The fourier-analysis tag has no usage guidance.

**12**

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380 views

### Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...

**12**

votes

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2k views

### Tanh version of a Fourier Transform?

I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...

**11**

votes

**0**answers

395 views

### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...

**10**

votes

**0**answers

490 views

### Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.
A famous result of Polya says if $f$ is an entire function of ...

**9**

votes

**0**answers

150 views

### Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...

**9**

votes

**0**answers

345 views

### Uncertainty principle in Entropy terms

Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
...

**9**

votes

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607 views

### Is the Fourier-Transform a bounded operator on Lorentz spaces L(2,q)?

It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^n)$ continuously into $L^\infty(\mathbb{R}^n)$ and $L^2(\mathbb{R}^n)$ continuously into $L^2(\mathbb{R}^n)$.
Then, by ...

**8**

votes

**0**answers

174 views

### Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...

**8**

votes

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1k views

### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...

**7**

votes

**0**answers

282 views

### Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...

**6**

votes

**0**answers

449 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 ...

**6**

votes

**0**answers

775 views

### injectivity of Fourier transform: is there algebraic proof?

Let $L$ be the Banach algebra of $L^1$-functions from $\mathbb{R}$ to $\mathbb{C}$ with $L^1$-norm and convolution as algebra multiplication. Assume that we knew that the homomorphisms from $L$ to ...

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votes

**0**answers

429 views

### Phase perturbations in oscillatory integrals

I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...

**6**

votes

**0**answers

189 views

### How big is the Fourier transform of the log of a polynomial over the p-adic numbers

Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...

**5**

votes

**0**answers

105 views

### Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm.
A nonclassical polynomial of degree $d$ is a ...

**5**

votes

**0**answers

51 views

### Value of prolate speroidal wave function at 0

I have a very basic question about prolate speroidal wave functions that can be defined as eigenfunctions to the following integral equation:
$$
\lambda\cdot \psi(x) = \int_{-1}^{1}\frac{\sin ...

**5**

votes

**0**answers

332 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...

**5**

votes

**0**answers

106 views

### Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?
...

**5**

votes

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118 views

### Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?
Examples for what I have in mind:
Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and ...

**5**

votes

**0**answers

273 views

### converse of Weyl criterion

Let $f∈L^1([0,1))$,suppose for all equi-distributed sequence $\{a_n\}^∞_{n=1}$ in $[0,1)$,we have
$$\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{k=1}^Nf(a_k)=\int_0^1f.$$
Do we have that $f$ is Riemann ...

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votes

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357 views

### Fourier theory of characteristic functions

Here is a question which (up to some translation) I have been asked by an electrical engineer. Let $f:\mathbb{R}\to[0,1]$ be a smooth function with $f(x+1)=f(x)$. I would like to approximate $f$ in ...

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votes

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582 views

### some questions about properties of harmonic measure

The original post
The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...

**4**

votes

**0**answers

233 views

### Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'

I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that ...

**4**

votes

**0**answers

406 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

**4**

votes

**0**answers

230 views

### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...

**4**

votes

**0**answers

238 views

### $f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in ...

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votes

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106 views

### Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area

In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...

**4**

votes

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2k views

### Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?

**3**

votes

**0**answers

134 views

### Proof without distributions

I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...

**3**

votes

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61 views

### Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...

**3**

votes

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234 views

### The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...

**3**

votes

**0**answers

96 views

### Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on ...

**3**

votes

**0**answers

100 views

### Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
...

**3**

votes

**0**answers

207 views

### What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi ...

**3**

votes

**0**answers

166 views

### An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...

**3**

votes

**0**answers

131 views

### Estimates of a bilinear oscillatory integral

Consider the operator $T_\lambda f(x)= \int_{\mathbb{R}^3}{\frac{\sin\lambda|x-y|}{|x-y|}\phi(x-y) f(y) dy}$, where $\phi\in C_0^{\infty}$, $\phi(x)=1$, when $|x|<1$ . My question is that do we ...

**3**

votes

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159 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

**3**

votes

**0**answers

91 views

### Does the following inequality hold under Zygmund condition?

Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, ...

**3**

votes

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153 views

### 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 ...

**3**

votes

**0**answers

192 views

### Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?

Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...

**3**

votes

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284 views

### On a differential inequality

The question has probabilistic origins, but it would take too long to elaborate. $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\eS}{\mathscr{S}}$
Fix a nonnegative ...

**3**

votes

**0**answers

208 views

### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...

**3**

votes

**0**answers

896 views

### estimates of exponential polynomials

Let $ p(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t}$ be an exponential polynomial.
In the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty ...

**2**

votes

**0**answers

66 views

### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...

**2**

votes

**0**answers

86 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
...

**2**

votes

**0**answers

66 views

### Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind
$$
I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...

**2**

votes

**0**answers

88 views

### Error term for a Fourier integral

There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable,
$$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$
So it should be that
...

**2**

votes

**0**answers

90 views

### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
...

**2**

votes

**0**answers

153 views

### Largest Fourier coefficient of sparse boolean function

Consider a Boolean function $f: GF(2)^n \rightarrow \{0, 1 \}$. I would like to show that if $f$ is sparse, i.e. $\sum f(i) \leq t$, then $f$ must have a large Fourier coefficient. (A Fourier ...

**2**

votes

**0**answers

66 views

### How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem.
Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...