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4
votes
1answer
133 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
4
votes
0answers
145 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 ...
1
vote
0answers
132 views

Bounding the norm of the Dirichlet kernel as a matrix function

Consider the Dirichlet kerel: $f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$. Now, given a diagonalizable real matrix $A$, one can consider $f(A)$, using the standard notation of matrix functions. Namely, $f(A) ...
1
vote
2answers
102 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
2
votes
0answers
98 views

Positive Fourier coefficients for a function $f:\{+1,-1\}^n \to \mathbb R$

This is from my research in computer science where the Fourier transform over $GF(2)^n$ is a tool to study functions on the Boolean hypercube. For example, the majority function on 3 variables is ...
5
votes
1answer
202 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
0
votes
1answer
168 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
91 views

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; ...
2
votes
1answer
121 views

Fourier inversion

I have certain doubts about a classical Fourier inversion theorem. According to it (this is a theorem from "Panorama of Harmonic Analysis" by Krantz), if $f$ and $\hat{f}$ are both in $L_1(R)$ and ...
0
votes
0answers
82 views

Fourier Analysis in Kahane and Żelazko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Żelazko (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 ...
1
vote
0answers
57 views

Counting frequencies of occurrence of patterns within a sequence using harmonic analysis?

Assume that we are given a sequence $\mathbf x := X_1,\dots,X_n \in \mathbb N^n$ for some $n \in \mathbb N$. I am interested in calculating the frequency of occurrence of some fixed sequence $\mathbf ...
0
votes
0answers
95 views

Localized convolution operators

Take our favorite differential operator $\Delta$ on $\mathbb{R}^3$. Module a constant, we can represent its inverse (in some sense) by a convolution operator, namely $$\Delta^{-1}f=f*\frac{1}{|x|}.$$ ...
5
votes
3answers
444 views

Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here. Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on ...
2
votes
2answers
116 views

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 $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
1
vote
1answer
88 views

norm of trigonometric polynomials under arbitrary change of signs

Let $\mathcal P$ be the collection of trigonometric polynomial with $L^p[0,1]$ norm one, $p>2$, then by changing the sign of coefficients of polynomials in $\mathcal P$ randomly, can we produce ...
1
vote
1answer
108 views

Carleson-Hunt inequality: changing order of summation

The well-known Carleson-Hunt inequality states that for any $p>1$ and any $f \in L^p(0,2\pi)$, the $L^p$-norm of $sup_n |S_n|$, where $S_n$ is the n-th partial sum of the Fourier series of $f$, is ...
4
votes
0answers
88 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 $o(1/n)$? ...
0
votes
0answers
56 views

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 ...
-3
votes
1answer
154 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

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 ...
1
vote
1answer
130 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
88 views

Relationship between Fourier series & DFT

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT. However, isn't there a flaw? When I integrate over the continuous ...
1
vote
1answer
255 views

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 ...
8
votes
0answers
188 views

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 $L^1$ ...
3
votes
1answer
143 views

What does a Zonal sphere harmonic look like?

Let $\mathcal{H}_k\subset L^2(S^n)$ be the space of sphere harmonics of degree $k$, i.e., they are the eigenfunctions of $\Delta_{S^n}$. And let $\Pi_k:L^2(S^n)\to \mathcal{H}_k$ be the orthogonal ...
0
votes
1answer
119 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
3
votes
1answer
123 views

Real-analytic variant of theorem 4.2.5 of Duistermaat's “FIO”, 1996

Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states: Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical ...
1
vote
1answer
110 views

Absolute convergence of multi-dimensional Fourier series

For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true that the multi-dimensional Fourier series converges absolutely? In other words, $\sum_{k\in ...
1
vote
1answer
179 views

An L1 function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
7
votes
2answers
515 views

Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question

Given $N$ pairs of distinct real numbers $t_i, t'_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in ...
0
votes
0answers
58 views

Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $||e^{if}||=e^{r}$?

Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put, $\ell^{1}(\mathbb Z)= ...
1
vote
1answer
105 views

A question which belongs to a class of Zygmund functions

Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, ...
2
votes
0answers
65 views

riemann mapping theorem for skew-fields of quaternions and beyond

Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it? In a less focused way, how far does the main ...
9
votes
1answer
262 views

Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 ...
3
votes
0answers
121 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 ...
0
votes
0answers
54 views

Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution ) $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...
0
votes
0answers
117 views

Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle , $\mu$ be a Borel probability measure on $S^1$ and $T:S^1\to S^1$ is a measure-preserving map (not necessarily invertible) with respect to $\mu$. The Fourier ...
1
vote
0answers
113 views

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 ...
2
votes
1answer
175 views

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 ...
0
votes
0answers
55 views

Fourier coefficients of a non-atomic measure on the unit cirle

Let $0<\lambda<1$ and $\pi:\Pi_{\mathbb{N}} \{0,1\}\to S^1$ be the continuous map from the product space of $\{0,1\}$ over the set of natural numbers $\mathbb{N}$ onto the unit circle $S^1$. Let ...
0
votes
1answer
86 views

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 ...
3
votes
0answers
164 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
2answers
341 views

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 ...
1
vote
1answer
95 views

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 ...
1
vote
1answer
245 views

Fourier Transform of compactly supported $L^1$ functions

Background Given a (translation bounded) positive definite measure $\gamma$ lets say on $\mathbb R^d$, its Fourier transform as a tempered distribution is a positive measure $\widehat{\gamma}$. I am ...
0
votes
1answer
158 views

Problem with understanding an equation

I have read the article Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets and I don't know how to interpret Equation (10) on page 630, because this equation ...
9
votes
0answers
118 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 ...
11
votes
0answers
347 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 ...
1
vote
0answers
121 views

Are the two functions equal? If not, what relation are they?

We have $$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$ $$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi ...
1
vote
1answer
91 views

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- ...
2
votes
1answer
229 views

Let $f \in S(\mathbb R)$. Can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$, $\hat{f} (y) : = ...