Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

2797 questions
1
vote
0 answers

integrability of Hilbert transform of a function

The problem is : Let $\varphi \in \mathcal S(\mathbb R)$ (Schwartz space) with $\int \varphi \ dx = 0$. Then the Hilbert transform of $\varphi$ belongs to $L^1(\mathbb R)$. I believe this helps $\lim_{|x| \rightarrow \infty} x H(\varphi) (x) =…
math student
  • 4,566
1
vote
0 answers

Fitting Sounds Waves with Sines/Cosines

I am trying to model sound waves with a series of sines and cosines but I am not sure what the best way to find the best deterministic sine/cosine combination that best fits the data. What are some methods to find, $cos(2{\pi}x/f+shift) $ the shift…
jessica
  • 1,002
1
vote
0 answers

Locally Compact Hypergroups

I am reading a Paper by ROBERT I. JEWETT Spaces with an Abstract Convolution of Measures. On page 51, he defines $K = \{ a,b,c\}$. The convolution operations are defined as \begin{align*} p_ap_a &= \frac 14p_e+\frac1{20}p_a+ \frac 7{10}p_b;\\ p_ap_b…
Milan Amrut Joshi
  • 1,109
1
vote
0 answers

A problem about the convergence of convolution

Let $G$ be a topological group. Let $f_{n}$ and $g_{n}$ be two sequences in $L^{p}(G)$ that are convergent to $f$ and $g$, respectively. Let $f * g \in L^{p}$. Is $f_{n} * g_{n}$ convergent to $f * g$? why? ($f_{n}*g_{n}\in L^{p}$ for all $n\in…
hassan zaki
  • 41
1
vote
0 answers

Several quesions about proof of Theorem mentioned in the book "A course in abstract harmonic analysis" by B.Folland

In chapter 4, sub-chapter 4.5 (closed ideals) in the book there is the following Theorem: Let $G=\mathbb{R}^{n}$ with $n\geq3$, and let $S$ be the unit sphere in $\mathbb{R}^{n}$. There is a closed ideal $\mathcal{J}$ in…
pelegbarsever
  • 163
1
vote
1 answer

Classical Hardy inequality from the modern form of the Hardy inequality.

This is well known Hardy operator $$(Hf)(x)=\frac{1}{x}\int_{0}^{x}f(t)dt. (1)$$ We have the classical Hardy inequality $$\int_0^{\infty}\left(\frac{1}{x}\int_0^x f(t)dt\right)^pdx\leq \left( \frac{p}{p-1}\right)^p\int_0^{\infty}f^p(x)dx, (2)$$…
Mokhmad-Salekh Khekhaev
  • 745
1
vote
0 answers

On strictly decreasing functions and Fourier Transform properties and Riemann's Zeta function

Consider a Gaussian function $f(t) = e^{- \pi t^2}$ which is strictly decreasing from $t=[0, \infty]$. It is a Fourier transformable, positive and even symmetric analytic function. Do such strictly decreasing functions have specific properties in…
Sriram
  • 11
1
vote
1 answer

Bounding an Oscillatory Integral Operator involving the phase $|x-y|$

I am currently studying Chapter 9 of Stein's "Harmonic Analysis", in particular Section 2.2.2. Consider the oscillatory integral operator $$ T_\lambda f(x) = \int a(x,y) e^{2 \pi i \lambda |x - y|} f(y)\; dy $$ The standard oscillatory integral…
Jacob Denson
  • 2,129
1
vote
0 answers

Integral form of a "generalized" Bessel potential

The Bessel potential is defined by $(I-\Delta)^{-\alpha/2}$ and it has an integral kernel $K$ such that $(I-\Delta)^{-\alpha/2}f = K*f$. Is there a way to generalize this to something like $$ (g-\Delta)^{-\alpha/2}f = \tilde{K}* f$$ where $g$ is…
Jakob Elias
  • 1,375
1
vote
0 answers

Integration of a complex exponential

I would like to know how to integrate a function that looks like $$ \int_{a<|x|
Hazelnut
  • 73
1
vote
1 answer

maximal function bounded below by original function

Is is true that for a locally integrable function, we always have $Mf(x)\geq |f(x)|$ a.e.? I think that is true but I can not find any reference for that.
Tongou Yang
  • 1,985
  • 12
  • 22
1
vote
0 answers

The Fourier transform in the sense of tempered distribution is not a function

Suppose $f\in L^p,p>2$, we can regard it as a tempered distribution. Then we can define the Fourier transform of the tempered distribution as $\hat{f}(\phi)=f(\hat{\phi})$, where $\phi$ is in the Schwartz. I want to prove that there exists $f$ such…
89085731
  • 7,614
1
vote
0 answers

$f\in$ BMO, $A$ invertible linear transform. Is then $f\circ A$ in BMO?

The space of functions of bounded mean oscillation on $\mathbb R^n$ is defined by $$ BMO(\mathbb R^n) = \left\{f\in L^1_{\rm loc}(\mathbb R^n) : \sup_Q\frac 1{|Q|}\int_Q|f(x)-f_Q|\,dx < \infty\right\}. $$ Here, the supremum is taken over all bounded…
amsmath
  • 10,633
1
vote
0 answers

estimate series by $log(e+x)$

I want to prove this inequality but it does not work with me, need help…
W.Vicky
  • 11
1
vote
0 answers

$L^p$ version of Square Function

I have taken an introductory course into Harmonic analysis and in the course, we went over square functions $s_\phi f(x) =\left(\int_0^\infty |f *\phi_t (x)|^2 \frac{dt}{t}\right)^{1/2}$ where $\phi \in \mathcal{S}$ is Schwartz with average zero:…
welshman500
  • 491
1 2 3
4
5 6 7 8