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 $\mathbb{R}^n$.

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Fourier transform of convolution and Plancherel's theorem

For some function $h \in \mathcal{S}(\mathbb{R})$ and $\theta, \eta \in \mathbb{R}$, we defining the Fourier transform as \begin{equation}\label{fourier} \tilde{h}^{\pm}(\theta) := \frac{1}{2 \pi} \int dp h(p)e^{\pm i \theta p}, \end{equation} the…
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A sequence of functions such that $\hat{f}_\alpha(\gamma) = 1$ and $\lim_{\alpha \rightarrow \infty} f_\alpha * f = 0$ if $\hat{f}(\gamma)=0$.

The following definition comes from my professors notes. $\hat{f}$ refers to the Fourier transform of $f$. Given $\gamma \in \mathbb{R}^d$, a bounded family of functions $(f_\alpha)_{\alpha > 0}$ is called a $\gamma$-net in $L^1(\mathbb{R}^d)$ if…
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L2 norm inequality respect to convolution on $S^1$

I am trying to show if $f,g\in C^0(S^1)$, then $$||f\ast g||^2_{L^2(S^1)}\leq ||f\ast f||_{L^2(S^1)}||g\ast g||_{L^2(S^1)}.$$ But nothing comes out in mind, any ideas?
Jingeon An-Lacroix
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Question about Sets of Divergence

In Katznelson's Introduction to harmonic Analysis there is a theorem which follows from a lemma on p. 64 chapter 3 Theorem: E is a Set of Divergence for B ( B is a homogenous Banach Space) if and only there exists an $ f \in B$ so that, $$…
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Hausdorffness on Topological Quotient Group

Let $G$ is a topological group i.e. operation of the group and inverse are continuous wrt topology. I should show that if $G$ is Hausdorff then $G/N$ is Hausdorff where $N$ is a normal subgroup of $G$. Is it valid with only normalness of $N$? My…
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Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$

I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows. Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a measure space $(X,\mu)$ such that $f_n \to f$ in $L^{q,s}$. That…
Suugaku
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Use Riemann-Lebesgue Lemma to show the pointwise convergence of a partial sum

Let $f:\mathbb{R}\longrightarrow\mathbb{C}$ be a $2\pi-$periodic function. Denote $e_{k}(x):=e^{ikx}$, and then we know that the Dirichlet Kernel has forms…
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Density of $C_c^\infty(\Omega)$ in $W^{k,p}(\Omega)$.

Is is true that if $1 \leq p < \infty$, $k \geq 0$, and if $\Omega$ is an open subset of $\mathbf{R}^d$, then $C_c^\infty(\Omega)$ is dense in $W^{k,p}(\Omega)$? I seem to have a proof of this statement, but the care with which Evans handles the…
Jacob Denson
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Relation between Stein-Tomas adjoint restriction estimate and the Helmholtz equation

Let $d\sigma$ denote the surface measure on $\mathbb{S}^2$. For each function $f\in L^2(\mathbb S^2)$, the Fourier transform $\widehat{fd\sigma}$ is defined as the integral $$ \int_{\mathbb S^2} f(\xi)e^{ix\cdot \xi}\, d\sigma(\xi), \qquad x\in…
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Fourier transform and convolution product

Find the Fourier transform of the function: $$ I(x) = \int^{1/2}_{0} e^{-(x-t)^2} dt$$ using the theorem about convolution products. The theorem states that $\mathcal{F} \{f *g\} = \mathcal{F} \{f\} \mathcal{F}\{ g \}$. I am given the following…
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Techniques for studying harmonic functions in dimension $ > 2$

Given $y\in\mathbb{R}^N$, I am trying to show that the function $$V(x) = \frac{\lvert y\rvert^2-\lvert x\rvert^2}{\lvert y-x\rvert^N}$$ is harmonic in $\mathbb{R}^N\setminus\{y\}$. While it is always possible to try to compute the derivatives…
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Basic question about Bony decompositions - summation indices

I'm trying to understand some sort of inequality in a larger calculation. I believe my only issue is in counting correctly, so I've also tagged combinatorics. Suppose I have a function $f$. Let $\triangle_q$, $q\ge -1$ be the Littlewood-Paley…
Calvin Khor
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Fourier transform using convolution product

The question asks to find the Fourier transform of the function: $$ I(x) = \int^{1/2}_{0} e^{-(x-t)^2} dt$$ using the theorem about convolution products. I know that the theorem states that $\mathcal{F} \{f *g\} = \mathcal{F} \{f\} \mathcal{F}\{ g…
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A few questions concerning shearlets

I am currently reading Compactly supported shearlets are optimally sparse by Kutyniok and Lim and have a few small questions, which I have to give a talk about. At our university, this seminar is custom so one can take a dive into scientific…
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Limit of inner product of Gaussian convolutions

I am trying to prove that the Fourier transform on $L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$ preserves the inner product (from $L^2$). I have already proved the result for $f,g\in L^1\cap L^2$ such that $\widehat{f},\widehat{g}$ are also in $L^1\cap…
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