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, ...

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Involution in $L^{1}(G)$ is isometric.

(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.) Let $G$ be a locally compact group, and ...
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Is the zero set of a non zero real valued harmonic function discrete?

It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete. By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real ...
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Hardy-Littlewood-Sobolev inequality for $p=1$

Let $\mu $ be a positive Borel measure on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some $n\in (0,d]$ ...
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Simply connected domain and harmonic function

Let $\Omega$ be a simply connected domain that is properly contained in $\mathbb C$, and $u(x,y)$ is harmonic on the unit disk $\mathbb D $, then there is a funtion $f(z)$, that is one-one and ...
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Radial limits of harmonic conjugate and Hilbert transform

Let $\mu$ be a real measure on the circle $\mathbf{T}$. Then the function $$f(z)=\int_\mathbf{T} \mathrm{Im}\left(\frac{\zeta+z}{\zeta-z}\right) d\mu(\zeta)$$ is harmonic on the unit disc and its ...
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On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
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Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
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Properties of subharmonic functions

A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset ...
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Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
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Bound on the relative measure of $\delta$-neighbourhoods of compact sets using the Hardy-Littlewood maximal theorem

Notation: $|A|$ is the Lebesgue measure of $A \subset \mathbb{R}^d$, and $A_\delta = \{ x : \text{dist}(x,A) \leq \delta \} $ is the $\delta$-neighborhood of $A$. I want to show that there is a ...
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Do we have the pointwise bound $\left|\tilde{f}\right| \lesssim_d Mf$?

Edit. I know I am missing a mean value on all integrals, but unfortunately I do not know if it is possible to make an integral sign with a horizontal slash through it with MathJax. For any locally ...
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Direct sum decomposition of $L^2(\mathbb{R})$ using Fourier Transform

Let $L_+^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^+}\}$ and $L_-^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^-}\}$, where $\hat{f}$ denotes the Fourier ...
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Is Rudin correct here? Fubini's theorem and product measures

Let $X, Y$ be locally compact Hausdorff spaces with nonnegative regular measures $\mu, \lambda$. By definition (in the book I'm reading) a regular measure is a Borel measure for which every Borel set ...
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128 views

Convergence of the integral of a product of functions.

Let $\phi:\mathbb{R^n}\to\mathbb{R}$ be a Lebesgue-measurable function, with the property that for every $n$-dimensional cube $Q$ in $\mathbb{R^n}$, we have $$ \left|\int_{Q}\phi(x)dx ...
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Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
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restriction of unitary operator is unitary?

Let $\mathcal{U}: \mathcal{H} \rightarrow \mathcal{H}$ be a unitary operator on a Hilbert space $\mathcal{H}$. If $\mathcal{K}\subset \mathcal{H}$ is a closed subspace such that ...
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question regarding Fourier restriction estimates

Thanks for reading my post. I am trying to prove the following claim: If we have \begin{equation*} \left\|\hat{f}\right\|_{L^q(N_{1/R}(S))}\lesssim R^{\alpha-1/q}\left\|f\right\|_{L^p(B(0,R))} ...
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Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
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Example of an unbounded operator

Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does ...
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Consequences of Pontryagin Duality?

What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
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Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
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Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and ...
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$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
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Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates ...
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(Exponential) Growth of Operator Norm of Uncentered Maximal Function

Define the uncentered Hardy-Littlewood maximal operator $M$ by $$Mf(x):=\sup_{x\in B}\dfrac{1}{\left|B\right|}\int_{B}\left|f\right|,$$ where we the supremum is taken over all (open) balls $B$ ...
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Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
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Lebesgue differentiation theorem for Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...
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What classical conditions give unique Laplace equation solutions on a half-plane?

Solutions of the Laplace equation on the upper half plane of $\mathbb{R}^{2}$ are not unique: $$ \frac{\partial^{2}}{\partial x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u = ...
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Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space ...
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inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} ...
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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Odd form of controlling derivatves

In Muscalu, Schlag - Classical and Multilinear Harmonic Analysis (Cambridge Universitv Press 2013), page 299 there is a rather odd estimate for wich I cannot find any justification: Functions used: ...
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Decay of the Fourier transform of the surface measure of the sphere via uncertainty

I'm working through Tao's Recent Progress on the Restriction Conjecture notes (http://arxiv.org/abs/math/0311181). Currently, I'm working on problem 2.4, which will eventually allow us to compute the ...
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How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
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Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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Upper Bound for Operator Norm in Marcinkiewicz Interpolation Theorem

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and ...
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Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
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Computing the modularity function of upper triangular matrices

Put $B_p := \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(Q_p) : a, b, c \in Q_p \right\}$ the subgroup of upper triangular matrices in $GL_2(Q_p)$, $Q_p$ denoting the $p$-adic ...
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Fejer kernel applied to a measure

Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that $$\int_{\mathbb R} T \text{sinc}^2(Tx) d\mu(x) \sim\frac{\mu([-1/T,1/T])}{1/T}, \text{ as } T\to\infty?$$ Here ...
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Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
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Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
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Preduals of Banach spaces and in particular of $\text{BMO}(\mathbf R^d)$

In general the predual of a Banach space is not unique. If there are multiple ones must they be isomorphic? More specifically is $H^1(\mathbf R^d)$ the only predual of $\text{BMO}(\mathbf R^d)$ or ...
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Steinhaus theorem in topological groups

Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq ...
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Why isn't the parallel between the Fourier transform and the Laplace transform complete?

I mean the question in the following sense. For Fourier, we can do it on compact intervals and then we get a sequence of coefficients. We can do it continuum-style, and then we get a superposition ...
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Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
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What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...