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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English version of the paper - L’algèbre de Fourier d’un groupe localement compact

Help regarding finding the English translation of the paper below required: Pierre Eymard. L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France, 92:181–236, 1964
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Do we have for $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ exists for $u$ subharmonic?

Let $u:\mathbb{C}\to \mathbb{R}$ be a subharmonic function. Do we have that the limit $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ converges to a (possibly infinite) ...
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Can the real part of an entire function be bounded above by a polynomial?

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial. If $p(z)$ is constant, then this can be shown by ...
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references for abstract harmonic analysis

I am currently studying a basic course in abstract harmonic analysis. I do have a good background in abstract algebra and functional analysis but I have not done a course in Fourier analysis. Is it ...
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Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
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Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
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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 ...
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Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
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26 views

$L^{p}$ identity for the maximal operator.

This is probably a very easy question, but I'm failing to understand it. Given a function $f \in L^{p}(\mathbb{R}^n)$, $1<p\leq \infty$, we define the uncentered maximal function of $f$ as $$ ...
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1answer
18 views

Number of copies of irreducible unitary representation in $L^2(G)$ for compact group $G$?

Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is ...
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Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
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Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
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Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t ...
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Commutator Characterization of $BMO(\mathbb{R})$

Let $a:\mathbb{R}\rightarrow\mathbb{C}$ be a locally integrable function, and let $H$ denote the Hilbert transform. Suppose that the commutator operator $[a,H]$ defined by $[a,H]f:=aH(f)-H(af)$ is ...
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1answer
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Invariance of noneuclidean laplacian

In a book I'm reading it says: Putting $f(x,y)=F(u,v)$ with $\gamma(x+yi)=u+iv$ and using Cauchy-Riemann equation for $\gamma(z)$, we have $$\Delta f(x,y) ...
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An operator is linear and bounded on a hilbert space

an operator linear and bounded on a hilbert space Let H be the Hilbert space L^2(R), and assume that the continuous function g satisfies 0
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Extension to a continuous linear functional

Extension to a continuous linear functional May this functional be extended to a continuous linear functional on these Hilbert spaces?
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Non-compactness of support of linear KdV equation solution

The last question in Linares and Ponce's 'Introduction to Nonlinear Dispersive Equations's first chapter asks the reader to prove that, if the following IVP is given: $$\begin{cases} \partial_t u + ...
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35 views

How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to ...
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A question about general Marcinkiewicz interpolation theorem

The general Marcinkiewicz interpolation theorem states as following: If $T$ is a linear operator of weak type $(p_0,q_0)$ and of weak type $(p_1,q_1)$ where $q_0\neq q_1$, then for each ...
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60 views

Questions about delta function.

Let $z \neq 1$ be a complex number. Then \begin{align} \frac{1}{1-z} = \sum_{n=0}^{\infty} z^n. \end{align} We have \begin{align} \frac{z^{-1}}{1-z^{-1}} = \sum_{n=1}^{\infty} z^{-n}. \end{align} ...
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Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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Equivalence of $H^{1/2}(S^{1})$ norm with integral

I've been attempting to show that the Sobolev space norm $$\|f\|^{2}_{H^{1/2}} := |\hat{f}(0)|^{2} + \sum_{n \neq 0} n |\hat{f}(n)|^{2}, $$ for $f$ on the circle, is equivalent to the integral $$I(f) ...
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Proving Div-Curl Lemma through Paraproducts

I want to to prove the classical div-curl lemma of Coifman-Lions-Meyer-Semmes in the following form: Div-Curl Lemma. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Schwartz function such ...
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Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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Fourier series calculation

I had an exam question today that stated something along the lines of the following: "Let $f$ be an even function given by $f(x)=x$ on $[0,\pi]$ and extend $f$ to $\mathbb{R}$ by $2\pi$-periodicity. ...
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How come $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$ and how to calculate it's eigenvalues

How to prove Euler's Formula $e^{i\theta} = (cos\theta + isin\theta)$? I know this is kind of basic and I am familiar with this equality for a long time. But, how do I prove it? And another ...
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Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
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Laplacian of a radial function on a riemannienne symmetric space

I would like to know : Is that the Laplacian $L$ of any radial function $f$ on a riemannienne symmetric space $X$ is a radial function? Thank you in advance.
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The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq ...
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How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...
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Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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On a property of $(Mf)^{\delta}$

For each positive $C$, define a set $$A_C=\left\{g\ge0: \frac{1}{|I|}\int_Ig\le C\inf_{x\in I}g(x) \text{ for any interval } I\right\}$$ In other words, elements in $A_C$ are non-negative functions ...
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The support of $f(x)= \cos(x)$

The support of a function is the closure of the set of points where the function has non zero values. The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in ...
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Gradient Estimate for Logarithmic Cutoff Function

In Notes 4 on regularity of harmonic maps, T. Tao asserts the following lemma, which in short, allows us to localize a map $u:\Omega\rightarrow S^{m-1}$ (here, $\Omega\subset\mathbb{R}^{2}$ is a ...
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Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
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Properties of the Fejer kernel

The Fejer kernel $k_m : \mathbb{R} \to \mathbb{C}$ is defined by $k_m (t) = \frac{1}{2\pi (m+1)} \sum^m_{n=0} \sum^n _{k=-n} e^{ikt}$ One of the properties of the Fejer kernel is For any $\delta ...
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Interpretation of $L_2 ([-\pi , \pi])$

What is the interpretation of $L_2 ([-\pi , \pi])$ in laymans terms? How is $L_2 ([-\pi , \pi])$ different to $L_1 ([-\pi , \pi])$? Does $L_2 ([-\pi , \pi])$ just mean that the function is twice ...
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Inverse short time Fourier transform

The short time Fourier transform $S: L^2(\mathbb{R})^2 \rightarrow L^2(\mathbb{R}^2)$ can be defined as $$S(g,f)(a,b):=\int_{\mathbb{R}}f(x) \overline{g(x-a)} e^{-i b x} dx.$$ Now a natural question ...
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Interpolation of a linear operator acting on a sequence of functions

Let $\mathbf{f} = \{f_{n}\}$ be a sequence of Schwarz functions and suppose $T$ is a linear operator which sends a given sequence of Schwarz functions to a given function in $L^{p}(\mathbb{R}^n)$ for ...
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Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$?

Suppose $f$ is a smooth function compactly supported in some ball of radius $R$. Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$ where $B_{1/R}$ is any ball ...
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Showing $\hat{\tilde{f}}=\tilde{\hat{f}}$ where $\hat{f}$ is the Fourier transform, and $\tilde{f}(x) = f(-x)$

I'm trying to prove that $\hat{\tilde{f}}=\tilde{\hat{f}}$ for any integrable function $f$, where $\hat{f}$ denotes the Fourier transform of $f$ and $\tilde{f}$ denotes the mapping $f(x)\to f(-x)$, ...
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Proof of an inequality with using maximal operator

I want to prove an inequality such that $$ \int_{B}|f(y)|dy\leq |B|^{1-\frac{1}{p}}\|f\|_{L^p(B)}, $$ where $B\subset\mathbb{R}^n$ is a ball, $p>1$ and ...
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Question related to decay of Fourier transform and smoothness

Suppose $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ Let $$g(x) = \frac ...
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Additive combinatorics in Hamming weights of addition of numbers modulo $2^n$ with prescribed Hamming weight

I wonder if anyone could point me to a reference about the following type of combinatorics problem: Fix $n $. For an integer $k \in [n] = \{1, \ldots, n\}$, let $A(k)$ be the set of integers in $[0, ...
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Question regarding constructing a function via its Fourier transform

Let $\varepsilon>0$. I was interested in understanding the justification of defining the following function $\phi$ via its Fourier transform, satisfying the following properties: (1) ...
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Proving Holders inequality for the sequence space $l_p (\mathbb(N)$

We first look at when $p=1$ and $q=\infty$ And we look at the non trivial case when the sequences $x=(x_k)_{k \in \mathbb{N}}$ and $y=(y_k)_{k \in \mathbb{N}}$ are both not equal to zero. We first ...
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Parition of unity argument in a Fourier analysis paper

I am currently reading a paper "The proof of the $\ell^2$ decoupling conjecture" by Bourgain and Demeter. I was wondering if someone could explain me one of the arguments in their paper. First I will ...