Tagged Questions

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-...

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Decay property of oscillatory integrals in $\mathbb{R}^n$

We know that an oscillatory integral in $n$ dimensions is an integral of the form \begin{equation*} I(\lambda)=\int_{\mathbb{R}^n}e^{i\lambda\phi(x)}f(x)dx \end{equation*} where $\phi\in C^{\infty}$ ...
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Concluding that if $f\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$ and $\mathcal{M}(f)\in L^{1}(\mathbb{R}^{n})$ then $f=0$ a.e.

Let $\mu$ be a Lebesgue measure, $\mathcal{M}$ be the Hardy-Littlewood Centered Maximal Function, and the rest as in the title. Then to this end I define $f_{R}(x):=f(x)\chi_{|x|\le R}$ where $\chi$ ...
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Folland real analysis 9.11

This comes from question 9.11 of Folland's Real analysis textbook. Unfortunately, I have no idea to how to start with this question. So can some one help me with part $a$? For part $a$, I can not ...
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Is set of adjoint derivations on a Banach algebra to Banach bi-module closed subspace?

Let $A$ be a Banach algebra and $B$ be a Banach space. Then $B$ is called left( resp. right) $A$- module if there is continuous representation(resp. anti-representation) $T : A \to BL(B)$ (Bounded ...
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Fourier transform of a Gaussian

I am trying to solve the following exercize: Show that Fourier transform of a Gaussian (a function of the form $Ae^{-\frac{x^2}{\sigma^2}}$) is also a Gaussian. So I did the required calculation (I ...
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Question on the existence of the limit in S. Petermichl's paper

I have read S.Petermichl's paper Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol many days and got trouble in the proof of Lemma 2.1 in this paper. Let us see Lemma 2....
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Inverse transform of digamma function

Let $g$ be in $C_c^\infty(\bf R)$, and $h$ its Fourier transform. There is the following equation: \begin{align}\frac{1}{2\pi}\int^\infty_{-\infty}h(r)\frac{\Gamma'}{\Gamma}(1+ir)dr=&-\gamma g(0)+...
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Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
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Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty$) of: $$f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases}$$ My ...
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Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
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Question while reading “Harmonic analysis as the exploitation of symmetry — a historical survey”: what is the meaning of “identity map”?

I am reading through the following article: Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. Bull. Amer. Math. Soc. 3 (1980), no. 1, part 1, 543–698 This is a ...
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Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
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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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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: $f(t)=v_1*cos(t) + v_2*sin(2t) + v_3*cos(2t)+v_4*sin(4t)+... 2answers 26 views Harmonic Motion - Fourier Approximation What does this mean below? There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ... 2answers 75 views Estimate of a Fourier Multiplier Operator Let$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, $$\| T_t f\|_{\... 0answers 20 views Commutator Characterization of BMO(\mathbb{R}) Let a:\mathbb{R}\rightarrow\mathbb{C} be a locally integrable function, and let H denote the Hilbert transform. Suppose that the commutator operator [a,H] defined by [a,H]f:=aH(f)-H(af) is ... 1answer 16 views Invariance of noneuclidean laplacian In a book I'm reading it says: Putting f(x,y)=F(u,v) with \gamma(x+yi)=u+iv and using Cauchy-Riemann equation for \gamma(z), we have$$\Delta f(x,y) \overset{(*)}=-y^2(u_x^2+v_x^2)(F_{uu}+F_{vv}... 1answer 38 views 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: $$\begin{cases} \partial_t u + \... 1answer 35 views How to prove that f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda where f_{\lambda} is the eigenfunctions of \Delta On Euclidean space \mathbb R^n, how to prove that$$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$where \Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) , whith \Delta is the ... 0answers 28 views Laplacian on the sphere (proof of a proposition)$$ 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 \mathbb{R}\;defined\;on\;an\;open\;subset\;of\;{{\... 0answers 53 views 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$\theta\...
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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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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 ...