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Results tagged with harmonic-analysis
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user 120369
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
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On Defining the Fourier Transform and Performing Changes of Variable on Quotient Subgroups o...
Much to my dismay, in my work the more number-theoretic side of harmonic analysis (ex: the fourier transform on the adeles, on the profinite integers, etc.), I have found myself struggling with techni …
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Poisson Summation Formula for Square-Integrable Functions on Locally Compact Abelian Groups
The Poisson Summation Formula (PSF) is most often stated with the requirement that the functions in question be in $L^{1}$. However, after doing some searching, I found that there is a paper by R.P. B …
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Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct p...
Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to …
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The formula for (and computation of) the inverse p-adic mellin transform
So, after scouring the entirety of the internet, I managed to find one (and, so far, only one) source that actually explains how to invert the $p$-adic mellin transform:
$$\mathscr{M}_{p}\left\{ f\ri …
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How to projectivize an infinite-dimensional $L^{2}$ space and do fourier analysis on said pr...
I'm doing work with repelling (that is, non-contracting) linear operators on hilbert spaces, and I wondered if it might be worth my while to study my operators on a projectivization of my hilbert spac …
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A different kind of weighted Hardy space
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$, let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space of all complex-valued functions which are holomorphic on $\mathbb{D}$, and …
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A.C.M. van Rooij's *Non-archimedean functional analysis* (1978) is very out-of-print! Anyone...
(This is a literature/reference question.)
So... long story short:
(1) In my present research, I needed a theory of continuous functions from the $p$-adic integers to the $q$-adic integers. Unable t …
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Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\ …
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The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence ...
Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, …
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Function Spaces on the Open Unit Disk defined by Hardy Space norms
I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations …
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Approximate identities on the unit disk and going beyond a power series' radius of convergence
Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. Additional …
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