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Results tagged with harmonic-analysis
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user 75422
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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interpretation of a singular integral
I think the "finite part interpretation" is what makes these formulas valid, including the third one: it amounts to computing the distributional Fourier transform of the "pseudo-function" (and tempere …
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Accepted
The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator
Here is an example where $T(f,g)\notin l^2$ while $m\in L^2$ and $f=g\in l^2$:
Let $a(n)=|n|^{-1+s}$ with $\frac38<s<\frac12$ (so that $a\in l^2$), $m(\xi,\eta)=\hat{a}(\xi)\hat{a}(\eta)$. Take also …
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The convolution between weighted $L^1$ space and normal $L^1$ space
Let $N=1$ and $\omega(x)=1+|x|^{-m}$ with $0<m<1$ (it satisfies your conditions, with $C=2^m/(1-m)$, if I'm not mistaken -- I've checked it for intervals in $\mathbb R^+$ only...).
Let $u$ vanish out …
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Hardy space, Lebesgue space for $p<1$,
The Hardy space $H^p$ is defined as the space of those $\rho\in\mathcal{S}'$ such that the maximal function $M_\rho(x):=\sup_{t>0}|\rho_t(x)|$ satisfies $\int M_\rho(x)^p\ dx<\infty$. This is stronger …
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3
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Sampling Theorem for non-bandlimited Functions
No. Example $f(x)=e^{-cx^2}\sin(\pi x)$, chosing $c$ in the proper range...
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Accepted
Sampling Theorem for non-bandlimited Functions
No as well to the edited question (if I understood it...): $f(x)=e^{-cx^2}\sin(\pi x/\alpha)$
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4
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Differentiability of the logarithmic potential
I think there are necessary and sufficient conditions in the literature, but here is a simple sufficient condition : $d\mu=h\ dt$ with $h\in L^2([a,b],dt)$, because then the (distributional) derivativ …
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Accepted
Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables ...
Let $1=a_0=a_1>\ldots >a_n>\ldots >0$, and let $e_n$ be the "triangle" function that vanishes outside $(a_{n+1},a_{n-1})$, equals $1$ at $a_n$, and interpolates linearly from $0$ to $1$ on $[a_{n+1},a …
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Completion of $\mathcal{S}(\mathbb{R})$ for a given norm
In the general case of a normed linear space $X$ and a larger quasi-complete Hausdorff topological vector space $E$ (as $\mathcal S'$ is) with $X\subseteq E$ : it can be completed within $E$ iff its u …
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