All Questions
Tagged with convolution fourier-transform
19
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Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
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Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as
$$(f*g)(R) = \...
- 2,226
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Is there an generalisation of convolution theorem to integral transforms
Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
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Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
- 392
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Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$
$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
- 1,091
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Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
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What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
I searched on Google but couldn't find a satisfiable answer.
Thanks in advance :)
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2-Wasserstein metric on convolution of probability distributions
I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
- 161
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Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
- 273
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Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
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Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
- 791
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452
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Convolution, Fourier transforms, and area preservation [closed]
Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
- 111
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Characterization of convolution operators via the Fourier transform
Let $\mathcal{L}$ be a linear and continuous operator from the space of tempered distributions $\mathcal{S}'(\mathbb{R})$ to itself. The Fourier transform of a tempered distribution $f$ is denoted by $...
- 2,226
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Particular Ehrenpreis factorization for covariance function
Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite).
Is there a compactly supported function $g:\mathbb{R}^d\...
- 311
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Wavelet momentum identity
I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31):
\...
