Questions tagged [fourier-transform]
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Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
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Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
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Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?
I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
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Plancherel's Theorem in wider space
By Planchere's Theorem, we have for $f$, $g$ in $L^2(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n}fg=\int_{\mathbb{R}^n}\hat{f}\hat{g}$$
Also, by distribution theorey, we can define the Fourier ...
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Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
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Function samplable from the past and Hardy spaces
What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...
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Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
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Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?
Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere.
Why
$\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$.
$\hat{f}$ is the Fourier transform fora function f.
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How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
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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
$$(f*g)(R) = \...
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Is there an generalisation of convolution theorem to integral transforms
Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
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Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
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Construction of an analytic function whose Fourier transformation has compact support [closed]
Is there a non-constant real analytic function $f$ on $\mathbb{R^2}$ satisfying the following properties?
$f$ vanishes on $x$-axis and $y$-axis;
the Fourier transformation $\hat{f}$ of $f$ has a ...
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Spectral theory: a key to unlocking efficient insights in network datasets
In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
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Finding specific coefficients of product of high-dimensional Fourier series faster than FFT
I need a fast algorithm to perform a specific Fourier-type computation in my physics research. Suppose I have the following two Fourier series in three dimensions
$$
a(t_1,t_2,t_3)=\sum_{j=-n}^{n}\...
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