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Tagged with convolution harmonic-analysis
66
questions
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Convolution between $L^1$ function and a singular integral kernel
I meet a problem when reading Modern Fourier Analysis(3rd. Edition) written by L.Grafakos. On pg.82 he writes:
Fix $L\in\mathbb{Z}^+$. Suppose that $\{K_j(x)\}_{j=1}^L$ is a family of functions ...
3
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Properties of fourier series on $SO(3)$
With standard fourier series we can use some identities like convolution theorem and Parseval's theorem:
(convolution theorem) Fourier series of the convolution of $f$ and $g$ is the point-wise ...
- 555
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1
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33
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Complex exponential Fourier coefficients of a convolution involving the exponetial function
In the book "Elementary Classical Analysis", by Marsden, the following is proposed as a worked example:
Let $f:[0,2\pi]\to\mathbb{R},g:[0,2\pi]\to\mathbb{R}$ and extend by periodicity. ...
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The fact that the space of matrix coefficients is a 2-side ideal in $C(G)$ implies Schur orthogonality.
Suppose that $G$ is a compact group and $C(G)$ is the space of continuous functions on $G$. For $f_1$, $f_2\in C(G)$, define the convolution by
$$(f_1*f_2)(g)=\int_Gf_1(gh^{-1})f_2(h)\mathrm{d}h=\...
- 430
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$L^1(G)$ is a Banach *-Algebra [duplicate]
Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
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If $f \in L^p(\Bbb R)$ for $1 < p\le 2$, then $f\ast \frac{\sin \pi t}{\pi t} \in L^p(\Bbb R)$.
I'd like help with the following problem:
Let $f\in L^p(\Bbb R)$ for $1 < p\le 2$. Show that $f(t)\ast \frac{\sin \pi t}{\pi t}$ is defined, and $f(t)\ast \frac{\sin \pi t}{\pi t} \in L^p(\Bbb R)$....
- 14.2k
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45
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Convolution of class functions in terms of integrals over maximal torus
Let $G$ be a connected compact Lie group. If $f : G \to \mathbb C$ is a class function ($f(ghg^{-1})=f(h)$ for any $g,h \in G$), the integral of $f$ can be computed in terms of an integral over a ...
- 3,100
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32
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Autocorrelation of a time-limited, wide-sense stationary stochastic process
Problem
Let $\{X_t\}$ be a real-valued, wide-sense stationary, continuous stochastic process and consider the autocorrelation of samples of $\{X_t\}$ taken in time windows of width $T$:
$$R_{X; T}(\...
2
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1
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How to show a convolution $f \ast g$ vanishes at infinity, where $g(x)=\frac{1}{|x|}$?
Let $f$ be a smooth function defined on $\mathbb{R}^3$ with $f(x) = O(1/|x|^{2+\epsilon})$, and let $g(x)=\frac{1}{|x|}$. I want to show that $f\ast g (x) \to 0$ as $|x| \to \infty$.
(Maybe this is ...
- 1,075
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Bounding $\widehat{G_{m+1}}\ast\widehat{H_m}(0)$ when $\frac{1}{2}\leq\widehat{H_m}(0)\leq\frac{3}{2}$ and $H_m,G_{m+1}$ are smooth over $\mathbb{T}$
To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that
$$\left|\widehat{H_m}(k)\right|\leq C\log(\left|...
- 1,030
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86
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Extending the definition of convolution product on $\mathcal{F}(C(G))$, $G$ abelian compact group
Let $G$ be a compact abelian group.
We know that we can define the convolution product on its convolution algebra $L^1(G)$ in a natural way:
\begin{equation}
(f_1*f_2)(g):=\int\limits_Gf_1(h)f_2(h^{-1}...
- 123
1
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1
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214
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Fourier transform of a time-variant convolution
For a time-invariant convolution, given the convolution theorem, we know that $\mathcal{F}\{(h*x)(t)\}=\hat{h}(\omega).\hat{x}(\omega)$. My question is what if the convolution is time-variant.
Let $(h*...
- 338
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Convolution on a locally compact group is associative
Consider the following fragment from Folland's book "A course in abstract harmonic analysis" (question is below the image).
Can someone explain why the boxed equality is true? Don't we need ...
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$K(x,t)=2\pi\sum_{n=1}^\infty n\sin(n\pi x)e^{-n^2\pi^2t}$ satisfies some "pseudo-good-kernel" properties
Let $f\in C([0,\infty))$. Define
$$K(x,t)=2\pi\sum_{n=1}^\infty n\sin(n\pi x)e^{-n^2\pi^2t},\qquad x\in[0,1],\ t>0.$$
Show that
$$\lim_{x\to0^+}\int_0^tK(x,t-\tau)f(\tau)\,d\tau=f(t), \qquad t>...
- 13.7k
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Can convolution between functions on a finite group be represented as a Toeplitz matrix?
Let $f$ and $g$ be complex-valued functions on a finite group $G$. Left convolution by $f$ can be realized as an operator $L_f(g) = f * g$, so it follows that $L_f$ can be represented as a matrix ($f$ ...
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