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3
votes
0answers
80 views

Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative. It is easy to show that $f *g$ is defined pointwise when ...
2
votes
0answers
53 views

Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as ...
0
votes
0answers
40 views

How to solve a multidimensional integral equation of convolution type 2 with real coefficients?

Can someone suggest suitable reference for solving a 4*4 integral equation of convolution, and whether the following equation has a closed form solution? I really appreciate your help. where y, K, ...
5
votes
3answers
373 views

Deconvolution of sum of two random variables

Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$ I have observations of $Z_i$'s and thus can ...
0
votes
1answer
129 views

Bounded convolutions with binomial coefficients

I need to figure out a nice family of decaying functions such that $\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$ How can I figure out what good candidates could be? Any ...
0
votes
0answers
167 views

Value of convolution integral of Gaussian function and curvature of a circle segment

Is there an analytical expression for the following integral, assuming $\alpha\in(0,\pi)$ and $\sigma>0$? $$-\frac{1}{\sqrt{2\pi} \sigma} \int_{-\sin(\alpha/2)}^{\sin(\alpha/2)} \exp(-\tfrac12 ...
3
votes
2answers
197 views

Banach algebra for measures induced by Haar measures

It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral ...