# Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

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### convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
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### Analycity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$

Posted also on MO with a bounty Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\backslash\{0\}$ but non-...
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### Algorithm to compute a convolution recursively

Let $$f(t) = \int_0^t k(t-s)g(s) \, ds.$$ Assume that $g$ is only given in a grid $t_j = j\delta_t$, and that we wish to compute similarly $f$ on the same grid. What's an efficient algorithm to ...
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### Clarification Needed on 1D Convolution and Kernel Purpose

I am confused about the definition of 1D convolution. Given $a = [-\frac{1}{2}, \frac{1}{2}]$ and $b = [1, 1, 1, -1, -1, -1]$, what will be the result of the convolution $( a * b )$? From my ...
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### Convolution of slightly multivariate Gaussians slightly modified

Starting with $p(a) = \int p(a|b) p(b) db$ replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$ where $\tilde{D}$ is an additive diagonal covariance. Assuming ...
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### Convolution preserve the boundary condition

Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
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### What's the necessary and sufficient condition for a real sequence to be written as the self-convolution of another real sequence?

Definition For a sequence $a_0,a_1,\cdots,a_n$, the corresponding self-convolution is another sequence $\displaystyle b_m=\sum\limits_{i+j=m}a_ia_j$ where $0\leq m\leq 2n$. Calculating the self-...
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### Associativity of Convolutions

In Folland's real analysis textbook, there are the following propositions: Assuming that all integrals in question exist, we have $$(f*g)*h=f*(g*h)$$ The proof is based on the Fubini's theorem.But ...
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### Help with understanding combination of probability distributions

I have two probability mass functions (PMFs) across the surface of a sphere. They are localised Gaussians (a few degrees in expanse), whose centres have arbitrary positions though they are quite close ...
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### Inverse Fourier Transform - convolution of exponential and rectangular window

I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
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### What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?

I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.'' Here's an attempt at a ...