# Tagged Questions

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

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### Reverse of convolution theorem

If I have a convolution $$z(t) = x(t) * y(t)$$ where I know $x(t)$ and $z(t)$, is there a way to determine $y(t)$? Is there a "reverse" convolution theorem for this? I know there are numerical ...
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### How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - u, ...
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### Help in finding a function in $L^p$ such that $f*g=||g||_1 f$ with $g\in L^1(\mathbb{R})$ non negative and fixed

I consider a non negative function $g\in L^1(\mathbb{R})$. I want to find a function $f\in L^p(\mathbb{R})$ such that $$f*g=||g||_1 \:f ,$$ where $*$ is the convolution. I would be very thankful if ...
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### Why is a convolution of an $L^1$ and an $L^p$ functions well-defined?

I was reading the wikipedia article about convolution and I found this one: If $f \in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$, then $||f\ast g||_p \leq ||f||_1||g||_p$. But to $f\ast g$ ...
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### what is the convolution of $sin(bx)$ and $e^{-a|x|}$?

my textbook says that $F[f*g] = F[f] \cdot F[g]$ but what is $F[sinbx]$? It doesn't exist right? So how should I solve it? $F$ is fourier transform
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### Show that these two differential equations have the same solution

Question: Show that the problems $ax'' + bx' + cx = f(t); x(0) = 0, x'(0) = v_0$ and $ax'' + bx' + cx = f(t) + av_0 \delta(t); x(0) = x'(0) = 0$ have the same solution for $t \gt 0$. Thus the effect ...
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### Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have u*g=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau=2\int_{t-2}^{t}(e^{-\...
I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
### How to show $|f*g|_{1} \le |f|_{1}|g|_{1}$ [closed]
Well, as it is stated in the titel. I have to show $\|f*g\|_1 \le \|f\|_1\|g\|_1$. thank you already now for your help.