# Tagged Questions

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### Does the Convolution of Two Series Require Absolute Convergence?

Let $A=(a_n)_{n=0}^\infty$ and $B=(b_n)_{n=0}^\infty$ be sequences of real numbers. Let $C = (c_n)_{n=0}^\infty$ be the sequence such that $$c_n = {a_0}{b_n} + {a_1}{b_{n-1}} +\cdots+ {a_n}{b_0}.$$ ...
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### Obtaining Impulse Response from Graph

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
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### How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
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### a matrix inverse laplace transform problem

Let $\mathcal {L}^{-1}[\cdot]$ be an inverse Laplace transform. Let $A$ be a square matrix, and $I$ an identity matrix. Based on the fact that $\mathcal {L} ^{-1} [{(sI-A)}^{-1}] = e ^{tA}$, how can ...
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### convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)}$$ where $u(t)$ is ...
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### Integrability of the function $f_1(y)f_2(x-y)$ for almost all $x$. (convolution)

What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or:  F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ ...
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