# 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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### Proof of Rudin's Theorem 8.14, RCA

In Rudin's proof of Theorem 8.14, which states that convolutions of Lebesgue integrable functions over the real line are Lebesgue integrable, he first proves the result for Borel measurable functions, ...
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### How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
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### Help with integral $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$

How should I proceed to work out following convolution integral: $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$ for real $\alpha$ > 0. It is the convolution of a powerlaw decaying impulse response ...
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### Calculating the convolution of $\sin(t)/t$

So how would I go about calculating the convolution of: $$1* \sin(t)/t$$ This will be a simple looking integral, however $\sin$ or $\cos$ are not defined for infinity and negative infinity.
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### Integral Equation from zero to infinity

Is there anyone could help to solve the following problem: Suppose $\,h\left(x\right)\,$ is a known function and $\,y\left(x\right)\,$ is unknown, you may assume these two are nice functions. I am ...
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### finding image and kernel from convolution result

given a one dimension image/(signal) I and three kernels K,L,J. given the results of the convolution of I with each kernel, for example \begin{matrix}I*K =[ 1/3&2/3&1&1&1&1&1&...
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### Can any integral of this form be written as a sum of convolutions?

Does the second equality always hold? $$I(x) \equiv \int dy F(y,x-y) = \sum_{i=1}^{N}\int dy f_i(y)g_i(x-y)$$ Motivation: The first integral is not obviously a convolution that I could calculate ...
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### Find the integral $\int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}}$

My problem is to calculate auto-convolution of $f(x) = \frac{1}{1+x^2}$. I know that $$(f \star f)(x) = \int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}} = \frac{2 \pi}{x^2+4},$$ but I ...
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### Solve a function consist of two functions in a sigma summation

I'm still new in here. I have been struggling to solve an equation. The equation is like: f(T)= $\sum_{i=0}^t [g(T-i)*d(i)]$ It is an equation describing the relationship between f(T), g(T) and d(t)....
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### Convolution form of the renewal density

In Asmussen's Applied Probability and Queues, Proposition 2.7 makes a claim about the form of the renewal density in terms of the density of the interarrival distribution $F$, namely: The renewal ...
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### Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
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### Interchanging integral and derivative operations in the context of Duhamel's formula

I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...
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### Help understanding the math of a basic convolution

We are learning about convolutions and I am totally totally lost. I think I understand the idea behind them, basically you're summing the output of the two signals at every point in time. Its the math ...
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### Convolution of convex function and Gaussian is convex

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function, i. e., for all $x_1, x_2 \in \mathbb{R}$ and $t \in [0, 1]$, $$\qquad f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).$$ I want to prove that the ...
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### How to calculate the volume of intersection of two cylinders which have radius $r$ and $R$ $(r\le R)?$

The cylinders have their axis meet at right angle. I try to use integral to derive the equation but I have a hard time visualizing the edge of intersection. Could someone help?