# 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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### Calculate the convolution of the product of two simple functions. (5.6-12)

Synopsis: I cannot duplicate the answer given in a very reputable online symbolic integral calculator as shown in this link ($x$ is $\tau$) although my answer does appear very similar. This tells me ...
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### How to get the output of a signal given the input and impulse

Here is my question So given the input and impulse response how would I go about getting the output? The output is given in solution (in teal) but I have no idea how to get it . thanks
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### Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
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### Convolution of a piecewise function with itself

Convolution of a function with itself I was going through this question. In the answer, the limits of the integral was transformed from (-infinity)-(+infinity) to 0 to x. Can anyone explain how this ...
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### Convolution of a function with itself n times convergence to bell curve [duplicate]

If we have a piecewise function defined as $f(x) = \begin{cases} 1, & \text{0$\lex\le$1} \\ 0, & \text{otherwise} \end{cases}$ Explain how the convolution of $f$ with itself for $n$ ...
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### Relation between Correlation and Convolution

We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following: Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
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### Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
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### Bizarre binomial sum

It is many times that we need to compute discrete convolutions. Driven by this need we have discovered a following formula: \sum\limits_{l=0}^k \binom{l+A_1}{A_2} \binom{-2 \beta (k-l)...
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### Sum of Independent Levy RVs is Levy RV [closed]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
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### Convolution of normal distribution not equal to product with constant?

Convolution of a normal distribution says: If, $X \sim \mathcal{N}(\mu, \sigma^2)$, then $X+X\sim\mathcal{N}(\mu+\mu, \sigma^2+\sigma^2)=\mathcal{N}(2\mu,2\sigma^2)$ However, Multiplication of a ...
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### Prove that $u\leq v$ everywhere.

Let $u$ be a subharmonic function on an open set $U$ in $\mathbb{C}$, and let $v$ be an upper semicontinuous function on $U$ such that $u\leq v$ almost everywhere. Prove that $u\leq v$ everywhere. ...
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### Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
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### Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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### Solving the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform, without missing solutions

I'm supposed to solve the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform and then explain if I got the most general solution. First of all, I don't know what "solve" means here ...
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### Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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### Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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### Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
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### Convolution of two rectangular pulses

Determine the shape of the following function$$\int^\infty_{-\infty} \Pi(4\tau) \Pi(t-\tau) d\tau$$ Attempt: This function is a convolution of two rectangular functions. I know that the result has ...
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### Approximate n^th power convolution

What is the approximation for $n^{th}$ convolution power (n-fold convolution) $g(x)= \underbrace{p * p * p * \cdots * p * p}_n$ with respsect to $p(x)$, where $p(x)$ is a probability density function?...
Consider a non-uniform ("generalized"?) convolution operator: $$A_h[f](t) = \int f(x)h(x,t)dx$$ I would like determine the eigenfunctions. In the "stationary" case where $h(x,t) = h(x-t)$ we have ...
I was struggling with this question, can use some help. given that $a\not=0$ $$f_a(x) =\frac{1}{x^2+a^2}$$ I'm trying to find k and c dependent on a and b $$(f_a ∗ f_b) (x) = kf_c(x)$$ I know ...