# 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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### Non-linear Systems, Impulse Responses, and Convolution

In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we ...
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### Meaning of limits, $\int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds$?

What does it mean to have $\max(0,t-3)$ and $\max(0,t-5)$ in the limits? Is it a abbreviation? $$\int_{t-5}^{t-3} e^{-3s}u(s) \, ds = \int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds$$ Source of ...
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### How to evaluate chirp transform in O(nlgn) time? [duplicate]

The question says to evaluate chirp transform in O(nlgn) time using the equation in the hint. But I'm unable to get any idea on how to prove the chirp transform from it. Any help is appreciated.
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### estimate a probability

Let $X_1....X_{48}$ be independent random variables, each follows a uniform probability distribution over [0,1]. What is the best way to estimate P($\Sigma_{i=1}^{48} X_i > 20)$?
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### Why convolution regularize functions?

There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
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### Properties of the mollification operator (Sohr - Navier-Stokes equations)

My question refers to Sohr - The Navier Stokes Equations p.66/67. I am trying to understand the statement (1.7.16). Since p.66 is not available via Google-books I will formulate the problem in the ...
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### Convolution is uniformly continuous and bounded

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function. The definition of ...
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### How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
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### Showing that a “convolution” operator is associative

I am dealing with the following operator $*$ : $(A*B)(t) = \inf\limits_{\tau\in\mathbb{R}} (A(\tau) + B(t-\tau))$. I would like to show that it is associative, i.e : $((A*B)*C)(t) = (A*(B*C))(t)$ . I'...
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