# 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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### 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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### 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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### 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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### Convolution involving the inverse Fourier transform

Suppose $$F(k) = \frac{1}{2\pi}\int f(x) e^{ikx} dx$$ and $$G(k) = \frac{1}{2\pi}\int g(x)e^{ikx} dx$$ Where $F(K),G(K)$ are Fourier transforms. Then how can I write the convolution of $F$ and $G$ ...
According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ... 1answer 42 views ### Convolution of Gaussian and error function I am trying to evaluate the following integral:$$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$where$$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$I have ... 2answers 23 views ### Alternative integration limits in a Laplace transform The unilateral Laplace transform of f(t) is \int_0^\infty e^{st} f(t) \mathrm{d}t. If we define the transform as \int_{a}^\infty e^{st} f(t) \mathrm{d}t, would it conserve all the nice ... 2answers 28 views ### DFT and windows I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following ... 1answer 35 views ### Continuity of characteristic function Problem: Let G be an open subset of \mathbb{R}. Show that \chi_G is continuous on G\cup(\mathbb{R}\backslash\overline{G}). Consequently, \chi_G is continuous a.e. on \mathbb{R}. My ... 0answers 19 views ### Can 2d convolution been represented as matrix multiplication? Discr. convolution on a discrete periodic signal can be represented as multiplication of input with matrix M. Where M is presented a special case of Toeplitz matrices - circulant matrices. The ... 0answers 19 views ### Verify the Green's function for Helmholtz equations It is well known that$$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$is the Green's function for Helmholtz equation$$ (\Delta+k^2)f=0 $$in \mathbb{R}^3. My question is, given v\in C^0_b(\... 4answers 3k views ### Why convolution regularize functions? There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ... 0answers 26 views ### Laplace transform of a convolution-like function Is there a way to calculate the Laplace transform of the following function?$$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0.  Thanks in advance.
I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$ and p and p' are ...