# 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.

39 views

### 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 ...
15 views

38 views

43 views

### 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 ...
32 views

21 views

### 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?...
55 views

### Eigenfunctions of non-uniform convolution

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 ...
31 views

8 views

### any way to simplify discrete convolution $g[n]x[n] *g[n]x[n]$ to the form $Gp(x[n])$

Is there any way to simplify discrete convolution $g[n]x[n] * g[n]x[n]$ to the form $Gp(x[n])$ where $G$ is a matrix and $p$ is a feature map which can be computed quite fast. Note $p(x[n])$ must be ...
42 views

### Convolution and Fourier transform problem

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 ...
60 views

### Convolution proof

If I have two functions in a convolution like $$X*Y=1$$ $$X*Z=1$$ then it means (trivially) $Y=Z$. Is this correct or are there subtleties in the convolution theorem where $Y=Z$ isn't always true?
20 views

### derivative of convolution integral

I'm confused for a derivation related to the derivative of convolution. Given that $$C_{im}(x,t)=\omega e^{-\omega t}*C_m(x,t)+C_{im}(x,0)e^{-\omega t}$$ By taking derivative of the above equation ...
41 views

### Is the convolution integration zero?

$$F(\omega)=\frac{\sin \omega}{\omega}$$ $$G(\omega)=\frac{\sin \omega}{\omega}e^{-j2\omega}$$ \begin{align} F(\omega )*G(\omega)&=\int^{+ \infty}_{-\infty} \frac{\sin \tau}{\tau}\frac{\sin (\...
23 views

### Convolution of the cumulative normal distribution and the uniform distribution [closed]

What is the resulting function of convolving the cumulative normal distribution and the continuous uniform distribution?
86 views

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 ...
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 ...
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 ...
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 ...
121 views

### Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e &...