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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2
votes
3answers
2k views

Convolution of triangle function with itself

I'm trying to find the convolution $(A \ast A)(x)$, where $$A(x) = \begin{cases} 1 + x, & -1\leq x \leq 0,\\ 1 - x, & 0\leq x \leq 1,\\ 0, & \text{otherwise}, \end{cases}$$ so far ...
4
votes
1answer
349 views

“anti-gaussian” 2D convolution kernel

What is the 2D kernel, k, that when convoluted with a 2D signal, f, that is convoluted again with a gaussian 2D kernel, g, produces a result that is closest to the original signal, f'. Something like ...
5
votes
3answers
1k views

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard ...
2
votes
2answers
193 views

Commutativity of convolutions implies $\int\limits_{-\infty}^\infty f(t)dt = \int\limits_{-\infty}^\infty f(t-a)dt$?

As I understand it, to convolve $f$ and $g$ means to find $\displaystyle \int_{\mathbb R} f(a)g(t-a)da$, which is also apparently commutative, and therefore $\displaystyle \int_{\mathbb R}f(a)g(t-a)da ...
2
votes
1answer
171 views

Preservation of Lipschitz Constant by Convolutions

The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function ...
5
votes
2answers
502 views

Limits and convolution

Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in ...
0
votes
2answers
433 views

A convolution identity (revision of the question “Is this convolution identity known?”)

I have deleted the content of the original post. The following exercise is inspired by answers to the questions this and this, but no knowledge of probability theory is required. The solution is ...
2
votes
1answer
316 views

Convolution on group with measure

I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain. For convolution on Lebesgue-integrable real-valued ...
5
votes
3answers
718 views

Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
1
vote
1answer
128 views

behaviour of discontinuities after convolution

$f$ is smooth (derivatives of all orders exist) except at $x_1$,$x_2$,$x_3$ where derivatives exist only upto orders of $k_1$,$k_2$,$k_3$ respectively, where $k_i$ is a natural number. Same is the ...
13
votes
4answers
670 views

How this operation is called?

This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients: $$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k) $$ Particularly, $$a^n\star ...
5
votes
1answer
362 views

Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?

Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?
13
votes
4answers
3k views

Meaning of convolution?

I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give ...
3
votes
3answers
841 views

Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$

From any textbook on fourier analysis: "It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = ...