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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1answer
175 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
9
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1answer
455 views

The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = ...
13
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4answers
621 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 ...
3
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3answers
2k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
5
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3answers
667 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 ...
4
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3answers
317 views

Alternating sign Vandermonde convolution

The well-known Vandermonde convolution gives us the closed form $\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$. For the case $r=s$, it is also known that $\sum_{k=0}^n (-1)^k {r \choose ...
2
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1answer
88 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
2
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2answers
453 views

Convolution of integer sequences

I read this somewhere: $U$ and $V$ are defined on the set $\mathbb{Z}$ of integers. The convolution of $u$ and $v$, noted as $(u*v) (k)\,\theta$ or $ (u \otimes v) (k)\theta$, is a new sequence whose ...
2
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1answer
592 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
1
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1answer
66 views

Sums of normal CDF's

This is my following problem: $$ CDF_A:F_A(x)=\Phi(x)^2 $$ $$ CDF_B:F_B(x)=1-\Phi(-x)^3 $$ $$ Defining: X=A+(-B) $$ I have those two CDF's and I want to calculate the probability that X is smaller ...
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2answers
236 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
11
votes
3answers
611 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 ...
1
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1answer
319 views

What's the difference between convolution and crosscorrelation?

What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing ...
1
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1answer
276 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
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
1answer
72 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
0
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1answer
209 views

Using Convolution Theorem to find the Laplace transform

In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet ...
5
votes
2answers
484 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 ...
3
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1answer
64 views

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ ...
3
votes
2answers
141 views

How to prove that operator is not compact in $L_2 (\mathbb{R})$

I have the operator $(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be ...
2
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1answer
58 views

Does negative distributive property of convolution over cross correlation holds?

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property ...
2
votes
0answers
184 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
2
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1answer
110 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
2
votes
1answer
119 views

The digit base and the NTT convolution

Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
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0answers
28 views

$f_{X^2}(x)$ VS $f_X(x^2)$ [duplicate]

Sorry, this time the format should be accurate. In probability, when we try to describe a pdf, we write it as $f_X(x)=1/x$, which means the random variable is X and the x is the specific variable in ...
1
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1answer
147 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
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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 ...
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2answers
74 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...