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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0
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2answers
40 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
1
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1answer
52 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
0
votes
1answer
33 views

Fast convolution with striding

I want to convolve two discrete functions $f$ and $g$ using convolution stride size $a$ to get the result as $s_{a, i}$: $$s_{i,a} = \sum_i g_k f_{ai-k}$$ I know that simple convolution with $a=1$ ...
0
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0answers
59 views

How is deconvolution done?

This is a purely theoretical and hopefully simple question I would like to get an answer to. So I know that by polarity reversing the impulse response a signal can be deconvolved. Here is a simple ...
0
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0answers
45 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
0
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1answer
41 views

Bound on uniform norm of convolution of $L^p$ functions

This is Proposition 8.8 in Folland's Real Analysis: If $p$ and $q$ are conjugate exponents, $f \in L^P$, and $g \in L^q$, then $f*g(x)$ exists for every $x$, $f*g$ is bounded and uniformly ...
0
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0answers
32 views

Intuition for convolution in min-plus algebra

These days I'm looking a bit into min-plus algebra, in function of network calculus. In min-plus algebra, the sum is replaced by the minimum operator and the product is replaced by the sum. For a ...
1
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1answer
29 views

Question about mollifiers.

So here is my problem, Let $\rho \in C^\infty (\mathbb{R}^n,R)$ with $\rho\geq 0$, $\rho(x)= 0 \; \forall \|x\|\geq 1$ and $\int_{\mathbb{R}^n}\rho(x)dx=1$. Further, consider the linear map ...
0
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1answer
27 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
1
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1answer
44 views

proof that$ L^1 (G)$ is a subspace of $M(G)$

Let G be a locally compact group, and let $M(G)$ be the space of complex Radon measures on G. I identify the function f with the measure $f(x) \rm dx$ . but How do I prove this inclusion?؟ . .
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2answers
75 views

Solving a differential equation using the laplace transform involving convolution

The problem is the following The thing that puzzles me here is the integral on the right hand side, so: How to take the laplace transform on the right hand side? Any help to get me going would be ...
1
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0answers
35 views

Question about the principal value of some integral

So here is my problem, is it possible that $$\int_{[0,1]}f(y)\cot(\pi(x-y))dy= p.v \int_{[0,1]}f(y)\cot(\pi(x-y))dy$$ I see that the left integral is singular for $x=0$ but since I never worked with ...
5
votes
3answers
246 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
1
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1answer
195 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
1
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1answer
49 views

convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
2
votes
1answer
34 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
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0answers
26 views

Corrections and Normalization for Power Spectrum Calculation

So I'm hoping I can get some help. I have a 2d image and need to get the 1d power spectrum. I know the basic steps: take fft, take fft^2 to get power, then take average power in radial bins to get 1d ...
0
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1answer
48 views

Differentiation of convolution integral

Following are the piece wise polynomial function for input plasma $$ C_a(t) = \begin{cases}0& t\leq t_d\\ \displaystyle\sum_{n=1}^3\frac{a_n}{t_\max-t_d}(t_-t_d)& t_d\leq t\leq t_\max\\ ...
0
votes
1answer
30 views

Most computationally efficient way to find convolution of a matrix kernel with impulse response?

Let say if we wish to filter an input sequence x[n1, n2, n3] of NxNxN points using an Linear Shift Invariance system with impulse response h[n1, n2, n3], where the filter is a separable sequence, ...
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0answers
38 views

Convolution of exponential family distributions

Is there a general form for the convolution of two exponential family distributions?
0
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1answer
38 views

Proper convolution notation

What would be the correct way to write down the convolution in "star" notation for these two functions; $h(t)$ and $\delta(t-x)$. $\delta$ is the Dirac delta function. The integral notation should ...
2
votes
1answer
62 views

An issue of applying Fubini's theorem in Fourier transform on Schwartz space.

Let $\hat{f}$, $\hat{g}$ be the the Fourier transform of $f$ and $g$ respectively where $f$ and $g$ are the members of the Schwartz space $\scr{S}{(\mathbb{R}^{N})}$. Then in the process of ...
0
votes
0answers
74 views

Combining multiple 2D and 1D convolutions into a single 3D convolution

I have a 3D image (multiple 2D slices) and two kernel images (one 2D kernel and one 1D kernel). And I do the following operations: I convolve each 2D slice (x,y direction) of the 3D image with my 2D ...
1
vote
1answer
34 views

Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
1
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0answers
37 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
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0answers
33 views

Min $+$ convolution is associative

Although the following question was encountered in a Communication Networking textbook, the problem is still one of algebraic and analytic manipulation. Define the (min,+) convolution of two real ...
1
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1answer
44 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
0
votes
1answer
34 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
1
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0answers
35 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetical functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
0
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1answer
80 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
0
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0answers
15 views

Average Orders and Convolutions

If I know the average order of an arithmetic function $f=I*g$, where $I$ is the identity function defined by $I(n)=n$, is there a way to find the average order of $g$?
0
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1answer
28 views

Details about Generalized Convolution (Number Theory - Apostol)

In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution. Let F denote a real or complex-valued function defined on the positive real axis such that ...
0
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1answer
58 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
0
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0answers
27 views

The derivative of convolution

For a compact supported continuous function $\rho$ in $R^3$, consider the convolution $f(y)=\int_{R^3}\frac{1}{|x-y|}\rho(x)d x$, did the following communicate of derivative and integral holds? $$ ...
0
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0answers
14 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
0
votes
1answer
21 views

How to convolve two stair-case functions?

For the life of me, I haven't been able to grasp convolution for functions with multiple pieces. For example, $$ h(\lambda) = \left\{ \begin{array}{l l} 2 & \quad \ 0\leq \lambda < 1\\ ...
3
votes
1answer
45 views

Separate the variables of the function $\frac{x^2}{\sqrt{x^2+y^2}}$

Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable? This is becasue I want to apply a convolution whose ...
0
votes
2answers
44 views

convolution product of characteristic functions

Consider the characteristic function $f(x)=1_{[0,r]}(x)$. How to compute $(f*f)(x)$ where $r \in [0,1[$ and by definition $(f*g)(x)=\int_{\mathbb R} f(y)g(x-y) \ dy$ ? thanks.
0
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0answers
24 views

Sum of two independent random variable, Convolution.

I need the graphic of this two function to evaluate this correlation?
2
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0answers
53 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
0
votes
1answer
60 views

Difference between two independent geometric random variables

Let $\xi_1$ and $\xi_2$ be independent random variables: $\xi_1 \simeq Geom(1/2), \xi_2 \simeq Geom(1/6)$. How do you find the probability mass function of $\eta=\xi_1-\xi_2$ using convolution?
1
vote
1answer
100 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
1
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0answers
62 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
5
votes
1answer
66 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
0
votes
1answer
34 views

Convolution of fraction function

I know that convolution is defined: $$f*g=\int f(x-y)\cdot g(y) \, dy $$ How to develop below functions to convolution equation $$\int {f(x-y) \over g(y)} \, dy =\text{ ???}$$ and $$\int {f(x-y) ...
0
votes
0answers
30 views

Find optimize of gaussian regression

I have one gaussian regression and I want to find parameters to optimize the LOSS function: $E(\sigma,\mu,b_0)=\int K(x-y).||f-f_i||^2$ where $f=\epsilon+b_0 ;$ $\epsilon$ ~$N(\sigma,\mu)$ is ...
0
votes
0answers
124 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
0
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1answer
38 views

The signal $\cos(2 \pi t )$ is an eigenfunction of every LTI system?

for $\sin(2 \pi t)$: Apparently that it's not an eigenfunction real-valued impulse response $h(t)$ but it's a eigenfunction for real-valued and even impulse response $h(t)$ What gives?
12
votes
5answers
810 views

Definition of convolution?

Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it ...
1
vote
2answers
75 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...