4
votes
0answers
84 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
1
vote
2answers
81 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
1
vote
1answer
73 views

Hölder's inequality. Understanding proof?!

I know how most standard textbooks show that $||f*g||_r \le ||f||_p||g||_q$ with $\frac{1}{r}+1=\frac{1}{p}+ \frac{1}{q}$, but I found a book where the hint $|f(x-y)g(y)|\le ...
0
votes
1answer
80 views

How to calculate the threefold convolution $f*f*f$

Somehow this convolution is driving me crazy. I am trying to calculate for the indicator function $f:=1_{[0,1]}$ the threefold convolution $$f*f*f$$ But honestly, it does not work somehow. ...
0
votes
0answers
148 views

How to bring f(x) from denominator to numerator?

Is it possible to rewrite the following ratio in a way that $f(x)$, its powers or derivatives appears only as numerator. $$\frac{1}{\int_{0}^{c}(c-x)^{2}f(x)dx}$$ $c>0$ is a constant. $f(x)$ is ...
0
votes
2answers
117 views

What does triple convolution actually look like?

I have to prove associativity of the convolution of three functions. I'm having trouble picturing how the variables will look. For periodic functions $f$, $g$, and $h$, I have the definition $$ (f * ...
1
vote
1answer
50 views

Recovering function from convolution with a square function

Let $m : \mathbb{R} \to \mathbb{R}$ be a continuous function of compact support. Given $M$ as: $$ M(x) = \int_{\mathbb{R}} m(t) (t+x)^2\ \mathrm{d}t,$$ is it possible recover $m$? I thought it ...
1
vote
0answers
71 views

How to deconvolve from the result of a sort of double convolution integral?

Say that I have a probability density function defined on the unit circle, $f_{\Theta}(\theta)$, with $\theta \in \left[0,2\pi\right)$. I have a joint pdf, assuming independence, of ...
1
vote
1answer
55 views

expanding convoluted integrand

I have a function on the form $$g(y) = \int_{-\infty}^{\infty}{e^{-v^2}f(y-v)dv}$$ I know that $g(y)$ is linear around $0$, $g(y\approx 0)\approx yG$, and I am interested in finding this gradient $G$. ...
0
votes
1answer
40 views

Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
-2
votes
1answer
76 views

Prove that L[f' ' ](s)$ = $sL[f](s)

Can anyone prove this question ? Let $f$:$\mathbb{R}$$→$$\mathbb{C}$ be continuous function such that $f$$(0)$ $=$ $0$ and that $f'$ be a piecewise continuous function and absolutely integrable on ...
0
votes
1answer
212 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 ...
1
vote
0answers
94 views

Integrability and differentiability of convolution of the fundamental solution and an integrable function

Define a function $\Gamma(\cdot)$ as $$ \Gamma(x-y)=\frac{1}{2\pi}\log\|x-y\|,\quad x\neq y $$ where $x=(x_1,x_2),y=(y_1,y_2)\in R^2$, and $\|x-y\|^2=(x_1-y_1)^2+(x_2-y_2)^2$. Note that $\Gamma(x-y)$ ...
3
votes
2answers
120 views

$\frac{1}{a^2+x^2}\ast \frac{1}{a^2+x^2}=\frac{2\pi /a}{4a^2+x^2} (a>0)$

I.e. $\displaystyle\int_{\mathbb{R}} \frac{1}{a^2+(x-y)^2}\cdot\frac{1}{a^2+y^2} dy=\frac{2\pi /a}{4a^2+x^2}$. If anyone feels like giving me a brief hint about how to get started on this I'd be ...
-3
votes
1answer
901 views

Is this formula for product of integrals correct?

$$\int_0^x f(x)dx \int_0^xg(x)dx=\int_0^xf(-x)*g(-x)dx+\frac12 \int_0^xf(2x)*g(2x)dx $$ Where * means convolution.
0
votes
2answers
417 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 ...
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 ...