This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine ...

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7
votes
2answers
551 views

Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
3
votes
1answer
368 views

Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ I want to show ...
6
votes
1answer
349 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
1
vote
0answers
54 views

Do integral transforms with meromorphic kernels always have analytic continuation?

Do integral transforms with meromorphic kernels always have analytic continuation ? I think so, but I do not know how to prove it. For clarity with analytic continuation I assume it was already ...
1
vote
1answer
509 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2) $$ -y''(x)+q(x)y(x)=0 $$ (3) $$ y(0)=0=y(\infty) $$ $ q(x) $ here is ...
1
vote
2answers
78 views

Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$

I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into $\psi(0,\...
1
vote
0answers
74 views

Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.

Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial. Express $f(x)$ as an integral from $0$ to $\infty$. As an example we ...
18
votes
1answer
554 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: \...
3
votes
0answers
130 views

Gram's series for integral equation

The prime counting function $ \pi(x) $ satisfies the integral equation $$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$ and it has the solution in terms of Gram's ...
0
votes
1answer
148 views

Tonelli's theorem using in mean residual life definition

If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life ...
2
votes
1answer
161 views

Proof of the Direct mapping Theorem for Mellin transform.

I cannot understand an integration in the proof of the Direct mapping Theorem for the Mellin transform. A statement of the Theorem, together with an outline of the standard proof, can be found at page ...
3
votes
0answers
65 views

Can I solve for a unique integral kernel?

Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$, $$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$ Is it possible to solve for the integral kernel, $K(\...
4
votes
2answers
381 views

Mellin Transform of $\sin$

Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$? $$\int_0^\infty \sin(y) y^{s-1} dy$$ I've googled for a while, but I could not ...
3
votes
1answer
348 views

Is this a correct way to convert an convolution equation into differential/difference equation?

For functions $f,g,h$ that are defined over $\mathbb{R}$, suppose we have a convolution equation: $$ f = g * h. $$ I would like to convert it into a differential equation. Is it correct that $$ \frac{...
3
votes
0answers
116 views

Properties of Eigenfunctions of a Kernel

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. I've and Kernel function $K(x,y)$ $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
1
vote
2answers
231 views

calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$

Could someone please help me to calculate the integral of: $$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$ a and b both real, b>0. I have tried integration by parts, but I can't seem to ...
4
votes
0answers
929 views

Do kernel functions of integral transforms have any special properties?

From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform $T$ of the following form: $$ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $$ ...There are ...
2
votes
1answer
483 views

fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
1
vote
0answers
102 views

Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- \sum_{\rho}\int_{C}h(u)|u|^{\rho}d^{*}u+\int_{C}...
0
votes
1answer
65 views

Help me to understand the Gaussian blurring (2)

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$\begin{align*} p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\ \Delta_{i,j} &= \...
0
votes
1answer
48 views

Help me to understand the Gaussian blurring

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$p_{i,j}= \frac{\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$ I need to express the ...
2
votes
2answers
107 views

Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$

For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both ...
0
votes
1answer
192 views

Lipschitz constant on a functional

Let $C$ be the space of continuous and nondecreasing functions defined on $[0,1]$ and endowed with the sup norm. Let $T:C\rightarrow C$ be a continuous mapping, and consider the following expression: $...
4
votes
1answer
774 views

Is the Mellin transform useful to solve differential equations?

The Mellin transform is defined as: $$F(\mu)=\int_0^\infty f(x)x^{\mu-1}dx$$ The derivative of the Mellin transform is: $$F'(\mu)=-(\mu-1)F(\mu-1)$$ Applying this property, for example to the Bessel ...
2
votes
0answers
68 views

Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
6
votes
0answers
14k views

Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
11
votes
1answer
224 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation $$\frac{\partial}{...
1
vote
0answers
134 views

An integral transform.

Let's consider a complex function that can be represented in the following form: $$ K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha $$ Writing $z=re^{i\theta}$, we get: $$ K(re^{i\theta})=\int_{-...
1
vote
1answer
85 views

How to establish the smoothness class

Consider a smooth function $f(x)$ on $\mathbb{R}^{n}$ from some smoothness class $S_1$ and define $$ F(y) = \int\limits_{g(x,y)\leqslant 0}f(x)dx $$ where $y \in \Omega$, $\Omega$ is some domain in ...
4
votes
1answer
3k views

Fourier transform of $\log x$ $ |x|^{s} $ and $\log|x| $

Can anyone provide or give an expression in the sense of distribution theory for the functions $|x|^{s} , \log|x| $? I mean I would like to evaluate the Fourier transform $ \int_{-\infty}^{\infty}f(x)\...
28
votes
2answers
6k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty a_nz^{-n}$...
0
votes
1answer
106 views

Can asymptotic of a Mellin (or laplace inverse ) be evaluated?

I mean, given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $, can we evaluate this integral, at least as $ x \rightarrow \infty $? Can the same be made for $ \int_{c-i\infty}...
3
votes
2answers
292 views

Stueckelberg Feynman propagator computation

On page 35 of Itzykson-Zuber's textbook on quantum field theory, I am having trouble deriving equation (1-180): $\displaystyle G_F(0,r) = \frac{i}{(2\pi)^2 r} \int_m^\infty dp \frac{p}{\sqrt{p^2-m^2}}...
3
votes
0answers
237 views

Mellin inverse transform

would be possible to evaluate the Mellin inverse transform $ \int_{c-i\infty}^{c+i\infty}ds \frac{\zeta(s)}{2i \pi s}$ in terms of the zeros of the RIemann Zeta ??? i know how to compute the invers ...
2
votes
0answers
129 views

Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
7
votes
1answer
1k views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma x)\frac{\...
2
votes
0answers
506 views

Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: $\xi(\alpha,\tilde{\alpha},\beta)=\frac{J_{1}\left(\varrho_{0}k_{0}n\sqrt{\sin^{2}\alpha+\sin^{2}\tilde{\alpha}-2\sin\alpha\,\sin\tilde{\alpha}\cos\beta}\...
1
vote
0answers
198 views

Inverting an integral transform

This question is either very obvious or by nature unsolvable in a general case. I Google'd around for a solution to no avail. Given an integral transform of kernel K across some interval I as a ...
0
votes
1answer
185 views

Interchanging the order of integration.

Can someone explain why the following works please? $\int^t_0 f(s) \int^s_0 f(u) \,du\,ds = \int^t_0 f(s) \int^t_s f(u) \,du\,ds $ EDIT: All I know is that $f(x)$ is an integrable function. This is ...
3
votes
0answers
301 views

Integral Transform

I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following. $w(t)$ has the Fourier Transform $W(f)= \dfrac{(j\pi f)}{(1+j2\pi f)}...
2
votes
1answer
92 views

Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the ...
2
votes
1answer
706 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
2
votes
0answers
193 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
7
votes
1answer
925 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this concept,...
0
votes
1answer
312 views

How to apply the solution of $y(n) = (0.85)y(n-1) + x(n)$ to data

I learned how to solve difference equation $y(n) = (0.85)y(n-1) + x(n)$ using z Transform, and inverse z Transform, I get $h(n) = 0.85^n u(n)$ where $u(n)$ is unit step sequence. Now my ...
10
votes
4answers
401 views

Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and ...
4
votes
1answer
212 views

About Mellin convolution technique

Recently I was studying the Mellin convolution technique to solve definite integrals. I am just wondering is the technique valid only for any definite integrals (with any range $\int^b_a$)? Or only ...
2
votes
1answer
628 views

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
4
votes
2answers
166 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
0
votes
2answers
185 views

Fourier transform in $\mathbb R^3$

I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ ...