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Questions tagged [integral-transforms]

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 transforms.

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219 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
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796 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
6
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162 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in \...
5
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106 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider ...
4
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92 views

The Parseval theorem applied to 8 well known functions. Could this be extended to L-series?

In the shaded section below, I listed 8 well-known functions that nicely hang together: $$\displaystyle\frac{\beta(s)+\lambda(s)}{\kappa(s)+\alpha(s)}=\frac{\zeta_H\left(s,\frac12\right)+\eta_H\left(...
4
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68 views

Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$

In the article on the $\Gamma$ function in the Princeton Companion to Mathematics, the author states The Mellin transform is a type of Fourier transform, but it is defined for functions on the ...
4
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271 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, s\right)=\underset{k\geq1}{\sum}...
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141 views

Clarification on the absolute convergence of Mellin transform

I have a question I haven't been able to find a direct answer to that I presume is true but I am unable to show. We know these two following results on the mellin transform. If $$\int_0^\infty |f(x)|...
4
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83 views

Some properties of an analogue of the integral Fourier operator

Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = \...
4
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149 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 ...
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1k 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 ...
3
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77 views

About $ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $

We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$ Usually this is shown by using differentiation under the ...
3
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40 views

“Bilateral Mellin convolution”

The Mellin convolution of two functions, when it exists, is of the form $$ (f \ast_M g)(t) = \int_0^\infty f\left( \frac{t}{\tau} \right) g(\tau) \frac{\mathrm{d}\tau}{\tau} $$ and ...
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362 views

Solving a non-separable PDE with Mellin or other integral transform

I am trying to solve a PDE of the form $$\xi_t(t, x) + a(t) x^{f(t)} \xi(t, x) + b(t) x^{f(t)+1} \xi_x(t,x) + c(t) x^{f(t)+2} \xi_{xx}(t, x) = 0.$$ I have no idea if this is even possible. There ...
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101 views

How would one integrate $e^{\sin x}\csc x$?

I was trying to develop a random integral transform, but things happened, and I'm kinda confused as to what I can do. Anyway, here's an example of my in-development transform: $$\int e^{\sin t}\csc(t)...
3
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567 views

Fourier Transform Motivation/Derivation

this is my first ever Math SE question, and I am wondering how one can go about rigorously explaining the Fourier Transform. I believe it is connected to Fourier Series, but I can't comprehend the ...
3
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48 views

Can inversion integral of characteristic functions on a finte interval be bounded?

For a real-valued uni-variate r.v. $X$, with pdf $f(x)$ and absolute integrable cf $\varphi(t)$, we have the following transform:$$2\pi f(x)=\int_{-\infty}^{\infty}e^{-itx}\varphi(t)\,dt.$$ However, I ...
3
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103 views

How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
3
votes
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261 views

Radial Green's function

I would like to solve an equation of the form $$ \bigg(\frac{d}{dr^2} + m^2 \bigg)f(r) = g(r), $$ for $f(r)$. Normally I would just find the Green's function $G(r,r')$, which is defined by $$ \bigg(\...
3
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111 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
3
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178 views

A Mellin Transform of a generating function

I am trying to find the Mellin transform of the function $$ G(z) = \sum_{k \ge 1} C_k\left( 1- \exp \left( \frac{-z}{4^k} \right )\right), $$ where $C_k$ denotes the $k$-th Catalan number ($C_k = \...
3
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116 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t \...
3
votes
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45 views

Why do you need an integral to invert the Z-Transform?

With integral transforms both the transform and its inverse are integrals. In the case of the Z-Transform the transform is a sum. My question Why do you need an integral (instead of another sum) to ...
3
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85 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
3
votes
0answers
100 views

Uniqueness for Integral Transform

What can be said about the uniqueness of the following integral transformation: $ (Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$, where the kernel $G(z) \in [0,1]$ for all $z\geq0$,...
3
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0answers
94 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
3
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0answers
233 views

Existence of zeros of Mellin transform and properties of function to be transformed

Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by $$ f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}. $$ I consider only exponentially decreasing (there exist such ...
3
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0answers
350 views

Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous

Consider the following function defined by a singular integral \begin{equation} F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) \...
3
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0answers
73 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(\...
3
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126 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 $K(x,y)$ $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
3
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0answers
304 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 ...
3
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352 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)}...
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262 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 \int_{...
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31 views

Binomial sum for an arbitrary function

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary ...
2
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0answers
78 views

Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$ \int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k, $$ where $J_0(kr)$ is the Bessel function of ...
2
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0answers
28 views

How can the following be transformed in to a sum of complete elliptic integrals of the first and second kind

I have the following, that I known from a numerical implementation of the problem by a third party should be able to be transformed in to elliptic integrals of the first and second kind however I can'...
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50 views

Can integration contour of an inverse Mellin transform be deformed at will in fundamental strip?

Let $m(x)$ be inverse Mellin transform of $M(s)$: $m(x)=\frac{1}{2\pi i} \int\limits_{c-i\infty}^{c+i\infty}x^{-s}M(s)ds$ Mellin transform $M(s)$ is analytic on fundamental strip $a<\Re(s)<b$ ...
2
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36 views

Fredholm integral equation of the 2nd kind with a weakly singular convolution kernel

I've reviewed the literature but could not find a solution of the integral equation $$\int_{0}^{1} dx\frac{\phi(x)}{|x-y|^{\alpha}}+a\phi(y)=b.$$ for the function $\phi$ where $\alpha\in(0,1)$ and ...
2
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0answers
90 views

When an Hilbert–Schmidt operator is invertible? How to construct the kernel of the inverse?

This question is motivated by the following example. I have the following integral operator defined on $L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n) $ $$ T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \...
2
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0answers
44 views

Transform $\cosh(x)$ into $\exp(x^2/2)$

I'm looking for an integral transform that maps $\cosh(x)$ into $\exp(x^2/2)$. More specifically I'm looking for a kernel $K(y,x)$ and a domain $\Omega \subset \mathbb{R}$ such that $$ \int_\Omega dx ...
2
votes
0answers
58 views

About $ A(f(x),g(x)) = h(x) = 1 + \int_0^x f(x - t) g(t) dt $

Consider the transforms $$ A(f(x),g(x)) = h(x) = 1 + \int_0^x f(x - t) g(t) dt $$ $$ B(h(x),g(x)) = f(x) $$ $$ C(h(x),f(x)) = g(x) $$ Where the functions on the LHS are given. Notice $B,C$ are ...
2
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0answers
48 views

Inverse Mellin transform of $\sin(x)f(x)$

I am trying to write the inverse Mellin transform of $\sin(x)f(x)$ in terms of the inverse Mellin transform of $f(x)$. That is, suppose that $F(y)$ is the inverse Mellin transform of $f(x)$, $$f(x) = ...
2
votes
0answers
176 views

Why do we multiply by transpose of matrix to get discrete cosine transform

When doing discrete cosine transform (DCT) on a image signal M e.g assume it is 8x8 like in JPEG, we multiply it with DCT marix T and its transpose. The T just contains numbers we get from cosine ...
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65 views

Representation of Hilbert transform by a singular integral.

Hilbert transform defines as follow: $$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$ $$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$ Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
2
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0answers
26 views

Can you make a product of kernels?

Let's apply two integral transforms to a function $f(t)$. $$ F'(v) = \int\limits_{t_1}^{t_2} K_1(t,v) f(t) \mathrm{d}t $$ $$ F(u) = \int\limits_{v_1}^{v_2} K_2(v,u) F'(v) \mathrm{d}v $$ Can I ...
2
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0answers
126 views

Parseval's equality for the Fourier transform

Please help me prove this! f(x) is piecewise continuous on any interval, and $$\int_{-\infty }^{\infty } |f(x)|dx < \infty$$ Here $$f(x)=\int_{-\infty}^{\infty}[A(\alpha)\cos \alpha x + B (\...
2
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0answers
42 views

Approximate solution for integration of two Bessel Functions

I am an engineering student. I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$ which is a function of $(n-1)$. degree polynomial. My statement as follows: $$...
2
votes
0answers
274 views

Solving differential equation by transform with contour integral

In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
2
votes
0answers
44 views

Fourier transforms of Gaussians with rational functions

Consider functions of the form $$ f(x)= e^{- x^2/\sigma^2} \frac{a_0 + a_1 x + \cdots +a_n x^n }{b_0+ b_1 x + \cdots + b_m x^m } $$ Under which circumstances do these functions have closed forms for ...
2
votes
0answers
216 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...