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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the multidimensional Hilbert transforms or partial Hilbert transforms

The one-dimensional Hilbert transform can be defined by the convolution $Hf:=f*\frac{1}{\pi x}$, or can be given by Fourier multiplier $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
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Integral transforms and scaling properties

It's well-known that the Fourier transform plays nicely with scaling. Particularly if we define, for $\alpha >0$, $D_{\alpha}$ by $D_{\alpha}f(x) = \alpha^{-1/2} f(x/\alpha)$, then (for suitable ...
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2answers
67 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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47 views

Inverse Mellin transform of simple function

How should the following inverse Mellin transform integral be evaluated: $ f(x) = \displaystyle \frac{\alpha}{2\pi i}\int_{c-i\infty}^{c+i\infty}(\alpha ...
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25 views

Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
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Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
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Question about integral transforms

Are there any integral transforms for which integral transform of function $f(t)$ is linearly related to integral transform of same function shifted by amount $T$. (Something similar to Fourier ...
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397 views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function ...
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Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
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34 views

Integral transforms and uncertainty products

Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often ...
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49 views

Fourier transform of a function

I'm struggling with FT, I just can't grasp the concept of it. Can somebody explain it on an example Ex 1: $f(t) = e^{-|t|}$ EX 2: $x(t) = \cos(\pi t/T)$ where it's different from $0$ just on ...
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42 views

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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Solution to recursion relation using Mellin transform

I have been trying to solve the following recursive equation for $0<x_c<1$ for few a days: $$ P(x) = 2\mathbf{1}_{0\leq x\leq x_c} + 2\int_x^1 dy P\left(\frac{x}{y}\right)y^{-1} ...
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2answers
56 views

Laplace transform of $L({1-e^{-t}\over t})$

I have to find the Laplace transform of $$\mathcal{L}\left[\dfrac{1-e^{-t}}t\right],$$ then this is equivalent to $$\mathcal{L}\left[\dfrac{1}t\right]-\mathcal{L}\left[\dfrac{e^{-t}}t\right]$$ But ...
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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 ...
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how to solve this inverse mellin transform

We know from the Perron's formula that $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)}{2\pi i s}x^{s}ds=[x] $$ $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)^{2}}{2\pi i s}x^{s}ds=\sum_{n\le x}d(n) ...
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138 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
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36 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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Z- transform existence

under what circumstances does a function $ f(x) $ has a Zeta transform ?¿? is this enough that a) $ f(x) $ is continous and derivable b) $ f(x) \to 0 $ as $ x \to \infty $ or at least $ f(x) \to C ...
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52 views

Inverting the integral $f(x)=\int_x^a \frac{g(t)}{\sqrt{t-x}}dt$

I am curious if there is a way to invert the integral $$f(x)=\int_x^a \frac{g(t)}{\sqrt{t-x}}dt$$ to solve for g(x) when f(x) is a known function. The integral from x to a makes this problem seem a ...
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How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$?

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$? I have checked my transform table and there is not a suitable case for this expression.
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Splitting Mellin Transforms

Let's say we have two functions, $f(x)$ and $g(x)$, such that the Mellin transform of $f(x)$ converges on the strip $a < x < b$ and the Mellin transform of $g(x)$ coverges on the strip $c < ...
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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 ...
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Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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53 views

Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is ...
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20 views

Finding relation between $\omega$ and scaling coefficient of mexican hat wavelet

I am not looking for complete solution as it is a homework problem. I would like to know how to start about finding the relation between $\omega$ of a sine wave and the scaling coefficient $a$ of a ...
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1answer
67 views

Integrating the Fourier Transform

I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$ Shouldn't it be $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ ...
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Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
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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 ...
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Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
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Inverting probability generating function via mellin transform substitution.

The pgf is defined as: $E(z^k)= \sum_{k=0}^{\infty} p(k)z^k$ which is a discretised version of the transform: $\widetilde{p}(z) =\int_{-\infty}^{\infty} z^k p(k) \, \mathrm{d}k$ The Mellin ...
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151 views

Fourier transform in $\mathbb R^3$

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

This is the maths part of a physics problem - that of the solution of Schrodinger equation for a 3D harmonic oscillator in spherical coordinates. The solution is a product of associated Laguerre ...
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1answer
115 views

Fourier transform integral

I'm trying to calculate the 3D fourier transform of this function: $$\frac{1}{(x^2+y^2+z^2)^{1/2}}$$ Any help would be appreciated, thanks.
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133 views

Bessel function integral

How to solve the integral for $J_1{(2x\sin{\frac{\theta}{2}})}$ at $[0,\pi]$? If solving by Matlab, please provide me the source. Thank you!
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“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then ...
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117 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
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50 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 ...
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1answer
85 views

Fourier Transform for triangular wave

Could someone tell me if I've worked this out right? I'm unsure of the process, especially the final parts where I convert it to a sinc function. Please let me know if I've made mistakes anywhere ...
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Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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1answer
137 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
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1answer
87 views

Inverse of an integral transform

Suppose that in a certain domain of analyticity we're given a function $A(s)$ in terms of the integral : $$A(s)=\int_{0}^{\infty}\frac{a(t)}{t(t^{2}+s^{2})}dt$$ How can we recover $a(t)$?
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Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n ...
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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 ...
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124 views

A problem about Laplace transform and Parseval–Plancherel theorem

I am reading a paper about fractional differential equation. One of the piece said as follow: By applying the Parseval–Plancherel theorem we may show: \begin{equation} \int_0^\infty ...
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138 views

how is the inverse Hankel transform defined

The finite Hankel transform in my notes is defined as $$ f^*(k_i) = \int_{0}^{\infty} {r f(r)J_0(rk_i)dr}, $$ where $k_i$ is one of the positive roots of $J_0$. However, my notes don't say anything ...
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Hankel transform and Laplacian in cylindrical coordinates

My book solved a PDE containing the Laplacian in cylindrical coordinates. It doesn't really explain why the Hankel transform is useful in this case (symmetries etc..); just brute force math. So yeah, ...
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Abel and Radon Transform

I am learning Radon and Abel transforms. As far as I understood, basically both the transforms are projection of a 3D object onto a 2D plane. Then what is the difference between both transforms? Under ...
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43 views

Transformation formula for multidimensional integral

Let $A,B$ be positive definite symmetric $n\times n$ matrices. I stumbled upon the following identity and don't see why it should hold: $$\int_{\mathbb{R}^n}\frac{1}{\sqrt{\det A \det ...
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Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...