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 ...

learn more… | top users | synonyms

0
votes
0answers
31 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor, this question is kinda all over the place. If you have a set $B $ of $ N $ basis functions $ g_0(t), g_1(t), g_2(t), \dots, g_{N-1}(t) $ which are orthogonal over $[t_1, ...
20
votes
7answers
11k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
1
vote
0answers
47 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and ...
2
votes
1answer
32 views

Fourier transform dependent upon a parameter and $L^2$ convergence

Suppose I know the Fourier transform of a function depending upon a parameter, call it $f_\epsilon(x)$, and that I want to know the Fourier transform of a function $f(x)$. Furthermore, suppose I know ...
0
votes
0answers
64 views

What is the inverse Fourier transform of $|k|^{-\alpha}$?

What is the inverse Fourier transform, $\mathcal{F}^{-1}\{|k|^{-\alpha}\}$? I am specifically interested in the case where $1<\alpha<2$. To do this, I need to compute the integral ...
0
votes
0answers
12 views

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 $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
1
vote
0answers
15 views

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 ...
1
vote
2answers
84 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 ...
0
votes
0answers
48 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 ...
0
votes
2answers
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 ...
43
votes
3answers
843 views

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 ...
0
votes
0answers
37 views

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 ...
8
votes
2answers
412 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 ...
5
votes
2answers
53 views

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 ...
6
votes
1answer
39 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 ...
3
votes
1answer
43 views

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

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
1
vote
0answers
18 views

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} ...
0
votes
2answers
58 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 ...
26
votes
2answers
3k 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 ...
1
vote
0answers
24 views

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) ...
4
votes
1answer
152 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} ...
0
votes
0answers
38 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 ...
0
votes
0answers
10 views

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 ...
3
votes
1answer
55 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 ...
0
votes
2answers
42 views

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.
1
vote
0answers
17 views

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 < ...
3
votes
0answers
23 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 ...
1
vote
1answer
54 views

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 ...
0
votes
1answer
65 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 ...
0
votes
0answers
22 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 ...
2
votes
1answer
68 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}^{+ ...
0
votes
0answers
15 views

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 ...
2
votes
0answers
26 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 ...
6
votes
2answers
73 views

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?
0
votes
0answers
57 views

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 ...
0
votes
2answers
154 views

Fourier transform in $\mathbb R^3$

I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ ...
0
votes
1answer
69 views

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 ...
1
vote
1answer
117 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.
1
vote
1answer
142 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!
3
votes
1answer
40 views

“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 ...
0
votes
1answer
142 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: ...
0
votes
1answer
65 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 ...
0
votes
1answer
99 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 ...
1
vote
0answers
97 views

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 ...
0
votes
1answer
138 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 ...
3
votes
1answer
89 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 $a(t)$?
1
vote
2answers
32 views

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 ...
2
votes
0answers
81 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 ...
0
votes
1answer
136 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 ...
1
vote
2answers
158 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 ...