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

2
votes
1answer
336 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 ...
2
votes
1answer
39 views

Hankel transform of a Bessel function of different order

Here I found that $$ \int_0^\infty J_\nu(kr) J_\nu(sr) r dr = \frac{\delta(k - s)}{s} = \frac{1}{s^2}\delta\left(1 - \frac{k}{s}\right). $$ I wonder how can that be derived and if a similar method can ...
0
votes
0answers
32 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
3
votes
1answer
55 views

Inverse Radon Transform of a function in the Schwartz class

This question comes from reading through Stein and Shakarchi's Fourier Analysis, page 206. Consider the two Schwartz spaces $\mathcal{S}(\mathbb{R}^3)$ and $\mathcal{S}(\mathbb{R}\times S^2)$, where ...
26
votes
7answers
18k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformations 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
1answer
10 views

Two small questions concerning the Hankel transform

For an application, I was reading the wikipedia page on the Hankel transform and I was hoping somebody could clarify two things for me that I have not been able to find elsewhere as well: 1) ...
3
votes
1answer
40 views

Norm of an integral operator $L^1 \rightarrow L^\infty$

Let $T:L^1(\mathbb{R}^n)\rightarrow L^\infty(\mathbb{R}^n)$ be an integral operator, i.e. there exist $K:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that for all $f\in ...
8
votes
0answers
117 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 ...
1
vote
1answer
91 views

Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

Can anyone help me find the following indefinite integral: $$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$ I cannot think of any transformation...
3
votes
0answers
43 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 ...
0
votes
1answer
29 views

Reconstruction of a function from its moments

The n-th moment of a real valued function $f$ is defined as: $m_n(f)=\int_{-\infty}^{+\infty}x^nf(x)dx$. I heard that a function $f$ is uniquely determined by its moments. I would be quite surprised ...
1
vote
1answer
40 views

Dirichlet energy and Fourier transform

Is there a direct relationship between the Dirichlet energy of a function: $$E(f)=\int_{\Omega}\lvert\nabla f(\mathbf{x})\rvert^2\mathrm{d}V$$ and its Fourier transform ...
5
votes
2answers
7k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
3
votes
4answers
78 views

What does it mean when two functions are “orthogonal”, why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
0
votes
0answers
19 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
3
votes
3answers
194 views

A more elegant way to find the Fourier transform

Let $f$ be defined analytically as : $$f(x)=\arccos \left ( \sin \left ( 2x \right ) \right ), x \in\left (0,10 \right ], f(x)=0, x\notin\left ( 0,10 \right ]$$ Here is a graph of the above ...
2
votes
5answers
46 views

“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
1
vote
1answer
39 views

Is the Fourier Transform of the limit the limit of the Fourier Transform?

Assume you want to compute the Fourier transform of a function $f_\epsilon(x)$ given by \begin{align} \mathcal{F}(f_\epsilon)(k) = \int f_\epsilon(x) e^{-ikx}\, dx \end{align} Further assume, that ...
6
votes
2answers
1k views

Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$ where $ \displaystyle ...
0
votes
2answers
55 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
1
vote
1answer
33 views

Smoothness of Fourier transform of $\frac{1}{|x|^p}$

Consider the "function" (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the ...
2
votes
2answers
388 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: ...
2
votes
0answers
51 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$

Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier ...
-2
votes
1answer
25 views

Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
2
votes
1answer
117 views

Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series $$ F(\phi(x))= ...
0
votes
1answer
23 views

Probability transformation

I have a question regarding probability transformations. Could someone tell me wether I am doing it good or not? Consider $f_{X}(x)=3/2e^{-3x}+3e^{-6x}, x \geq 0$, 1) Calculate the pdf of ...
0
votes
0answers
26 views

Geometric Interpretation of Fourier Transforms

I'm interested in tsunami wave science and I've already got an engineering degree and a basic knowledge of signal processing. The courses I took were intensively computational and taught some skills ...
1
vote
0answers
51 views

Can someone explain this z transformation to me?

I have a signal $h[n]=\frac{1}{z+3}$ and the solution is $H(z) = (-3)^{n-1}\delta[n-1]$. Looking the solution up in a transformation table, I come to the conclusion that I need to transform $h[n]$ ...
4
votes
1answer
205 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 ...
3
votes
0answers
274 views

An inverse Mellin transform

Is it possible to compute the inverse transform of $$ \frac{1}{a^{-s}\cos( \frac{\pi s}{2})\Gamma (s)} $$ or similarly is it possible to compute the Inverse Mellin transform ?? $$ \frac{ \zeta ...
3
votes
1answer
226 views

How to find the inverse Mellin transform?

On the wikipedia page http://en.wikipedia.org/wiki/Mellin_transform The second formula is an integral transformation for the inverse Mellin transform. Being new to integral transforms, I wonder how ...
2
votes
0answers
11 views

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
1
vote
0answers
27 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...
0
votes
0answers
13 views

Identify functional form of expectation value

I have the equation: $g(z) = {\int_{\mathbb{R}^n}}\;f(x)\exp(-c(x)^Tz)dx = \mathbb{E}_X[\exp(-c(X)^Tz)]$ where $c:\mathbb{R}^n \to \mathbb{R}_+^m$, $z \in \mathbb{R}_+^m$ and $f(x)$ is a ...
2
votes
2answers
213 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
2
votes
0answers
52 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
votes
1answer
60 views

Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
1
vote
0answers
35 views

Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
0
votes
1answer
22 views

Graphing Fourier transforms on a frequency versus intensity plot (how to deal with complex numbers)

I am trying to understand how Fourier transforms are used to make HNMR plots. HNMR basically consists of hitting some molecules with some radiation and listening to the radio signal that results. ...
0
votes
0answers
17 views

Transforming a function - general methods

I have some rather general questions about transforming a function from one form to another. I have tried reading about integral transforms, but I wish to know more about the theory in general, as ...
8
votes
1answer
96 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
0
votes
1answer
28 views

Quick Question on Zeroes of Transfer Function

Sorry for not providing context here. Suppose I have an output $Y(z)=\frac{z-1}{z}$ and input $X(z)=\frac{z^2+3z+2}{z^2}$ to yield a transfer function ...
16
votes
1answer
504 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: ...
43
votes
1answer
1k views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm ...
1
vote
1answer
20 views

Find a function in terms of another

I need to express an in terms of f(x). I did it but I'm not sure if it is right. Consider \begin{cases} -u_{xx}+u=f(x), &0<x<\ell\\[0.5em] u_x(0)=0,\;u_x(l)=0& \end{cases} ...
0
votes
1answer
33 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
0
votes
1answer
45 views

Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure ...
2
votes
0answers
80 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),\, ...
4
votes
3answers
109 views

How to learn Integral Transform?

I major in Electronic Engineering when back in college. I learned the Fourier Transform, Laplace Transform, Z Transform and wavelet Transform. But I always feel a lack of thorough understanding of ...
1
vote
1answer
41 views

Fourier transform of a random variable

During my research i'm dealing with a stochastic partial differential equation. The random term appearing in my equation is a tensorial random variable: $\boldsymbol{\sigma}(\boldsymbol{x},t)$ Which ...