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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3
votes
1answer
335 views

Relation between Hankel transform and Fourier transform

As a physics student, I ran into the following problem. I left out a lot of context, if anything is unclear please ask me. I quote: The statistic that is observable is the angular correlation ...
4
votes
0answers
115 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),\, ...
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0answers
27 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
7
votes
2answers
328 views

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
0
votes
1answer
42 views

Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know ...
0
votes
1answer
47 views

Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
3
votes
1answer
47 views

Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
0
votes
1answer
99 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
0
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1answer
124 views

Good recommendations for solving PDE's by integral transforms

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't ...
5
votes
1answer
46 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
2
votes
1answer
96 views

dummy variable in Fourier transform confusion

In this text, why is it using different dummy variable for the integral of coefficients $a_n$ and $b_n$? I know that choosing the dummy variable does not affect integral but over here since we are ...
2
votes
0answers
31 views

Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...
0
votes
0answers
124 views

Hilbert transform of a Gaussian wave packet

For the following function $f$, function from real number to real number, with $\mu$ real, $k$ real, $\sigma$ real strictly positif, defined by: \begin{equation} f(x)=cos(k x ) ...
2
votes
1answer
58 views

Dirichlet Series and Asymptotic Expansions

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page "Zeta Function Regularization" of Wikipedia http://en.wikipedia.org/wiki/Zeta_function_regularization I ...
6
votes
0answers
78 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 ...
0
votes
2answers
46 views

How to calculate the Fourier transform?

If the Fourier transform is defined by $\hat f( \xi)=\int_{-\infty}^{\infty}e^{-ix \xi}f(x)dx$. How to calculate the Fourier transform of $$\begin{equation*} f(x)= \begin{cases} ...
2
votes
2answers
35 views

Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
2
votes
2answers
79 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
2
votes
0answers
35 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
9
votes
1answer
105 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
2
votes
1answer
65 views

Express $f(x)=\sin{x}$ as an even function

Express $f(x)=\sin{(x)}$, with $(0 < x< \pi )$ as an even function, $f(x+ 2\pi)=f(x)$ The topic is on Fourier Series. I have the following so far: Since $f(x)$ must be an even function, ...
3
votes
1answer
48 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
1
vote
1answer
58 views

Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function ...
2
votes
1answer
102 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[g(z) + \int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
0
votes
1answer
53 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
0
votes
1answer
179 views

PDEs for string deflection.

Okay, I have to find $u(x,t)$ for the string of length $L=\pi$ when $c^2=1$. I know: $$\text{wave equation}: \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ $$u(x,0)=\frac ...
0
votes
1answer
186 views

How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Find the laplace transform of $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$ The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$ This took me about an hour to ...
2
votes
0answers
161 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
2
votes
0answers
53 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
1answer
182 views

Kernel of an integral operator with Gaussian kernel function

Suppose we have the integral operator $T$ defined by $$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$ where $f$ is assumed to be continuous and of polynomial growth at most (just to ...
0
votes
2answers
33 views

Mellin Transform of $e^{-2 \pi t}$

I need to show that if $f(\tau)=e^{-2 \pi \tau}$ then: $$\{\mathcal{M}\,f\}(s)=(2 \pi)^{-s} \Gamma(s)$$ where : $$\Gamma(s)=\int_{0}^{+\infty} e^{-t}t^{s}\frac{dt}{t}$$ and ...
4
votes
1answer
106 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
1
vote
0answers
23 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) ...
2
votes
2answers
58 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
0
votes
1answer
29 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
1
vote
1answer
22 views

Inverse Laplace Transform Table, Absolution of Form

Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...
1
vote
1answer
49 views

Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ ...
2
votes
1answer
106 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor. If you have a countable orthonormal basis $B$ for a Hilbert space $H$ , then any function $f \in H$ can be expressed as $$ f(t) = \sum\limits_{g \, \in \, B} \langle f, ...
1
vote
0answers
63 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
41 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 ...
1
vote
1answer
52 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 ...
2
votes
2answers
4k 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
2answers
33 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 ...
4
votes
2answers
77 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
66 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
145 views

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

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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vote
0answers
43 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
67 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 ...
1
vote
0answers
55 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) ...
21
votes
1answer
194 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...