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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1answer
77 views
arguing away - complex analysis
Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis.
I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
1
vote
1answer
47 views
How to establish the smoothness class
Consider a smooth function $f(x)$ on $\mathbb{R}^{n}$ from some smoothness class $S_1$ and define
$$
F(y) = \int\limits_{g(x,y)\leqslant 0}f(x)dx
$$
where $y \in \Omega$, $\Omega$ is some domain in ...
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votes
1answer
51 views
Condition for the inverse laplace transform of a function to exist and bromwich integral
Given any function, is there any way of determining from the nature of the function, if it is the laplace transform of a piecewise continuous function of exponential order? For e.g. say the function ...
0
votes
1answer
73 views
can asymptotic of a Mellin (or laplace inverse ) be evaluated?
i mean given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $ can we evaluate this integral, at least as $ x \rightarrow \infty $
can the same be made for $ ...
4
votes
0answers
46 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) = ...
3
votes
0answers
20 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
votes
0answers
51 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, ...
3
votes
0answers
82 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 $K(x,y)$
...
3
votes
0answers
169 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
votes
0answers
2k views
Relationship Between The Z-Transform And The Laplace Transform
Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
3
votes
0answers
199 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
$$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
2
votes
0answers
46 views
Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous
Consider the following function defined by a singular integral
\begin{equation}
F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) ...
2
votes
0answers
40 views
Mellin transform of digamma function
what is the Mellin trasnform of the Digamma function ??
from Ramanujan master theorem http://mathworld.wolfram.com/RamanujansMasterTheorem.html
y believe it should be equal to
$$ ...
2
votes
0answers
52 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
$$ F(\phi(x))= ...
2
votes
0answers
49 views
Gram's series for integral equation
The prime counting function $ \pi(x) $ satisfies the integral equation
$$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$
and it has the solution in terms of Gram's ...
2
votes
0answers
64 views
Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?
I would like to know if this formula is true:
$$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$
I have used the ...
2
votes
0answers
127 views
Mellin inverse transform
would be possible to evaluate the Mellin inverse transform
$ \int_{c-i\infty}^{c+i\infty}ds \frac{\zeta(s)}{2i \pi s}$ in terms of the zeros of the RIemann Zeta ???
i know how to compute the invers ...
2
votes
0answers
270 views
Fourier Transform of Bessel function with square root argument
Fourier Transform of the following function:
...
2
votes
0answers
147 views
Integral Transform
I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following.
$w(t)$ has the Fourier Transform $W(f)= \dfrac{(j\pi f)}{(1+j2\pi ...
2
votes
0answers
116 views
What does it mean to carry out calculations modulo $S^{- \infty}$
I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own:
" Oscillatory integrals are used for the study of singularities of ...
2
votes
0answers
72 views
Evaluating the limit $y \to 0^+$
Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$.
$\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$
This limit is given in the book Integral Transforms and Their ...
1
vote
0answers
21 views
Existence of zeros of Mellin transform and properties of function to be transformed
Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by
$$
f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}.
$$
I consider only exponentially decreasing (there exist such ...
1
vote
0answers
71 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
1
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0answers
41 views
Inverse Laplace transform is required
I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible.
$$
\frac{1}{s-a}\exp ...
1
vote
0answers
54 views
Mellin tranform of $\cos x$ using Ramanujan's master theorem
I've been messing around with Ramanujan's master theorem.
$\displaystyle \int_{0}^{\infty} x^{s-1} \cos x \ dx = \int_{0}^{\infty} x^{s-1} {}_0F_1 \left(-;\frac{1}{2};\frac{-x^{2}}{4} \right) \ dx $
...
1
vote
0answers
35 views
Understanding analyticity
Assume $\omega , \mu \in \mathcal{D}'(\mathbb{R})$ are distributions with $\operatorname{supp}\mu $ compact, that are related according to
$$
\omega = \varphi \ast \mu = \int (x-y)^{1/2}_+ \mu (y) \, ...
1
vote
0answers
26 views
Do integral transforms with meromorphic kernels always have analytic continuation?
Do integral transforms with meromorphic kernels always have analytic continuation ?
I think so, but I do not know how to prove it.
For clarity with analytic continuation I assume it was already ...
1
vote
0answers
63 views
Gelfand-Levitan-Marchenko equation
how can one solve the integral
$$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1)
so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2)
$$ -y''(x)+q(x)y(x)=0 $$ (3)
$$ y(0)=0=y(\infty) $$
$ q(x) $ here is ...
1
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0answers
67 views
Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.
Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial.
Express $f(x)$ as an integral from $0$ to $\infty$.
As an example we ...
1
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0answers
71 views
relationship between Connes trace formula and Weil's trace formula
Connes trace formula $$ Tr{U(h)}=2h(1)log\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$
Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
1
vote
0answers
84 views
An integral transform.
Let's consider a complex function that can be represented in the following form:
$$
K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha
$$
Writing $z=re^{i\theta}$, we get:
$$
...
1
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0answers
65 views
Different proofs of support theorem for Radon Transform
Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
1
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0answers
128 views
Inverting an integral transform
This question is either very obvious or by nature unsolvable in a general case. I Google'd around for a solution to no avail.
Given an integral transform of kernel K across some interval I as a ...
1
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0answers
134 views
Hankel Transform - Eigenfunctions and Inverse
I was reading Akhiezer's Lectures on Integral Transforms and in chapter nine, The Hankel Transform, he says that because the kernel of the Hankel transform is symmetric, its eigenfunctions ...
1
vote
0answers
235 views
What is the difference between resolvent kernel and iterative kernel of an integral equation?
As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
1
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0answers
99 views
Fourier transform integral
I'm trying to calculate the 3D fourier transform of this function:
$$\frac{1}{(x^2+y^2+z^2)^{1/2}}$$
Any help would be appreciated, thanks.
1
vote
0answers
136 views
Limits in Double Integration Question
I’m really having problems getting suitable limits when using double integration. For example:
Let E be the region defined by {(x,y): y-x <= 2, x + y >= 4, 2x + y <= 8}
Sketch the region E.
...
0
votes
0answers
18 views
Practical applications of the Fantappiè transform
The Fantappiè transform of function $f(x_1,x_2,\ldots,x_n)$, $x_1 \geqslant 0, \ldots, x_n\geqslant 0$, is defined by the formula
$$
(\Phi f)(y) = \int\limits_{\mathbb R^n_+} ...
0
votes
0answers
14 views
Lower estimates on Mellin transform
Let $f(t)$ be a smooth decreasing function on $[0,+\infty)$. Its Mellin transform is the function $f^\ast(z)$ given by
$$
f^\ast(z) = \int\limits_0^\infty x^{z-1} f(x) \, \mathrm dx.
$$
What are ...
0
votes
0answers
66 views
Triple Product Integral on Real Spherical Harmonics Basis Functions
Okay I know that Real Spherical Harmonics are given by
If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$
If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$
If $m \gt ...
0
votes
0answers
40 views
nonlinear integral equation
let be the integral equation for two functions $ f(x) $ and $ g(x) $
in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$
is valid to accept that in the sense of fractional calculus, the ONLY ...
0
votes
0answers
68 views
Inverse laplace transform - infinite residues
I need to compute the inverse transform of the following, $f(s)=
\dfrac{\sinh(k(l-x))}{\sinh(kl)}\dfrac{\omega}{\omega^2+s^2}$ where $k=\sqrt{\dfrac{s^2}{c^2}+n^2\pi^2},\ 0\leq x\leq l$. I used what ...
0
votes
0answers
130 views
Integral Transform with Hyperbolic Functions
I am at it with understanding the nitty-gritty of the integral transform suggested in a previous question of mine:
Length of a Parabolic Curve
To solve this integral, you can use the substitution
...
0
votes
0answers
34 views
Mathematical definitions of frequency transform and time-frequency transform?
I was wondering if there are mathematical definitions for an integral transform
$$(Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt $$
to be frequency transform or time-frequency transform?
An ...
-1
votes
0answers
722 views
Square of the sinc function integral
Considering the pulse function $\text{sinc}(t)$, show that :
$$\int_{-\infty}^\infty \text{sinc}^2(t) dt = 1$$
