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 ...
0
votes
1answer
26 views
proving a z transform
I am having trouble demonstrating the Z transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table.
I try using the definition of the z transform, but it comes out different than ...
1
vote
1answer
33 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 ...
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
$$
(\Phi f)(y) = \int\limits_{\mathbb R^n_+} ...
0
votes
1answer
80 views
Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform
This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below.
Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
1
vote
1answer
42 views
Evaluating improper integrals using laplace transform
I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only).
$$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$
I propose the following method. I plan to ...
0
votes
1answer
52 views
Condition for the inverse laplace transform of a function to exist and bromwich integral
Given any function, is there any way of determining from the nature of the function, if it is the laplace transform of a piecewise continuous function of exponential order? For e.g. say the function ...
5
votes
1answer
51 views
Solving an initial value ODE problem using fourier transform
I am a physics undergrad and studying some transform methods.
The question is as follows:
$y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$
I am having some ...
0
votes
2answers
64 views
Solving an integral equations using fourier transform
I have to solve the equation
$\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$
Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
0
votes
1answer
20 views
Probability integral transform: Is it integral transform? Can it be for discrete distribution?
From Wikipedia
the probability integral transform or transformation relates to the result that data values that are modelled as being random variables from any given continuous distribution can be ...
0
votes
1answer
23 views
Lambert transform of monomials
The Lambert transform of a function $f(x)$, is given by:
$$\int_{0}^{\infty}\frac{f(x)}{e^{xt}-1}dx\;\;\;\;(\Re(t)>0)$$
We wish for a closed form of the transform :
...
3
votes
0answers
20 views
fancy about some properties of kernel functions at infinity
Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not:
$1.$ If $K(x,t)$ is bounded ...
0
votes
0answers
14 views
Lower estimates on Mellin transform
Let $f(t)$ be a smooth decreasing function on $[0,+\infty)$. Its Mellin transform is the function $f^\ast(z)$ given by
$$
f^\ast(z) = \int\limits_0^\infty x^{z-1} f(x) \, \mathrm dx.
$$
What are ...
1
vote
0answers
21 views
Existence of zeros of Mellin transform and properties of function to be transformed
Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by
$$
f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}.
$$
I consider only exponentially decreasing (there exist such ...
1
vote
3answers
61 views
Fourier transform of $g(x)=x\frac{\partial f}{\partial x}$?
I have a problem with the Fourier transform of the function $g(x)=x\frac{\partial f}{\partial x}$. I need the transform to be itself a function of the Fourier transform of $f(x)$ and I don't know how ...
1
vote
0answers
71 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
0
votes
1answer
87 views
Solving a recurrence relation using Z transform
I'm trying to solve the following recurrence using Z transforms:
For $n\in \mathbb{N}^{*}$
$T(n)=1\ for\ n< 4$
$T(n)=T(\lfloor \frac{n}{4} \rfloor)+T(\lfloor \frac{3n}{4} \rfloor)+n\ for\ n\geq ...
2
votes
0answers
48 views
Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous
Consider the following function defined by a singular integral
\begin{equation}
F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) ...
1
vote
0answers
41 views
Inverse Laplace transform is required
I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible.
$$
\frac{1}{s-a}\exp ...
1
vote
0answers
55 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 $
...
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
$$ ...
1
vote
1answer
48 views
Fourier Transform of a function under an arbitrary coordinate transform [duplicate]
Consider a function $f(x)$ and its Fourier Transform $\tilde{f}(k)$ given by
$$
\tilde{f}(k) = \int_\mathbb{R}\!\!\!dx\; e^{-ikx}f(x).
$$
Now, lets have the coordinate transform $\xi = \tau(x)$ and, ...
1
vote
1answer
78 views
Dirac delta questions form Mellin transform
We know that
$$ f(s)= \int_{-\infty}^{\infty}f(x)\delta (x-s) d x$$
however, is there a similar delta function so for the Mellin transform
$$ f(s)=\int_{0}^{\infty}f(x)m(xs) d x$$ ?
That is a ...
5
votes
4answers
82 views
Change of variables Double integral
I have
$$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$
$A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$.
I initially thought to make the switch the polar coordinates, but the line ...
2
votes
1answer
102 views
Inversion formula for the Abel transform
I need an inversion formula for the Abel transform
$$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$
Hint: The inversion formula found on Wikipedia appears to be incorrect. The ...
3
votes
1answer
172 views
Hilbert transform and Fourier transform
Assume the following relationship between the Hilbert and Fourier transforms:
$$
\mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)),
$$
where $ \displaystyle ...
4
votes
0answers
46 views
Some properties of an analogue of the integral Fourier operator
Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = ...
0
votes
0answers
66 views
Triple Product Integral on Real Spherical Harmonics Basis Functions
Okay I know that Real Spherical Harmonics are given by
If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$
If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$
If $m \gt ...
2
votes
0answers
52 views
Theta series and Riemann Hypothesis
in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf
section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series
$$ F(\phi(x))= ...
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 ...
3
votes
1answer
68 views
Double Integral involving modifed bessel function
I'm try to derive a closed form of the following double integral:
$\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant.
Do you ...
1
vote
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 ...
2
votes
1answer
163 views
An inverse definite integral problem
I am seeking a function $f(x)$ that satisfies this condition:
$\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
-1
votes
1answer
29 views
When to use other transforms?
maple code
int(g*f, x=-infinity..infinity)
when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function
If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they ...
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
$$
\omega = \varphi \ast \mu = \int (x-y)^{1/2}_+ \mu (y) \, ...
0
votes
0answers
68 views
Inverse laplace transform - infinite residues
I need to compute the inverse transform of the following, $f(s)=
\dfrac{\sinh(k(l-x))}{\sinh(kl)}\dfrac{\omega}{\omega^2+s^2}$ where $k=\sqrt{\dfrac{s^2}{c^2}+n^2\pi^2},\ 0\leq x\leq l$. I used what ...
2
votes
2answers
58 views
Can an integral operator with negative kernel have a positive eigenvalue?
While reading "Integral equations - a reference text" (Zabreiko et al. eds) I came up with this question I cannot answer:
Suppose $A:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is a bounded linear integral ...
0
votes
1answer
72 views
Question about L2 Inner Product and Integrals
Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero:
...
5
votes
2answers
110 views
Usage of inverse Laplace transform
At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof?
...
2
votes
1answer
74 views
Decaying Fourier transform and smoothness
Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies
$$
|\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}.
$$
I want to show ...
4
votes
1answer
118 views
Convolution square root of $\delta $
I want to somehow classify the distributional solutions of the equation
$$
f \ast f = \delta
$$
where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
1
vote
0answers
27 views
Do integral transforms with meromorphic kernels always have analytic continuation?
Do integral transforms with meromorphic kernels always have analytic continuation ?
I think so, but I do not know how to prove it.
For clarity with analytic continuation I assume it was already ...
1
vote
0answers
63 views
Gelfand-Levitan-Marchenko equation
how can one solve the integral
$$ 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) $$
$ q(x) $ here is ...
1
vote
2answers
47 views
Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$
I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into ...
1
vote
0answers
67 views
Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.
Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial.
Express $f(x)$ as an integral from $0$ to $\infty$.
As an example we ...
0
votes
0answers
131 views
Integral Transform with Hyperbolic Functions
I am at it with understanding the nitty-gritty of the integral transform suggested in a previous question of mine:
Length of a Parabolic Curve
To solve this integral, you can use the substitution
...
14
votes
1answer
207 views
Is Fourier transform characterized by its diagonalization properties?
Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space:
$$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$
We then have the following properties:
...
2
votes
0answers
49 views
Gram's series for integral equation
The prime counting function $ \pi(x) $ satisfies the integral equation
$$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$
and it has the solution in terms of Gram's ...
0
votes
1answer
69 views
Tonelli's theorem using in mean residual life definition
If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life ...
2
votes
1answer
60 views
Proof of the Direct mapping Theorem for Mellin transform.
I cannot understand an integration in the proof of the Direct mapping Theorem for the Mellin transform. A statement of the Theorem, together with an outline of the standard proof, can be found at page ...
3
votes
0answers
51 views
Can I solve for a unique integral kernel?
Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$,
$$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$
Is it possible to solve for the integral kernel, ...
