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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2
votes
1answer
55 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
164 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 ...
2
votes
1answer
881 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 ...
1
vote
1answer
785 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 ...
4
votes
2answers
2k 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 ...
1
vote
2answers
1k views

Solving an integral equation using the 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
238 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
40 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
42 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 ...
2
votes
0answers
95 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
233 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 ...
2
votes
0answers
98 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
1
vote
0answers
262 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
380 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 ...
3
votes
0answers
146 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) ...
3
votes
0answers
142 views

Finding the Mellin tranform of $\cos x$ using Ramanujan's master theorem

I want to show that $$ \int_{0}^{\infty} x^{s-1} \cos x \ dx = \Gamma(s) \cos \left(\frac{\pi s}{2} \right) \ , \ 0<s<1$$ by specifically using Ramanujan's master theorem. $$ \begin{align} ...
7
votes
1answer
232 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
106 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
110 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 ...
6
votes
4answers
144 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
596 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 ...
5
votes
1answer
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 ...
4
votes
0answers
63 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) = ...
2
votes
0answers
88 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
70 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
103 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
97 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
374 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
36 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 ...
2
votes
2answers
137 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
130 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: ...
6
votes
2answers
301 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
264 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 ...
6
votes
1answer
264 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
43 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
1answer
277 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
71 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
73 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 ...
15
votes
1answer
465 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: ...
3
votes
0answers
106 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
121 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
117 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
60 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, ...
4
votes
2answers
295 views

Mellin Transform of $\sin$

Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$? $$\int_0^\infty \sin(y) y^{s-1} dy$$ I've googled for a while, but I could not ...
3
votes
1answer
229 views

Is this a correct way to convert an convolution equation into differential/difference equation?

For functions $f,g,h$ that are defined over $\mathbb{R}$, suppose we have a convolution equation: $$ f = g * h. $$ I would like to convert it into a differential equation. Is it correct that $$ ...
3
votes
0answers
111 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 $K(x,y)$ ...
1
vote
2answers
218 views

calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$

Could someone please help me to calculate the integral of: $$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$ a and b both real, b>0. I have tried integration by parts, but I can't seem to ...
4
votes
0answers
605 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 ...
1
vote
1answer
379 views

fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
1
vote
0answers
95 views

Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...