Questions tagged [integral-transforms]

Question 1

I will begin with the mathematical question at hand, and then describe technical details that were the background of the question, and then some possible approaches, although I clearly have not solved ...

Question 2

Let $p\geq 1$ and $u\in L^p(\Bbb R)$, for $t>0$ define $$U(t)= \frac12\int_{\Bbb R} (|u(x+t)-u(x)|^p+|u(x-t)-u(x)|^p) d x.$$ $$U_+(t)= \int_{\Bbb R} |u(x+t)-u(x)|^p d x.$$ Then we find that $(0,\...

Question 3

We know that applying the Laplace transform converts an ordinary differential equation into an algebraic equation, and we typically invert this transform to recover the solution. In most cases, the ...

Question 4

Many integral transforms have inverse kernels. The Fourier transfrom is an involution but others aren't. Differentiation is the inverse of integration, and integration of some functions like ...

Question 5

Consider the transform $T:L^1_{\mathrm{loc}}(\mathbb R)\to(0,\infty]^{\mathbb R}$ \begin{equation} (Tg)(x)=\int_0^\infty\exp\{-(\phi_t \ast g)(x)\}\,dt. \end{equation} where $\phi_t$ is the "tent&...

Question 6

Let $f : \mathbb{R} \to \mathbb{R}$ be a nice function (e.g. Schwartz, $L^2$, ...). I wonder if there's a name for the transform $$ T[f](y) := \int_{-\infty}^{\infty} e^{-yx^{2}}f(x) \mathrm{d}x. $$ (...

Question 7

Consider the following double integral transform, known as the Fourier-Kontorovich-Lebedev transform of a given function $f(r,z)$: \begin{equation} F(\alpha, \theta, k) = \mathscr{C}_{i\alpha} \{f\} =...

Question 8

Consider the following transform $T:g\mapsto f$, where $$ f(x)=\int_{0}^{\infty} \exp\left\{-(k_t * g)(x)\right\}\,dt $$ where $k_t$ is the "tent" function $$ k_t(u)=\begin{cases} t-|u|,&...

Question 9

Consider the following convolution $$ f(x)=(k * g)(x)=\int_{-\infty}^{\infty}k(t)g(x-t) \,{\rm d} t $$ where $g>0$, and $k$ is the "tent" function $$ k(t) = (1 - |t|)\chi_{[-1,1]}(t) $$ ...

Question 10

We define the Kontorovich–Lebedev transform as $$ F(\nu) = \int_{0}^\infty f(r) K_{i\nu} (r) \, r^{-1}\, \mathrm{d}r \, , $$ where $K_\nu$ denotes the $\nu$th order modified Bessel function of the ...

Question 11

I've been dealing with probability integral transforms a lot recently, and understand how to use them well enough at this point. For completeness, a probability integral transform pertaining to a ...

Question 12

Consider the following transform $g\mapsto f$, where $$ f(x) = \int_{0}^{\infty} \exp\left\{-\int_0^t\int_{x-\tau}^{x+\tau} g(y) \, dy \, d\tau\right\} \, dt $$ Assume $f,g>0$ are $C^\infty(\mathbb{...

Question 13

I've been reading this paper and matrix statistics and have run into a derivative that has completely stumped me. Can someone explain to me how these derivatives are being calculated? I've tried ...

Question 14

I have a function $$ f(x) = \int_{0}^{\infty} \exp\left\{-\iint_{R(x,y)} g(x',y') \, dx' \, dy'\right\} \, dy, $$ where $R(x,y) \subset \mathbb{R}^2$ is a given family of closed regions in the $(x', y'...

Question 15

For a function $f(t)$ of a single variable, I know the following property of the Dirac delta function: $\int_{-\infty}^\infty f(t) \delta(t-a) \, dt = f(a).$ But, what if we have a function of two or ...

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Questions tagged [integral-transforms]

On consequences of Titchmarsh theorem: can the analytical extension of the complex refractive index cross the negative real axis?

On Chebychev type inequality for Lp-first oder difference

What is the practical use of Laplace inversion formula?

Can the derivative be considered as a convolution / integral transform? [duplicate]

Is the integral exponential of a convolution injective?

Integral transform with Gaussian kernel

Determining the multiplication property of the Fourier-Kontorovich-Lebedev transform

How to invert the exponential of a convolution?

How to invert a "tent" convolution?

What is the Kontorovich–Lebedev transform of $K_0\left(\left(r^2+\rho^2+2r\rho\cos\alpha\right)^{\frac{1}{2}}\right)$?

Measure theoretic definition of the probability integral transform

Can we invert this transform?

Derivative of Stieljes Transform

How to infer properties of a function from an integral transform?

Sifting property of Dirac delta function

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