# Questions tagged [integral-transforms]

For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.

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### Is there a continuum-dimensional analog of complex numbers?

Consider the set of functions $f:\mathbb{R}\to\mathbb{R}$. This set is similar to the set $\mathbb{R}^n$ with Hadamard (element-wise) multiplication: ${a}{b}={a_n b_n}$, similar to $fg=f(x)g(x)$. Now, ...
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### The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
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### Most general space for the Wigner-Weyl transform

The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...
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### Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
1 vote
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### Convergence of oscillatory integrals

I'm considering integrals of the (Hilbert transform) type $$p.v.\int_{-\infty}^\infty\frac{f(r)}{r}\,dr$$ where $f(r)$ is periodic, say, with period $2\pi$. I'm assuming very little regularity on $f$. ...
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### Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation

Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
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### Inverse transform of $\int K(s,t) \rho(s) dS$

Assume that $V$ is a n-dimensional surface, or the disjoint union such surfaces. Let $K(s,t)$ be a function $V^2\to \mathbb R$. For the sake of simplicity, it can be assumed that $K$ is continuous, ...
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### Cauchy's integral formula and essential singularities

Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$ where $\gamma_{z_0}$ is a small circle around $z_0$. By ...
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### Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$?

I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$ where $a, b, c, d > 0$.
1 vote
I'm looking for a solution to $$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx.$$ Mathematica gives me the following solution, but I'd like to know/understand the steps involved in finding it. $$\int ... 3 votes 0 answers 65 views ### Injectivity of the Mellin Transform and discontinuities I am reading Stopple's A Primer of Analytic Number Theory. On page 234, the Mellin Transform \mathcal{M} f of a function f is defined as$$\mathcal{M} f (s) = \int_1^{\infty} f(x) x^{-s - 1} dx$$... 5 votes 2 answers 130 views ### Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals? Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$... 1 vote 1 answer 170 views ### Is there a solution to$\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$? I'm looking for a solution to the following integral. However, it seems it doesn't have a solution. $$\int\limits_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\... 0 votes 0 answers 67 views ### Differential properties of the Radon transform I am looking for a reference for the following fact: Let f \in L_1(\mathbb{R}^{d}) that has compact support and that is also in C^{s}(\mathbb{R}^{d}). We define the Radon transform of f as: \... 4 votes 0 answers 219 views ### A tricky integral equation In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.) Let f:\... 1 vote 0 answers 100 views ### Are Poisson integrals uniquely determined by their radial limits? Let \mu be a complex Borel measure on the unit circle, and suppose its Poisson integral u satisfies \lim_{r\to 1-}u(re^{i\theta})=0 for every \theta. Does it follow that \mu=0? This is of ... 2 votes 0 answers 100 views ### Hilbert's fourth problem and a non-linear integral transform The following nonlinear integral transform takes continuous functions defined on the cylinder \mathbb{R} \times S^1 to C^2 functions defined on the plane \mathbb{R}^2:$$ \mathcal{A}f (x,y) := \... 1 vote 3 answers 205 views ### Solution to$\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$Is there a solution to this integral? $$\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$ where$a > 0$and$d > 0. 0 votes 1 answer 70 views ### Unique zero solution to a difference equation via Laplace transform We want to prove that the unique solution to the following difference equation is the null one: $$au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1% }{2},1)}(x)u(x-\frac{1}{... 0 votes 1 answer 121 views ### Laplace transform injectivity for different values of p Let y\in L^{2}(0,1) and let \widetilde{y} be its extension on (0,\infty ). Assume that there exist p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion , p_{0}\neq ... 0 votes 0 answers 34 views ### Solving a difference functional equation by using Laplace transform Consider the operator T:L^{2}(0,r+1)\longrightarrow L^{2}(0,r+1): \begin{equation*} Tu(x)=:u(x)+a\mathbf{1}_{(0,1)}(x)u(x+r)+b\mathbf{1}_{(1,1+r)}(x)u(x-1),% \text{ }x\in (0,r+1), \end{equation*} ... 1 vote 0 answers 32 views ### Solving an equation containing Laplace transform Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where \mathcal{L} is the ... 1 vote 1 answer 275 views ### What is the integral representation of the exponential function e^{1/t} on (0,\infty)? A function q(x) is said to be completely monotonic on an interval I if q(x) has derivatives of all orders on I and (-1)^{n}q^{(n)}(x)\ge0 for x\in I and n\ge0. See Chapter 1 in the ... 2 votes 1 answer 99 views ### Solving an integral involving a Bessel function, Laguerre function and Gaussian We want to calculate the expectation value \langle q^2\rangle in polar coordinates which gives us the following integral, for integer values of p: \begin{equation}\int_0^\infty dq~q^3 \left(\int_0^... 2 votes 1 answer 196 views ### Is g(v)=\mathbb{E}[f(v+W)] a differentiable function of v when f is continuous and W is multivariate normal? Suppose f is a continuous function on \mathbb{R}^n, and W has a multivariate normal distribution on \mathbb{R}^n. If the expectation$$g(v)=\mathbb{E}[f(v+W)]is defined for all v \in \... 1 vote 0 answers 13 views ### Graph of the Hilbert transform of a function extended on Boehmians If a function is defined on set of Reals, we can draw the graph of function on plane, similarly we can draw the graph of its Hilbert transform. Now when we extend Hilbert transform to the space of ... 3 votes 1 answer 158 views ### Generalization(s) of variation diminishing property to multivariate case Let us first define the variation diminishing property for the Gaussian kernel. Consider a function f: \mathbb{R} \to \mathbb{R} that is sufficiently smooth and define \begin{align} F(x)= \int_{-\... 2 votes 0 answers 26 views ### finite fractional integral transform Several integral transforms are generalized by introducing their fractional counterparts,e.g., fractional Fourier transform is a very popular one. Similarly, their discrete versions are also ... 7 votes 1 answer 298 views ### Explicit isomorphism between L^2(\mathbb{R}^2) and L^2(\mathbb{R})? As Hilbert spaces, L^2(\mathbb{R}^2) and L^2(\mathbb{R}) are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I ... 3 votes 1 answer 155 views ### Fast computation of convolution integral of a gaussian function Given a convolution integral g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$where \varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\... 2 votes 0 answers 62 views ### On a question relating integral equation: I don't know if the following question qualifies as research level. If it isn't, sorry. Set the following terminology: \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c) \alpha_2=\alpha_2(s,x)=s(\tan^{-1}(... 0 votes 0 answers 41 views ### Inversion of integral transforms We have the following theorem statement of this Paper "An inversion formula for a Class of integral transforms, 1975, C.Nasim" page 8. Let \phi(x)= \frac{1}{x} \int_{0}^{x} \psi(t) dt, ... 2 votes 0 answers 63 views ### Reference request for type of specific integral equation in two variable: Consider the following integral equation:$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$Here, K(t,y) is a trigonometric kernel and \phi(t,x) is monotonic wrt x ( for fixed t). I want to find the ... 2 votes 0 answers 35 views ### Derivatives of G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt when h is positive-homogeneous Let h:\mathbb R \to \mathbb R be a continuous which is positive-homogeneous of order p \ge 1, and define G_h:[-1,1] \to \mathbb R by$$ G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt....
Let $z \in C$ and consider the following integral equation: $$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$...