# Questions tagged [integral-transforms]

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 transforms.

524 questions
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### Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
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### Laplace Transform of an integral function of a convolution

Making suitable assumptions wherever necessary, what is the Laplace Transform $\mathcal{L}(S(t))$ where $S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$. I tried using the Double ...
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### Laplace transforms with Fresnel(?) integrals

I've come into contact with this two part question, and the latter I'm not too sure how to go about; at least to me upon researching, I can't find anything remotely similar to what I've been asked. ...
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### What is purpose of wavelet scaling function and how is it derived for e.g Haar wavelet or Dabuchies wavelet?

Scaling function is also called father wavelet. I understand concept of mother wavelet but not father wavelet. In the continuous wavelet transform there is no concept of scaling function but only when ...
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### Why is the Fourier transform called a 'transform', and not a 'transformation'?

Why are the Fourier transform, Laplace Transform, etc called transforms, and not transformations? This is about linguistics or terminology in mathematics. I feel there should be a reason why the word ...
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### Is there a name for the square of a function plus the square of its Hilbert transform?

Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance ...
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### How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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### Boundary to boundary transformation of an integral

In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the ...
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### Binomial sum for an arbitrary function

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary ...
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### Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$\int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k,$$ where $J_0(kr)$ is the Bessel function of ...
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### How can the following be transformed in to a sum of complete elliptic integrals of the first and second kind

I have the following, that I known from a numerical implementation of the problem by a third party should be able to be transformed in to elliptic integrals of the first and second kind however I can'...
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### Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$

After weeks of going back and forth I've been able to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$$ To solve this I employ ...
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### Fourier and Mellin transforms of Hilbert Transform

I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications". If we define the Hilbert transform on the ...
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### Reference Request: n-dimensional Laplace Transform

I am looking for a reference, where the conditions for the existence of the n-dimensional Laplace transform are proven, i.e. when the laplace transform \begin{equation} F(\lambda_1, ..., \lambda_n) = ...
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### About $F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx$

We know for $u > 1$ $$\int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right)$$ Usually this is shown by using differentiation under the ...
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### Kernel of Hankel Transform

I need to solve a cylindrical diffusion problem that is defined in $[1,\infty]$. I would like to use Hankel Transform that has is defined on $[0,\infty]$. So in order to apply Hankel transform in my ...
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### Building an Integral Transform from an Orthonormal Basis on the L2 Circle

Background It is well known that the orthonormal basis for $L_{2}[-\pi, \pi]$ is $\Omega = \{ e^{-jmt} \}_{m \in \mathbb{Z}}$. We extend this to $L_{2}(\mathbb{R})$ via the Fourier Transform, which ...
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### Which transform makes correllation a multiplication?

In Fourier analysis, a central theorem for the Fourier Transform states: $$\mathcal F\{(f*g)(t)\}(\omega)=\mathcal F \{f(t)\}(\omega)\cdot \mathcal F\{g(t)\}(\omega)$$ In other words, convolution ...
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### Asymptotics of Hilbert transform from asymptotics of original function

Suppose we have a locally integrable function $f: \mathbb R^+ \to \mathbb R$ and we consider the 'Hilbert transformed' function $$h(t) := \int_0^\infty \frac{f(\tau)}{t -\tau} \mathrm d \tau.$$ We ...
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### Fix points of integral transfroms

After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type ...
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### Conditions for uniqueness of a Mellin transform

Let $f(x)$ and $F(s)$ be a Mellin pair, such that one is the Mellin inversion of the other in the fundamental strip $S_f$. Let $g(x)$ and $G(s)$ be a Mellin pair, such that one is the Mellin ...
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### Laplace transform of a “heat kernel”

This question is closely releted to this question: How do we solve the laplace transform of the Heat Kernel? Let $A>0$ and $$f(t) = \frac{A^2}{2\sqrt{\pi}t^\frac{3}{2}}e^{-\frac{A^2}{4t}}$$ ...
The Question: Solve the Heat Equation (for $u = u(x,t)$) $$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} \qquad u(x,0)=T(x)$$ by applying the Fourier Transform in the $x$ ...