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.

Integral transformations have been successfully used for almost two centuries in solving many problems in applied mathematics, mathematical physics, and engineering science. Historically, the origin of the integral transforms including the Laplace and Fourier transforms can be traced back to celebrated work of P. S. Laplace (1749–1827) on probability theory in the 1780s and to monumental treatise of Joseph Fourier (1768–1830) on La Théorie Analytique de la Chaleur published in 1822.

The integral transform of a function $~f(x)~$ defined in $~a ≤ x ≤ b~$ is denoted by $~\mathcal I \{f(x)\} = F(p)~$, and defined by $$~\mathcal I \{f(x)\} = F(p)~=\int_a^bf(x)~K(x,p)~dx$$where $~K(x,t)~$ is called the integral kernel of the transform. The operator $~\mathcal I~$ is usually called an integral transform operator or simply an integral transformation. The transform function $~F(p)~$ is often referred to as the image of the given object function $~f(x)~$ , and $~p~$ is called the transform variable.

Similarly, the integral transform of a function of several variables is defined by $$~\mathcal I \{f(x)\} = F(p)~=\int_Sf(x)~K(x,p)~dx$$where $~x=(x_1,~\cdots~,~x_n)~$,$~~p=(p_1,~\cdots~,p_n)~$, and $~S ⊂ \mathbb R^n~$.

A mathematical theory of transformations of this type can be developed by using the properties of Banach spaces. From a mathematical point of view, such a program would be of great interest, but it may not be useful for practical applications.

References:

https://en.wikipedia.org/wiki/Integral_transform

"Integral Transforms and Their Applications" by Lokenath Debnath, Dambaru Bhatta

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Possible bug LaplaceTransform in Mathematica

Let us consider in Mathematica 13.0 on Windows 10/Linux LaplaceTransform[DiracDelta[x - 2]*Exp[-x^2], x, s] E^(-2 (2 + s)) and then InverseLaplaceTransform[%, s, x] DiracDelta[-2 + x]/E^4 I was learned that DiracDelta[x - 2]*Exp[-x^2] should…
user64494
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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 inversion of the other in the fundamental strip $S_g$.…
anonymous
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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 use for? When to use them?
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