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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What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular frequency k, then it’s going to be something like…
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Inverse of integral transform $f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x$

Given $g(x)$ defined for positive reals, say $f(s)$ is defined as below $$f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x.$$ Is there a relationship to named integral transforms, or a generic approach to obtain $g$ from $f$? For instance, we can…
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What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition. Are there…
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Are Laplace Transforms a Special Case of Fourier Transforms?

A Laplace Transform is based on the integral: $F(\xi) = \int_0^{\infty} f(x) e^ {-\xi x}\,dx.$ In a roundabout way, a Fourier transform can get to $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx,$ Also, they both seem to use…
Tom Au
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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 confusing doing this using fourer transforms. I took the…
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A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation $$\frac{\partial}{\partial z}\mathrm F(z,\alpha)=\mathrm…
Pedro
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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 $$ \int_{0}^{\infty}dx\psi(x+1)x^{s-1}=\frac{-\pi}{\sin (\pi…
Jose Garcia
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Extending the domains of densely defined bounded integral transforms on $L^2(\Bbb R)$

This is a question I've contemplated for quite some time since it's pretty closely related to Fourier theory (particularly choosing the "right" space to define the Fourier transform on). However I've never been able to come up with anything…
Cameron Williams
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What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group structures by translating by group elements. The…
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Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma x)\frac{\mathrm{d}x}{x}= …
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What are the "right" spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace transform is at least bijective?
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Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and no longer relevant definitions from the problem…
nullUser
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Difficult Laplace Transform Type of Integral

Good afternoon. I have the following integral that I need help integrating; $$ \mathrm{F}\left(x\right) = \int_{0}^{\infty}\mathrm{e}^{s\left(j - 1/x\right)}\, \left[\mathrm{T}_{N}\left(s\right)\right]^{k - j}\,\mathrm{d}s $$ where…
Eleven-Eleven
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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 Feynman's Trick with Glasser's Master Theorom but I'm…
user150203
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Double integral with Hankel transform

Let's say we have a double integral in the following form: $$I=\int_0^\infty \int_0^\infty f(x) g(y) J_0(xy) x y dx dy $$ Using the definition of the Hankel transform, we can write: $$I=\int_0^\infty F(y) g(y) y dy=\int_0^\infty G(x) f(x) x…
Yuriy S
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