Integral Transform

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tags
Math, DiffEQ
sources
Elementary Differential Equations and Boundary Value Problems, https://en.wikipedia.org/wiki/Integral_transform

An integral transform maps a function \(f(\cdot)\) from its original Function Space to another Function Space via integration.

All integral transforms take the form:

\[ F(s) = (Tf)(s) = \int_{\alpha}^{\beta} k(s, t) f(t) dt \]

Where \(k(s,t)\) is called the kernel function, and \(\alpha < \beta\)

Some transforms have associated “inverse transforms” for invertible kernels

\[ f(t) = \int_{s_1}^{s_2} k^{-1}(s, t) F(s) ds \]

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