Integral Transformers

Integral transformers are kind of an higher order functions. We give another function with $t$ domain as input and we get a new function with a new $u$ domain as the output

Their Signature Definition

F(u) = T\{f(t)\} = \int_{a}^{b} f(t) \cdot K(t, u) dt

$K(t,u)$ known as "Kernel of the transformation"
The choice of kernel determines which domain you are moving into and what properties are preserved or simplified.
The domain will be transformed from $t$ to $u$

Important

In Intergral Transformers, the Kernel function is Independent from the transformed function. This means if you replace $f(t)$ with $f(at)$ , the $K(t,u)$ remains as it is. It will NOT be $K(at,u)$ I.e. The input is the $f(t)$ as a whole function, not $t$ variable.

Common Known transform types

Name	Kernel	What it does
Laplace Transform	$K(t,s) = e^{-st}$	convert a function of time, $f(t)$ , into a function of a complex frequency variable, $s$ .
Fourier Transform	$K(t, \omega) = e^{-j\omega t}$	It maps time to pure imaginary frequencies (ideal for steady-state waves and signal processing).
Mellin Transform	$K(x, s) = x^{s-1}$	It maps functions into a domain optimized for scaling operations, heavily used in computer science for analyzing the complexity of algorithms and in number theory.
Hankel Transform	Bessel functions as the kernel $K(r, k) = r J_n(kr)$	maps spatial coordinates into angular frequency spaces, which is incredibly useful for solving differential equations with circular or cylindrical symmetry (like vibrations of a drumhead or heat distribution in a pipe).

Integral Transformers

Their Signature Definition

Common Known transform types

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