An Operator-Based Framework for Integral Transforms

A framework that builds integral transforms from the Maclaurin coefficients of the input, letting each coefficient select a derivative of the kernel 1/s. The construction reproduces the Laplace transform, leads to Fourier and Mellin-type transforms through the choice of rank, and extends to general kernels, convolution, and fractional integration.

Transforms generated from one kernel

Textbooks usually introduce the Laplace, Fourier, and Mellin transforms separately, each with its own integral definition. Chapter 2 starts from the Maclaurin coefficients of the input instead. For $f(t)=\sum a_n t^n$, the book defines an operator-based transform in which each coefficient selects a derivative of the rational kernel $1/s$:

The power $t^n$ selects the derivative rank $n$, and writing $(-D_s)^n$ absorbs the sign $(-1)^n$. Using $n!/s^{n+1}=\int_0^\infty t^n e^{-st}\,dt$ and exchanging sum and integral, the series becomes the classical Laplace integral. The book states this as a result it calls Theorem 2.1.2 and presents it as an explanation of where the integral form comes from, not as a replacement for it.

Rank as the organizing parameter

Changing how the rank is encoded produces the other transforms. A bilateral extension with the symmetric damping $e^{-\sigma|t|}$ and $s=\sigma+i\omega$ pairs the kernels $1/s$ and $1/\bar s$, and recovers the Fourier transform in the distributional limit $\sigma\to0^+$. A Gamma-weighted kernel shifts the rank to $\alpha=n+\rho-1$:

Letting $s\to0^+$ in the integral, where that limit exists, gives the classical Mellin transform $\int_0^\infty f(t)\,t^{\rho-1}\,dt$. The book summarizes the pattern as $(-\partial_s)^{\alpha}(1/s)=\Gamma(\alpha+1)/s^{\alpha+1}$, with rank $\alpha=n$ for Laplace and Fourier, $\alpha=n+\rho-1$ for Mellin, and $\alpha=n+\nu$ for Hankel.

Seeing several transforms as one mechanism makes their shared rules visible. Shift, differentiation, and convolution properties become statements about how the kernel is shifted, differentiated, or divided. The chapter uses this to solve constant-coefficient linear ODEs with the operator transform alone, deriving rules such as $T\{y'\}(s)=sY(s)-y(0)$ from series manipulations.

General kernels, duality, and convolution

Section 2.5 replaces $1/s$ by a general generated kernel $G(s)=\mathcal{L}\{g\}(s)$. Because $(-D_s)^nG(s)=\int_0^\infty t^n e^{-st}g(t)\,dt$, the same coefficient series now produces a weighted transform:

Dividing the kernel by $s^k$ injects $k$-fold integration, which Cauchy's formula for repeated integration collapses to a single convolution. Replacing $k$ by a positive real $\alpha$ gives the Riemann–Liouville fractional integral, and dividing by a polynomial $P(s)$ gives a convolution with $\mathcal{L}^{-1}\{1/P\}$. The chapter closes with a kernel-based way to read off partial-fraction coefficients.

How it connects to the other areas

The framework grows out of the decay-kernel identity in higher-order derivatives. Mathematical geometry then reads the resulting Laplace pairs in polar and hyperbolic coordinates, root geometry supplies closed Taylor coefficients for rational inputs, and the Matrix Boundary Method recovers the Laplace transform in resolvent form $u^{\mathsf T}(sI-\Omega)^{-1}\Phi(0)$ for functions closed under differentiation.