List of Integral Transforms

Introduction

This article lists the integral transforms the corpus constructs and inverts. An integral transform is an operation that carries a function to a new function by integrating it against a kernel, and a row below names one transform, records the kernel that defines it and the inversion theorem that recovers the function, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The transforms are grouped by the family that introduces them: the Fourier transforms, on the line, on the circle, on a finite abelian group and on a locally compact abelian group; the Laplace and Mellin transforms of a half-line and of a multiplicative group; the singular and geometric transforms — the Hilbert and Riesz transforms, the Radon, Hankel and wavelet transforms; and the transforms attached to a group representation, among them the Gelfand and spherical transforms. The rows that record the boundary of the theory — a transform that does not converge absolutely, an inversion that needs a summability method, a kernel that fails the admissibility condition — stand beside the transforms as non-examples.

The Fourier Family

Object	Its kernel and its inversion theorem	Introduced in
the Fourier transform on $\mathbb R^n$	kernel $e^{-2\pi i\langle x,\xi\rangle}$; inversion $\hat f \mapsto f$ under $f \in L^1$ with $\hat f \in L^1$, and the $L^2$ form by Plancherel	Fourier Analysis on Euclidean Spaces
the Fourier series on the circle	kernel $e^{in\theta}$; inversion by the convergence of the partial sums, and by the Fejér and Cesàro means	Real Harmonic Analysis; Fourier Analysis on Euclidean Spaces
the Fourier transform on a finite abelian group	the characters as kernel; discrete inversion by the orthogonality relations	Harmonic Analysis on Groups
the Fourier transform on a locally compact abelian group	the characters of $\hat G$ as kernel; inversion and Plancherel for the Haar measure	Harmonic Analysis on Groups
the Fourier transform on a compact group	the matrix coefficients of the irreducible representations as kernel; inversion by the Peter–Weyl theorem	Analysis on Compact Groups
the Fourier transform on the adeles	the additive characters as kernel; inversion and the Poisson summation formula	Adelic Analysis
the Fourier transform on a hypercomplex system	the Fourier kernel of the system; inversion for the complex, quaternion, split-complex and dual-number planes	Harmonic Analysis over Hypercomplex Systems

The Laplace and Mellin Family

Object	Its kernel and its inversion theorem	Introduced in
the Laplace transform	kernel $e^{-st}$; inversion by the Bromwich integral, and the resolvent form $R(\lambda,A) = \int_0^\infty e^{-\lambda t}T(t)\,dt$	Semigroups and Evolution Equations; Split-Complex Harmonic Analysis; Markov Chains and Processes
the Fourier–Stieltjes transform $\hat\mu(\chi) = \int_G\overline{\chi(x)}\,d\mu(x)$	the transform of a finite measure on a group, recovering the classical transform for $G = \mathbb R^n$; inversion by Bochner's theorem and the uniqueness of the transform	Harmonic Analysis on Groups; Noncommutative Harmonic Analysis
the Mellin transform	kernel $x^{s-1}$; inversion on a vertical line, and its identification with the Fourier transform in logarithmic coordinates	Complex Harmonic Analysis
the Mellin convolution and its inversion	the multiplicative convolution and its transform rule $\mathcal M(f \star g) = \mathcal M f\,\mathcal M g$	Complex Harmonic Analysis

The Singular and Geometric Transforms

Object	Its kernel and its inversion theorem	Introduced in
the Hilbert transform on the line	kernel $1/(\pi(x-y))$; inversion by $H^2 = -I$ and the boundedness on $L^p$, $1	Fourier Analysis on Euclidean Spaces; Real Harmonic Analysis
the Riesz transforms $R_j$	symbols $-i\xi_j/\lvert\xi\rvert$; the model singular integrals of the Calderón–Zygmund theory, of which the Hilbert transform is the case $n=1$	Fourier Analysis on Euclidean Spaces; Real Harmonic Analysis
the Radon transform	integration over hyperplanes; inversion by the Fourier slice theorem and filtered back-projection	Complex Harmonic Analysis
the Hankel transform	kernel $J_\nu(\rho r)r$; inversion by the Hankel inversion theorem, the radial part of the Fourier transform	Octonion Harmonic Analysis
the wavelet transform	kernel $\psi((t-b)/a)/a$; inversion by the admissibility condition $\int \lvert\hat\psi(\omega)\rvert^2/\lvert\omega\rvert\,d\omega < \infty$	Complex Harmonic Analysis
the wavelet transform on a hypercomplex system	the same kernel with the system's scalings; inversion by the resolution of the identity	Biquaternion Continuous Harmonic Analysis

Transforms Attached to a Group or an Algebra

Object	Its kernel and its inversion theorem	Introduced in
the Gelfand transform	the characters of a commutative Banach algebra as kernel; inversion by the Gelfand representation, an isomorphism for a commutative C*-algebra	Harmonic Analysis on Groups; Holomorphic Functional Calculus
the spherical transform on a symmetric space	the spherical functions as kernel; inversion by the Plancherel theorem for the spherical transform	The Plancherel Theorem
the Fourier transform of a representation	the matrix coefficients as kernel; the Plancherel decomposition of a unitary representation	Noncommutative Harmonic Analysis
the character of a representation	the trace of the representation, the degenerate kernel that inverts the decomposition	Analysis on Compact Groups
the Wigner transform of a state	the matrix-element kernel $W(x,\xi)$ on phase space; its semiclassical defect measure is positive, carried by the energy surface and invariant under the Hamiltonian flow; inversion is the Weyl quantisation $\operatorname{Op}^w_\hbar$	Semiclassical Analysis; Pseudodifferential Operators

Transforms That Do Not Satisfy Their Inversion Theorem

Object	The failure	Introduced in
the Fourier transform of a function outside $L^1 \cup L^2$	the integral does not converge; only the tempered-distribution definition applies	Fourier Analysis on Euclidean Spaces; Distributions and Fundamental Solutions
the Fourier transform of a function with a singularity	the integral converges only as an improper integral or a principal value; otherwise the transform is defined as a tempered distribution	Fourier Analysis on Euclidean Spaces; Distributions and Fundamental Solutions
the Laplace transform with a divergent defining integral	the transform exists only in the half-plane $\operatorname{Re}\lambda > \omega_0$ of convergence of the defining integral, and the inversion is confined to it	Semigroups and Evolution Equations
a wavelet whose admissibility integral diverges	the transform has no inversion; the admissibility condition is the necessary hypothesis	Complex Harmonic Analysis
the Radon transform of a function that is not summable over the lines	the transform is not defined without a decay hypothesis on the measured set	Complex Harmonic Analysis
the Hilbert transform on $L^1$	the strong-type bound fails at $p = 1$; only the weak-type $(1,1)$ bound holds	Real Harmonic Analysis

Summary

This list gathers the integral transforms of the corpus: the Fourier transforms of the line, the circle, the finite, locally compact and compact groups and the adeles; the Laplace, Fourier–Stieltjes and Mellin transforms of the half-line and the multiplicative group; the singular and geometric transforms of Hilbert, Riesz, Radon, Hankel and wavelet type; and the Gelfand, spherical, representation-theoretic and Wigner transforms attached to a group or an algebra. Each row records the kernel and the inversion theorem, and the closing table records the cases in which the defining integral or the inversion fails.

Summary of Notation

The objects are named rather than denoted; the symbols in the tables are collected here.

Symbol	Meaning
$\hat f$, $\mathcal F f$	the Fourier transform of $f$
$\hat\mu$	the Fourier–Stieltjes transform of a measure
$\varphi_X$, $\chi$	a character, and a characteristic function in the probabilistic use
$R(\lambda,A)$	the resolvent, the Laplace transform of a semigroup
$\mathcal M f$, $f \star g$	the Mellin transform and the multiplicative convolution
$Hf$, $R_j$	the Hilbert transform and the Riesz transforms
$Rf$	the Radon transform
$W_\psi f$	the wavelet transform
$J_\nu$	the Bessel function of order $\nu$, the Hankel kernel