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| Makkovik–Pacquet transform | |
|---|---|
| Name | Makkovik–Pacquet transform |
| Introduced | 1983 |
| Area | Integral transforms |
| Field | Applied mathematics |
Makkovik–Pacquet transform. The Makkovik–Pacquet transform is an integral transform used in analytical problems in mathematical physics and signal analysis, combining features of Laplace, Fourier, and Mellin techniques. It serves as a tool in solving boundary value problems and spectral decomposition tasks arising in contexts connected to the works of Joseph Fourier, Pierre-Simon Laplace, Ralph Mellin, John von Neumann, and Norbert Wiener. Its development influenced research at institutions such as the Institute for Advanced Study, the Courant Institute of Mathematical Sciences, and the Max Planck Society.
The transform was introduced to bridge methods from Fourier transform, Laplace transform, and Mellin transform approaches for problems motivated by studies at Harvard University, Princeton University, University of Cambridge, Massachusetts Institute of Technology, and Stanford University. Researchers comparing techniques at the Royal Society, the American Mathematical Society, and the European Mathematical Society found it useful alongside classical tools such as the Z-transform, the Hankel transform, and the Radon transform. It has been applied in publications appearing in journals like the Annals of Mathematics, Communications on Pure and Applied Mathematics, and the Journal of Differential Equations.
The Makkovik–Pacquet transform maps a function f defined on an appropriate domain to an image F via an integral kernel K structured from combinations of kernels used by Fourier transform, Laplace transform, and Mellin transform. For suitable functions studied at the American Institute of Mathematics, the definition resembles integral formulas seen in works from G. H. Hardy, E. T. Whittaker, Salem, Norbert Wiener, and Andrey Kolmogorov. The kernel incorporates oscillatory and decaying factors related to constructions used by Ludwig Boltzmann, Hermann Weyl, David Hilbert, Stefan Banach, and John Littlewood.
Key properties include linearity, invertibility under conditions analogous to the Paley–Wiener theorem, and mapping theorems reminiscent of results by Plancherel, Parseval, Riemann, Weierstrass, and Cauchy. Unitarity results have been proved drawing on techniques from Harald Bohr, Frigyes Riesz, Marshall Stone, Norbert Wiener, and von Neumann. Spectral decomposition theorems relate to analyses by Atle Selberg, I. M. Gelfand, Israel Gelfand, L. D. Faddeev, and Mark Kac in spectral theory contexts at institutions including Columbia University and the Steklov Institute of Mathematics.
Examples include solving heat and wave equations analogous to problems addressed by Joseph Fourier, Sophie Germain, Bernhard Riemann, George Green, and Lord Kelvin in potential theory. Applications appear in inverse problems studied by Paul Erdős, André Weil, John Nash, Andrew Wiles, and Terence Tao in contexts of tomography, geophysics, and scattering theory linked to work at Lawrence Livermore National Laboratory, CERN, and National Aeronautics and Space Administration. The transform aids in time-frequency analysis alongside contributions by Dennis Gabor, Kurt Gödel, Claude Shannon, Richard Hamming, and Norbert Wiener.
Numerical implementations draw on quadrature schemes from literature at SIAM, IEEE, and ACM, and on fast algorithms inspired by the Fast Fourier Transform developed by James Cooley and John Tukey. Computational stability analyses reference techniques used by Alan Turing, John Backus, Grace Hopper, Donald Knuth, and Andrey Markov. Software implementations have been incorporated in libraries maintained by NumPy, SciPy, GNU Scientific Library, MATLAB, and tools developed at Los Alamos National Laboratory and Argonne National Laboratory.
The transform was named to honor contributions from researchers associated with expeditions and collaborative projects involving institutions such as Memorial University of Newfoundland, University of Toronto, McGill University, Université Laval, and Dalhousie University. Historical accounts connect development phases to conferences held under the auspices of the International Mathematical Union, the Royal Society of Canada, and the Canadian Mathematical Society, and to workshops where participants included figures from Princeton University, Yale University, University of Oxford, University of Edinburgh, and McMaster University.
Generalizations extend to families resembling the Laplace transform, Fourier transform, Mellin transform, Hankel transform, and Radon transform, and link to operator-theoretic frameworks developed by John von Neumann, H. Weyl, Issai Schur, Emmy Noether, and Sophus Lie. Extensions have been studied in stochastic analysis contexts related to work by Kiyoshi Itô, Paul Lévy, Kolmogorov, Andrey Kolmogorov, and W. Feller and in noncommutative geometry inspired by Alain Connes, Michael Atiyah, Isadore Singer, and Jean-Pierre Serre.