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JuliaApproximation / FastTransforms.jl

:rocket: Julia package for orthogonal polynomial transforms :snowboarder:
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Add oscillatory integral quadratures #42

Open MikaelSlevinsky opened 6 years ago

MikaelSlevinsky commented 6 years ago

According to R. Piessens and M. Branders, On the computation of Fourier transforms of singular functions, J. Comp. Appl. Math., 43:159-169, 1992, the modified Chebyshev moments of the complex exponential (multiplied by algebraic endpoint singularities) satisfy a many-term recurrence relation that is neither stable in the forward nor backward directions. How to proceed?

Combination of moment-based quadrature with fast orthogonal polynomial transforms to change bases is now irresistible: for example, the modified Legendre moments of the complex exponential are proportional to spherical Bessel functions that satisfy a three-term recurrence relation with well-known stability properties. By changing bases, we would simplify the recurrence and secure the stability by design.

dlfivefifty commented 6 years ago

Should probably go here

https://github.com/JuliaApproximation/OscillatoryIntegrals.jl

daanhb commented 6 years ago

For reference, see also https://github.com/AndrewGibbs/NSDpackage (this is a Matlab package and not related to moment-based quadrature)