Hans Munthe-Kaas

Title: On Multivariate Chebyshev Polynomials and spectral approximation 
theory for triangles and simplexes.

Abstract: The importance of (classical) Chebyshev polynomials in
computational mathematics is summarized by the quotation:
"The Chebyshev polynomials are everywhere dense in numerical analysis."
Important reasons for this is their optimal approximation properties, 
and availability of FFT-based fast algorithms.
Koornwinder (1974), and later Eier & Lidl (1982), Hoffman & Withers (1988) 
did generalize univariate Chebyshev polynomials to multivariate families 
of polynomials. The construction is based on affine Weyl groups, yielding 
caledoscopic mirror groups acting on R^n.
However, the mathematical literature on the multivariate versions consists 
of just a few isolated singular points, rather than being 'everywhere dense', 
and within numerical and computational mathematics domain, they are almost 
absent.

Our interest in these polynomials originates from the goal of
developing spectral (element) methods and Clenshaw-Curtis type
quadratures based on triangles and simplexes, as well as exploiting
discrete domain symmetries in computational algorithms for
differential equations. The multivariate Chebyshev families share the
excellent approximation properties of the univariate case, they allow
for FFT-based fast algorithms, and they live on domains that are
related to triangles and simplexes. In this talk we will present
recent developments, from new theoretical results to computational
algorithms and development of software libraries.

Joint work with Brett Ryland.