Han Feng, Christian Krattenthaler and Yuan Xu

Best polynomial approximation on the triangle

(15 pages)

Abstract. Let En(f)α,β,γ denote the error of best approximation by polynomials of degree at most n in the space L2(ωα,β,γ) on the triangle {(x,y): x, y >= 0, x+y <= 1}, where ωα,β,γ(x,y) := xα yβ (1-x-y)γ for α,β,γ > -1. Our main result gives a sharp estimate of En(f)α,β,γ in terms of the error of best approximation for higher order derivatives of f in appropriate Sobolev spaces. The result also leads to a characterization of En(f)α,β,γ by a weighted K-functional.

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