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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