## Han Feng, Christian Krattenthaler and
Yuan Xu

# Best polynomial approximation on the triangle

### (15 pages)

**Abstract.**
Let *E*_{n}(*f*)_{α,β,γ} denote the error of best approximation by polynomials of degree at most *n* in the space
*L*^{2}(*ω*_{α,β,γ}) on the triangle {(*x*,*y*): *x*, *y* >= 0, *x*+*y* <= 1}, where *ω*_{α,β,γ}(*x*,*y*) :=
*x*^{α} *y*^{β} (1-*x*-*y*)^{γ}
for *α,β,γ* > -1. Our main result gives a sharp estimate of *E*_{n}(*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
*E*_{n}(*f*)_{α,β,γ}
by a weighted *K*-functional.

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