Whitney's theorem for local anisotropic polynomial L_p-approximation, 0

Publication date: 11 December 2017

Abstract: Dinh D~ung and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in

L_{p} (Q)

for

1 l e p l e i n f t y

, where

Q

is a

d

-parallelepiped in

R R^{d}

with sides parallel to the coordinate axes. They considered the error of best approximation of a function

f

by algebraic polynomials of fixed degree at most

r_{i} - 1

in variable

x_{i}, i = 1, . . ., d

. The convergence rate of the approximation error when the size of

Q

going to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable to the case

0 < p < 1

. In the present paper, by a different method we proved this theorem for

0 < p l e i n f t y

.

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