Bernstein-Durrmeyer polynomials on a simplex (Q1185259)

The approximation behaviour of the Bernstein-Durrmeyer polynomials \(V_ n(f;x)\) of functions \(f\) on a simplex \(S\) is studied. The space of Lebesgue measurable functions \(f\) on \(S\) for which the norm \(\| f\|^ p_ p=\int| f|^ p\) is finite is denoted by \(L^ p(S)\). A saturation behavior of \(V_ n\) is shown. Conditions are found for \(f\) to belong to the saturation class, i.e. \(\| V_ nf-f\|_ p=O\left({1\over n+1}\right)\). The upper bound of \(\| V_ nf- f\|_ p\) is given. It is shown that the saturation class of \(D^ 2S\) \((D^ p(S)\) is a subspaces of \(L^ p(S))\) is equivalent to the Sobolev space. A constant exists which limits \(D^ 2_ if\). The approximation behavior of \(\{V_ n\}^ \infty_{n=0}\) on \(L^ 2S\) is completely described by the Peetre \(K\)-modulus.