Weighted moduli of smoothness and spline spaces (Q1288877)

A relationship between the moduli of smoothness on \(I=[0, 1]\) with variable step function and approximation by spline spaces with suitable knots is etablished. The class of moduli of smoothness corresponding to step-weight function \(\varphi(x)\sim x^{\beta(0)}(1-x)^{\beta(1)}\) is considered. The direct and converse results are obtained. It is shown that the orthogonal projections onto the spline spaces under consideration give the best approximation in all the \(L^p(I)\) norms.