Jackson’s theorem in Q p spaces

Abstract

A Jackson type inequality in Qp spaces is established, i.e., for any f(z) = Σj=0∞ajzj ∈ Qp, 0 ⩽ p 1, and k − 1 ∈ ℕ, $$ \left\| {f(z) - \frac{{\Gamma (k)}} {{\Gamma (k + a)}}\sum\limits_{j = 0}^{k - 1} {\frac{{\Gamma (k - j + a)}} {{\Gamma (k - j)}}a_j z^j } } \right\|_{Q_p } \leqslant C(a)\omega \left( {\frac{1} {k},f,Q_p } \right), $$ where ω(1/k, f, Qp) is the modulus of continuity in Qp spaces and C(a) is an absolute constant depending only on the parameter a.