Jackson’s theorem in Q _p spaces

Abstract

A Jackson type inequality in Q _p spaces is established, i.e., for any f(z) = Σ ^∞_j=0 a _j z _j ∈ Q _p , 0 ⩽ p < ∞, a > 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, Q _p ) is the modulus of continuity in Q _p spaces and C(a) is an absolute constant depending only on the parameter a.