Pointwise analog of the Jackson approximation theorem
Let \(X=X_Q\) be the set of real--valued, \(2 \pi\)--periodic, continuous functions on \(Q\), let \(E_n(f)_{X_{[x-\pi,x+\pi]}}\) denote the best approximation of \(f(t)\) by trigonometric polynomials of degree at most \(n\), and let \(\omega f( \delta)_X\) denote the modulus of continuity. A typical result is that if \(f \in X\), then there is a \(\delta > 0\) such that \(E_n(f)_{X_{[x-\delta,x+\delta]}} \leq K \omega f( {{\pi}\over {n+1}})_{X_{[x-\pi,x+\pi]}}\). Similar results are presented for Lebesgue integrable functions.
scientific article; zbMATH DE number 5992856