Approximation by polynomials in rearrangement invariant quasi Banach function spaces (Q437729)

The author deals with the approximation properties of certain linear polynomial operators in rearrangement-invariant quasi-Banach function spaces \(X\) with absolutely continuous norm under the assumption that \(X\) is \(p\) convex for some \(p\in(0,1]\). This class of spaces includes all Lebesgue spaces \(L^p\) for \(0<p<\infty\), all Lorentz spaces \(L^{p,q}\) for \(p,q\in(0,\infty)\), all Zygmund spaces \(L^p(\log L)^\alpha\) for \(p\in(0,\infty)\) and \(\alpha\in{\mathbb R}\). He obtains some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.