On some approximation properties of real Hardy spaces \((0<p\leq 1)\) (Q1066028)

In the real Hardy spaces Re \(H_ p\) defined on the one-dimensional torus T or on \({\mathbb{R}}\), \(0<p\leq 1\), the author proves some analogous inequalities of approximation theory known for \(L_ p(T)\), \(1\leq p<\infty\), using atomic decompositions and molecules. One can find e.g., a description of the Re \(H_ p\)-moduli of continuity by a corresponding K'-functional, a Jackson-type inequality for the approximation of u(t)\(\in Re H_ p(T)\) by the partial sums \(P^ m_ nu(t)\) of its Fourier series with respect to the periodic orthonormal spline systems \(F^{(m)}\) \((m=1,2,...)\) introduced by Z. Ciesielski and the approximation by Bochner-Riesz means of the Fourier integral of distributions belonging to Re \(H_ p({\mathbb{R}})\).