On equivalence of moduli of smoothness
It is known that if \(f \in W^k_p\), then \[ \omega_m(f,t)_p \leq t \omega_{m-1}(f',t)_p \leq \cdots. \] The inverse statement with constants independent of \(f\) is not true in general. \textit{Y. Hu} and \textit{X. Yu} [SIAM J. Numer. Anal. 32, No. 5, 1428-1435 (1995; Zbl 0841.41007)] proved that the inverse holds true for splines \(S\) with equally spaced knots; thus , \[ \omega_m(S,t)_p \sim t \omega_{m-1}(S',t)_p \sim t^2 \omega_{m-2}(f'',t)_p \cdots \] In this paper the authors extend Hu and Yu's results to splines with any given knot sequence, and further to principal shift--invariant spaces and wavelets under certain conditions. Some applications are also given.
scientific article; zbMATH DE number 1278845
scientific article