Univariate splines: Equivalence of moduli of smoothness and applications

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any 1 ≤ k ≤ r+1, 0 ≤ m ≤ r − 1, and 1 ≤ p ≤ ∞, the inequality n−νωk−ν(s(ν), n−1)p ∼ ωk(s, n −1)p, 1 ≤ ν ≤ min{k,m + 1}, is satisfied, where s ∈ Cm[−1, 1] is a piecewise polynomial of degree ≤ r on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian–Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.