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, 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

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... e simplification of various known proofs in the area of constrained approximation by polynomials and splines.