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Quantitative inheritance properties for simultaneous approximation by tensor product operators II: Applications

“…For the case of d dimensions see, e.g., [29], [33] where, however, no assertions concerning simultaneous approximation were made. This paper is a continuation of our recent paper [3] and focusses on applications. A more detailed presentation including full proofs was given in our technical report [2].…”

“…In this note we will first summarize how quantitative properties of certain univariate operators are inherited by the product of their parametric extensions, and this in a form as simple as possible. Details can be found in [2] and [3]. In order to illustrate what is meant by this we cite the following special case (for n = 1) of Theorem 4.1 in the article [31] on generalized n-th order blending as Theorem 1.…”

“…Proof. Can be found in [2], [3]. In the case of discretely defined operators we are in a much more convenient situation: it suffices to only consider one fixed partial of order (p, q) and its univariate predecessors.…”

“…The following result is similar to Theorem 2, but makes different assumptions for the univariate building blocks. Before discussing these further we state it as Theorem 4 (see Theorem 6 in [3]). Let the operators L and M be given such that…”

“…(iii) Both Theorems 2 and 4 are proved under the assumptions that information is available for the derivatives of order 0 ≤ k ≤ p ′ and 0 ≤ l ≤ q ′ . This is needed in order to apply Lemma 2 in [3]. If the operators L and M are discretely defined (like, for example, the Bernstein operators), then Theorem 3 is also applicable and a more efficient tool.…”