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Transactions of the American Mathematical Society

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On the degree of convergence of piecewise polynomial approximation on optimal meshes

Author: H. G. Burchard
Journal: Trans. Amer. Math. Soc. 234 (1977), 531-559
MSC: Primary 41A15
MathSciNet review: 0481758
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Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces $ {\mathbf{N}}_0^{p,n}$ (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in $ {\mathbf{N}}_0^{p,n}$ we define a derivative $ {D^{n,\sigma }}f$ in $ {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}$, which agrees with $ {D^n}f$ for smooth f, and prove several properties of $ {D^{n,\sigma }}$.

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