Degree of best approximation by trigonometric blending functions. (English) Zbl 0547.41011
The paper deals with the approximation of \(C_{2\pi}^{r,s}\)-functions (2\(\pi\) -periodic, bivariate functions h which have continuous derivatives \(D^{\rho,\sigma}h\), 0\(\leq\rho \leq r\), 0\(\leq\sigma \leq s)\) by spaces \(T_{m-1}\otimes C_{2\pi}+C_{2\pi}\otimes T_{n-1}\) of blending functions; here \(T_ k\) denotes the space of trigonometric polynomials of degree \(\leq k\) and \(C_{2\pi}\) is the vector space of univariate, continuous and 2\(\pi\) -periodic functions. The following approximation theorem is proved: For every \(h\in C_{2\pi}^{r,s}\) there exists an element \(w^*\in T_{m-1}\otimes C_{2\pi}+C_{2\pi}\otimes T_{n-1}\) such that \(\| h-w^*\|_ p\leq K_ rK_ s/m^ rn^ s\| D^{r,s}h\|_ p\) for all 1\(\leq p\leq\infty \), where \(\|.\|_ p\) denotes the \(L_ p\)-norm on \([0,2\pi]^ 2\) and the \(K_{\nu}\) are the so-called Favard constants. In addition, the order of approximation will not be changed if the approximating space is modified within a large scale. To do this, we use a result by M. von Golitschek and E. W. Cheney [Contemp. Math. 21, 125-136 (1983)].
Reviewer: M.G.de Bruin
MSC:
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 42A10 | Trigonometric approximation |
| 41A25 | Rate of convergence, degree of approximation |
| 41A30 | Approximation by other special function classes |
| 41A15 | Spline approximation |
| 41A50 | Best approximation, Chebyshev systems |
| 41A10 | Approximation by polynomials |
Keywords:
blending approximation; estimates of Favard type; \(L_ p\)-approximation; convolution; blending functions; order of approximationReferences:
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