A note on best selection of quasi Descartes systems

Kazuaki Kitahara

doi:10.33993/jnaat382-910

Authors

Kazuaki Kitahara Kwansei Gakuin University, Japan

DOI:

https://doi.org/10.33993/jnaat382-910

Keywords:

Descartes Systems, best approximations

Abstract views: 201

Abstract

Let \(\{u_0, \dots, u_n\}\) be a quasi Descartes system of \(C[a, b]\) and \(p\) a positive number with \(1 < p < \infty\) or \(\infty\). In this note, we search for an \(m (\leqq n)\) dimensional subspace that possesses the least distance from \(u_n\) among all \(m (\leqq n)\) dimensional subspaces of \({\rm Span}\{ u_0, \ldots, u_{n-1}\}\).

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References

Borosh, I., Chui, C. K. and Smith, P. W., Best uniform approximation from a collection of subspaces, Math. Z., 156, pp. 13-18, 1977, https://doi.org/10.1007/bf01215125 DOI: https://doi.org/10.1007/BF01215125

Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities, Springer, New York, 1995, https://doi.org/10.1007/978-1-4612-0793-1 DOI: https://doi.org/10.1007/978-1-4612-0793-1

Karlin, S. and Studden, W. J., Tchebycheff Systems: With Applications in Analysis and Statistics, Wiley-Interscience Publ., New York, 1966, https://doi.org/10.2307/2283712 DOI: https://doi.org/10.2307/2283712

Kitahara, K., On Tchebysheff systems, Proc. Amer. Math. Soc., 105, pp. 412-418, 1989, https://doi.org/10.1090/s0002-9939-1989-0943794-4 DOI: https://doi.org/10.1090/S0002-9939-1989-0943794-4

Kitahara, K., Spaces of Approximating Functions with Haar-like Conditions, Lecture Notes in Mathematics, 1576, Springer, 1994, https://doi.org/10.1006/jath.1995.1101 DOI: https://doi.org/10.1007/BFb0091385

Kitahara, K., Some results related to Descartes' rule of signs, East J. Approx., 14, pp. 467-484, 2008.

Pinkus, A. and Ziegler, Z., Interlacing properties of the zeros of the error functions in best Lp approximations, J. Approx. Theory, 27, pp. 1-18, 1979, https://doi.org/10.1016/0021-9045(79)90093-5 DOI: https://doi.org/10.1016/0021-9045(79)90093-5

Shi, Y. G., The Chebyshev Theory of a variation of L approximation, J. Approx. Theory, 67, pp. 239-251, 1991, https://doi.org/10.1016/0021-9045(91)90001-q DOI: https://doi.org/10.1016/0021-9045(91)90001-Q

Smith, P. W., An improvement theorem for Descartes systems, Proc. Amer. Math. Soc., 70, pp. 26-30, 1978, https://doi.org/10.1090/s0002-9939-1978-0467118-7 DOI: https://doi.org/10.1090/S0002-9939-1978-0467118-7

Zalik, R., Čebyšev and Weak Čebyšev Systems in Total Positivity and Its Applications, M. Gasca and C. A. Micchelli (eds.), Kluwer Academic Publishers, Dordrecht, pp. 301-332, 1996, https://doi.org/10.1007/978-94-015-8674-0_15 DOI: https://doi.org/10.1007/978-94-015-8674-0_15

Zielke, R., Discontinuous Čebyšev Systems, Lecture Notes in Mathematics, 707, Springer, 1979, https://doi.org/10.1007/bfb0071032 DOI: https://doi.org/10.1007/BFb0071032