Yudin, V. A. Polynomials of least deviation from zero. (English. Russian original) Zbl 1115.41027 Math. Notes 78, No. 2, 279-284 (2005); translation from Mat. Zametki 78, No. 2, 308-313 (2005). The author studies the problem of approximating a homogeneous polynomial of degree \(n\) on the unit disk \(B\), i.e., a polynomial of the form \[ F(x,y) = \sum_{k=0}^n a_k x^k y^{n-k}, \] by algebraic polynomials of smaller degree, i.e., polynomials of the form \[ P(x,y) = \sum_{k+\ell \leq n-1} x^k y^{\ell}. \] He obtains lower bounds for best approximation in the spaces \(L_p(B)\) for \(1 \leq p \leq \infty\). Reviewer: Richard A. Zalik (Auburn University) Cited in 2 Documents MSC: 41A50 Best approximation, Chebyshev systems 41A63 Multidimensional problems Keywords:algebraic polynomial; polynomial of least deviation from zero; polynomial of best approximation; Chebyshev polynomial; harmonic polynomial; the space \(L_p(B)\) PDFBibTeX XMLCite \textit{V. A. Yudin}, Math. Notes 78, No. 2, 279--284 (2005; Zbl 1115.41027); translation from Mat. Zametki 78, No. 2, 308--313 (2005) Full Text: DOI References: [1] P. L. Chebyshev, Complete Collection of Works [in Russian], Akad. Nauk SSSR, Moscow-Leningrad, 1948. · Zbl 0041.48503 [2] W. B. Gearhart, ”Some Chebyshev approximations by polynomials in two variables,” J. Approx. Theory, 8 (1973), 195–209. · Zbl 0282.41009 [3] T. J. Rivlin and H. S. Shapiro, ”A unified approach to certain problems of approximation and minimization,” J. Soc. Indust. Appl. Math., 9 (1961), 670–699. · Zbl 0111.06103 [4] M. Reimer, Constructive Theory of Multivariate Functions with an Application to Tomography, Bibliographisches Institut, Mannheim, 1990. · Zbl 0717.41009 [5] S. N. Bernshtein, Complete Collection of Works [in Russian], vol. 1, Akad. Nauk SSSR, Moscow, 1952. [6] N. N. Andreev and V. A. Yudin, ”Polynomials of least deviation from zero and cubature formulas of Chebyshev type,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 232 (2001), 45–57. · Zbl 1001.41016 [7] M. Reimer, ”On multivariate polynomials of least deviation from zero on the unit ball,” Math. Z., 153 (1977), 51–58. · Zbl 0334.41009 [8] L. Bos, ”On Kergin interpolation in the disk,” J. Approx. Theory, 37 (1983), 251–261. · Zbl 0533.41001 [9] U. Majer, ”On best approximation of the monomials on the unit ball of \(\mathbb{R}\)r,” J. Approx. Theory, 72 (1998), 74–81. [10] A. N. Korkin and E. I. Zolotarev, ”Sur un certain minimum,” in: Complete Collection of Works of E. I. Zolotarev [in Russian], vol. 1, Akad. Nauk SSSR, Leningrad, 1931, pp. 138–153. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.