Geodesic
Best approximation of functions in $L_p$ by polynomials on affine system
Sbornik. Mathematics, Tome 202 (2011) no. 2, pp. 279-306 Cet article a éte moissonné depuis la source Math-Net.Ru

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Estimates of the best $L_p$-approximation of functions by polynomials in an affine system (system of dilations and translations), which are similar to well-known estimates due to Ul'yanov and Golubov for approximations in the Haar system, are obtained. An analogue of A. F. Timan and M. F. Timan's inequality is shown to hold under certain conditions on the generating function of the affine system; this analogue fails for the Haar system for $1. Bibliography: 10 titles.
Keywords: Haar system, system of dilations and translations, affine system, best approximation. 
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P. A. Terekhin. Best approximation of functions in $L_p$ by polynomials on affine system. Sbornik. Mathematics, Tome 202 (2011) no. 2, pp. 279-306. http://geodesic.mathdoc.fr/item/SM_2011_202_2_a4/

[1] A. Haar, “Zur Theorie der orthogonalen Funktionensysteme Erste Mitteilung”, Math. Ann., 69:3 (1910), 331–371 | DOI | MR | Zbl

[2] N. K. Bari, “O nailuchshem priblizhenii trigonometricheskimi polinomami dvukh sopryazhennykh funktsii”, Izv. AN SSSR. Ser. matem., 19:5 (1955), 285–302 | MR | Zbl

[3] P. L. Ulyanov, “O ryadakh po sisteme Khaara”, Matem. sb., 63(105):3 (1964), 356–391 | MR | Zbl

[4] B. I. Golubov, “Series with respect to the Haar system”, J. Soviet Math., 1:6 (1973), 704–726 | DOI | MR | Zbl | Zbl

[5] B. I. Golubov, “Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials”, Math. USSR-Sb., 16:2 (1972), 265–285 | DOI | MR | Zbl | Zbl

[6] B. Golubov, A. Efimov, V. Skvortsov, Walsh series and transforms. Theory and applications, Math. Appl. (Soviet Ser.), 64, Kluwer Acad. Publ., Dordrecht, 1991 | MR | MR | Zbl | Zbl

[7] P. A. Terekhin, “Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces $L^p[0,1]$”, Math. Notes, 83:5 (2008), 657–674 | DOI | MR | Zbl

[8] V. I. Filippov, P. Oswald, “Representation in $L_p$ by series of translates and dilates of one function”, J. Approx. Theory, 82:1 (1995), 15–29 | DOI | MR | Zbl

[9] P. A. Terekhin, “Multishifts in Hilbert spaces”, Funct. Anal. Appl., 39:1 (2005), 57–67 | DOI | MR | Zbl

[10] P. A. Terekhin, “Riesz bases generated by contractions and translations of a function on an interval”, Math. Notes, 72:3–4 (2002), 505–518 | DOI | MR | Zbl