Bernstein type inequalities of different metrics for splines defined on the real domain

V.F. Babenko; V.A. Zontov

doi:10.15421/241303

Bernstein type inequalities of different metrics for splines defined on the real domain

V.F. Babenko (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0001-6677-1914
V.A. Zontov (Oles Honchar Dnipropetrovsk National University)

Abstract

New sharp Bernstein type inequalities of different metrics in spaces of integrable functions for non-periodic splines of order m and minimal defect, having equidistant nodes, are obtained.

Keywords

non-periodic splines; Bernstein type inequalities of different metrics

Full Text:

PDF (Русский)

References

Korneichuk N.P. Exact constants in approximation theory, Nauka, Moscow, 1987; 424 p. (in Russian)

Tikhomirov V.M. "Set widths in functional spaces and theory of the best approximations", Uspekhi mat. nauk, 1960; 15(3): pp. 81-120. (in Russian)

Subbotin Yu.N. "On the piecewise-polynomial interpolation", Mat. zametki, 1967; 1(1): pp. 24-29. (in Russian) doi:10.1007/BF01221723

Babenko V.F., Pichugov S.A. "Bernstein type inequalities for polynomial splines in $$$L_2$$$ space", Ukrainian Math. J., 1991; 43(3): pp. 420-422. (in Russian) doi:10.1007/BF01670081

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, Kyiv, 1992; 304 p. (in Russian)

Ligun A.A. "Sharp inequalities for spline-functions and the best quadrature formulas for certain classes of functions", Mat. zametki, 1976; 19(6): pp. 913-926. (in Russian) doi:10.1007/BF01149933

Ligun A.A. "On inequalities between norms of derivatives of periodic functions", Mat. zametki, 1983; 33(3): pp. 385-391. (in Russian) doi:10.1007/BF01686326

Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Kofanov V.A. "On sharp inequalities of Bernstein type for splines", Ukrainian Math. J., 2006; 58(10): pp. 1357-1367. (in Russian) doi:10.1007/s11253-006-0152-z

Magaril-Il'yaev G.G. "On the best approximation of functional classes by splines on the real domain", Trudy Matematicheskogo instituta RAN, 1992; 194: pp. 148-159. (in Russian)

Babenko V.F., Zontov V.A. "Inequalities of Bernstein type for splines defined on the real domain", Ukrainian Math. J., 2011; 63(5): pp. 603-611. (in Russian) doi:10.1007/s11253-011-0536-6

Babenko V.F., Spektor S.A. "Inequalities of Bernstein type for splines in $$$L_2(\mathbb{R})$$$ space", Res. Math., 2008; 16(6/1): pp. 21-27. (in Russian)

Hermander L. Linear differential operators with partial derivatives, Mir, Moscow, 1965; 380 p. (in Russian)

Volevich L.R., Paneyakh B.P. "Certain spaces of generalized functions and inclusion theorems", Uspekhi mat. nauk, 1965; 20(1): pp. 3-74. (in Russian) doi:10.1070/RM1965v020n01ABEH004139

Mikhajlets V.A., Murach A.A. Hermander spaces, interpolation and elliptic problems, Kyiv, 2010; 372 p. (in Russian)

Strelkov N.A. "Projectively-net widths and lattice packings", Mat. sb., 1991; 182(10): pp. 1513-1533. (in Russian) doi:10.1070/SM1993v074n01ABEH003346

Strelkov N.A. "Universally optimal wavelets", Mat. sb., 1997; 188(1): pp. 147-160. (in Russian) doi:10.1070/SM1997v188n01ABEH000192

Chui K. Introduction to wavelets, Moscow, 2008. (in Russian)