Interpolation of maps with values in semilinear space

V.F. Babenko; M.V. Polishchuk

doi:10.15421/241306

Interpolation of maps with values in semilinear space

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

Abstract

Several known results about exact errors of piecewise linear interpolation of maps from $$$f\colon {\mathbb{R}}^n \rightarrow {\mathbb{R}^m}$$$ class are generalized to classes of set-valued mappings with given convex majorant of moduli of continuity.

Keywords

set-valued maps; interpolation; interpolation error; optimization

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References

Vitale R.A. "Approximations of convex set-valued functions", J. Approxim. Theory, 1979; 26: pp. 301-316.

Artstein Z. "Piecewise linear approximations of set-valued maps", J. Approxim. Theory, 1989; 56: pp. 41-47. doi:10.1016/0021-9045(89)90131-7

Dyn N., Farkhi E., Mokhov A. "Approximations of Set-Valued Functions by Metric Linear Operators", Constructive Approximation, 2006; 25: pp. 193-209. doi:10.1007/s00365-006-0632-9

Dyn N., Farkhi E. "Approximations of set-valued functions with compact images — an overview, approximation and probability", Banach Center Publ., 2006; 72: pp. 1-14.

Babenko V.F. "Interpolation of continuous maps by piecewise-linear ones", Matem. zametki, 1978; 24(1): pp. 43-50. (in Russian) doi:10.1007/BF01812980

Babenko V.F. "On the optimal piecewise-linear interpolation of continuous maps", Researches on modern problems of summation and approximation of functions, and their applications, 1980; pp. 3-7. (in Russian)

Malozemov V.N. "On deviation of broken lines", Vestnik LGU, 1966; 7: pp. 150-153. (in Russian)

Babenko V.F., Ligun A.A. "On interpolation by multifaceted functions", Matem. zametki, 1975; 18(6): pp. 803-814. (in Russian) doi:10.1007/BF01099983

Borisovich Yu.G., Gel'man B.D., Myshkis A.D., Obukhovskij V.V. "Set-valued maps", Itogi nauki i tekhniki. Mat. analiz, 1982; 19: pp. 127-230. (in Russian)

Gruber P.M., Wills J.M. Aspects of Approximation of Convex Bodies. Handbook of Convex Geometry, 1993; pp. 319-345.

Price G.B. "The theory of integration", Trans. Amer. Math. Soc., 1940; 47: pp. 1-50. doi:10.2307/1989999

Rockafellar R.T. Convex analysis. Tran. from Eng., Mir, Moscow, 1973; 472 p. (in Russian)

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)