On hyperbolic summation and hyperbolic moduli of smoothness. (English) Zbl 0857.41007

For the functions of one variable, the relation between smoothness of a function and the order of the best approximation by trigonometric polynomials or spline functions with given knots is well understood. The situation is different in the multidimensional case of tensor products, as we have several possible orders. This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions on \([0,1]^d\). The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established.


41A15 Spline approximation
41A63 Multidimensional problems
Full Text: DOI


[1] Z. Ciesielski (1975):Constructive function theory and spline system. Studia Math.,53:277–302. · Zbl 0273.41010
[2] Z. Ciesielski (1979):Equivalence, unconditionality and convergence a.e. of the spline bases in L p spaces. In: Banach Center Publications, vol. 4, Approximation Theory. Warsaw: PWN, 55–68.
[3] Z. Ciesielski (1983):Bases and K-functionals for Sobolev Spaces over C compact manifolds. Trudy Mat. Inst. W. A. Steklova AN USSR,164:197–202 (Russian). · Zbl 0565.58015
[4] Z. Ciesielski, J. Domsta (1972):Construction of an orthonormal basis in C m(Id) and Wpm(Id). Studia Math.,41:211–224. · Zbl 0235.46047
[5] Z. Ciesielski, J. Domsta (1972):Estimates for the spline orthonormal functions and for their derivatives. Studia Math.,44:315–320. · Zbl 0215.46901
[6] R. A. DeVore, S. Konyagin, P. P. Petrushev, V. N. Temlyakov,Hyperbolic wavelet approximation, in preparation.
[7] Din’Zung (1988):Approximation by trigonometric polynomials of functions of several variables on the torus. Math. USSR-Sb.,59(1):247–267. · Zbl 0652.42001
[8] A. Kamont (1994):Isomorphism of some anisotropic Besov and sequence spaces. Studia Math.,110(2):169–189. · Zbl 0810.41010
[9] C. A. McCarthy (1964):Commuting Boolean algebras of projections II, Boundedness in L p. Proc. Amer. Math. Soc.,15:781–787. · Zbl 0127.33003
[10] W. N. Temlyakov (1986): Approximation of the functions with bounded mixed derivative, Trudy Mat. Inst. W. A. Steklova AN USSR, vol. 178. Moscow, Pauka (Russian): English translation Proceedings of the Steklov Institute of Mathematics, 1989, issue 1. · Zbl 0625.41028
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.