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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.

MSC:

41A15 Spline approximation
41A63 Multidimensional problems
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