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Analog of the Akhiezer-Krein-Favard sums for periodic splines of minimal defect. (English) Zbl 1050.41016

Let n,r,m, mr, W p (r) be the space of 2π-periodic functions whose (r-1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L p , and S 2n,m be the space of 2π-periodic splines of order m of minimal defect over the uniform partition kπ/n (k). The author constructs linear operators X n,r,m :L 1 S 2n,m such that

sup fW (r) f-X n,r,m (f) f (r) =sup fW 1 (r) f-X n,r,m (f) 1 f (r) 1 =K r n r ,

where

K r =4 π l=0 (-1) l(r+1) (2l+1) r+1 ·

The operators X n,r,m are constructed using the interpolation of Bernoulli kernels. As is proved, the operators X n,r,m converge to polynomial Akhiezer-Krein-Favard operators as m.

MSC:
41A35Approximation by operators (in particular, by integral operators)
41A15Spline approximation