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Statistical approximation for periodic functions. (English) Zbl 1065.41041

Let A:=(a nk ), n,k=1,2,, be a nonnegative regular summability matrix. The sequence x:={x k } is called A-statistically convergent to L, notation st A -limx n =L, if for every ϵ>0,

lim n |x k -L|ϵ a nk =0·

Let C * denote the space of 2π-periodic functions on the real line with the usual sup-norm, and let {L n } be a sequence of positive linear operators mapping C * into itself. The author proves the following Korovkin type theorem

Theorem: Let A and {L n } be as above. Then necessary and sufficient condition in order that

st A -limL n (f,·)-f=0,(1)

is that (1) holds for the three functions, 1, sinx, and cosx.

An application is given describing sufficient conditions on a matrix of coefficients {ρ k (n) }, 1kn=1,2, ensuring that the operators

T n (f,x):=a 0 2+ k=1 n ρ k (n) (a k coskx+b k sinkx),

where the a k ’s and b k ’s are the Fourier coefficients of f, converge to f in the sup-norm.

41A36Approximation by positive operators
41A10Approximation by polynomials