Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  682.41031
Autor:  Erdös, Paul; Vértesi, P.
Title:  On certain saturation problems. (In English)
Source:  Acta Math. Hung. 53, No. ½, 197-203 (1989).
Review:  In the present paper, the author studies certain saturation problems on the interpolatory linear operators

(Lnf)(x) = sum0 \leq k \leq nf(k/n)|x-k/n|-r/sum0 \leq k \leq n |x-k/n/|-r;   0 \leq x \leq 1,  n \geq 1,

where r > 2 is a fixed real number, f in C[0,1], and gives two theorems on it. His main result is as follows: ``Theorem: Let 0 < x0 < 1 be a fixed irrational number, {yr}oor-1 be an arbitrary sequence with yr\ne x0, r = 1,2,..., limr ––> ooyr = x0. Further let 0 < p* \leq 1/3 (real), p,q > 0, (p,q) = 1 (integers), 0 \leq \gamma < p, 0 \leq \delta < q (reals) be fixed numbers. Then there exist a sequence {xk}\subset {yr} and positive integers {\ellk}ook = 1 and {nk}ook = 1 with 1 < n1 < n2 < .... i.e. limk ––> oonk = oo such that relations

|x0-(p\ellk+\gamma)/(qnk+\delta)| = o(1/nk),   k = 1,2,... ,


p*/2nk \leq |xk-(p\ellk+\gamma)/(qnk+\delta)| \leq (2p+2)p/nk,   k = 1,2,...,

hold true.
Reviewer:  S.P.Singh
Classif.:  * 41A40 Saturation
                   41A05 Interpolation
                   41A36 Approximation by positive operators
Keywords:  diophantine equation; modulus of continuity; saturation problems; interpolatory linear operators

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