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On Jackson’s theorems in \(L_ 2\). (English. Russian original) Zbl 0737.42001

Math. Notes 48, No. 4, 1071-1074 (1990); translation from Mat. Zametki 48, No. 4, 152-157 (1990).
Estimates for \(E_ n(f)\) by means of iterated differences of derivates are investigated and characterizations of finite sets are given on which the differences are considered. In particular, if \(\Delta^ mf^{(r)}, r>0\), is taken into account and \(1/q\leq r/m<1/(q-1)\), \(q\) an integer, then it is shown that the cardinality of the sets in question is estimated by \(q+1\) from below and there is no analogue for \(r=0\).
Reviewer: M.Krbec (Praha)

MSC:

42A10 Trigonometric approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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References:

[1] N. I. Chernykh, ?On Jackson’s inequality in L2,? Transactions of Math. Institute of Academy of Science of the USSR,88, 71-74 (1967).
[2] N. I. Chernykh, ?On the best approximation of periodic functions by trigonometric polynomials in L2,? Matem. Zametki,2, No. 5, 513-522 (1967).
[3] V. A. Yudin, ?On Jackson’s theorems in L2,? Matem. Zametki,41, No. 1, 43-47 (1987).
[4] J. W. S. Cassels, Introduction to Diophantine Approximations, Cambridge University Press (1957). · Zbl 0077.04801
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