Degree of approximation by linear operators (Q1173642)

Let \(\{p_ n\}\) and \(\{q_ n\}\) be sequences of positive constants such that \(P_ n=\sum^ n_{k=0} p_ k\), \(Q_ n=\sum^ n_{k=0} q_ k\), and \(R_ n=\sum^ n_{k=0} p_ k q_{n-k}\), where \(R_ n\to\infty\) as \(n\to \infty\). Let \(\sum a_ n\) be an infinite series with the sequence \(\{s_ n\}\) of its partial sums. The generalized Nörlund operators are defined by \(N^{p,q}_ n= R^{-1}_ n \sum^ n_{k=0} P_{n-k} q_ k s_ k\). Let the Fourier series of a \(2\pi\)- periodic function \(f\) be given by \(f\sim (a_ 0)/2+\sum^ n_{k=1} (a_ k\cos kx+b_ k\sin kx)\). If \(f\in L_ p[a,b]\), \(p>1\), then \(\omega_ p(f;t)=\sup_{a\leq h\leq t}\bigl[\int^ b_ a | f(x+h)- f(x)|^ p\bigr]^{1/p}\) is the integral modulus of continuity of \(f\). The authors prove some interesting results on the degree of \(L_ p\)-approximation. We mention a sample result: for \(f\in L_ p[0,2\pi]\), \(p>1\), and for sequences \(\{p_ n\}\), \(\{q_ n\}\), \(\{R_ n/n\}\) nonincreasing, we have \[ \| N^{p,q}_ n- f\|_ p=O\left[ R^{- 1}_ n\sum^ n_{k=1} k^{-1} R_ k \omega_ p(f;\pi/k)\right]. \] {}.