Equiconvergence of some lacunary trigonometric interpolation polynomials (Q1097441)

IF \(f\in C_{2\pi}\) let \(Q_ n(f,x)\) \((n=2m+1)\) be the trigonometric polynomial of order \((3n-1)/2\) that interpolates f in the sense that \(Q_ n(f,x_{kn})=f(x_{kn})\) and \(Q_ n''(f,x_{kn})=Q_ n'''(f,x_{kn})=0\) for \(x_{kn}=2\pi k_ n\), \(k=0,1,...,n-1\). (The existence of such polynomials follows from a result of \textit{A. Sharma} and the first author [Ann. Pol. Math. 21, 51-58 (1968; Zbl 0175.351)].) In this paper the authors study the problem of equiconvergence of the polynomials \(Q_ n(f,\cdot)\) and the trigonometric polynomials \(T_ m(f,\cdot)\) of order m interpolating f (in usual sense) at the nodes \(x_{kn}\), and prove that \[ \lim_{n\to \infty}[f(x)-Q_ n(f,x)- \mu_ n(x)(f(x)-T_ m(f,x))]=0 \] uniformly on \([0,2\pi]\), with some \(C_{2\pi}\) functions \(\mu_ n\) independent of f. Hence in particular, by a well-known theorem of Marcinkiewicz, for each \(p>0\) we get \(\lim_{n\to \infty}\int^{2\pi}_{0}| f(x)-Q_ n(f,x)|^ pdx=0\).