Trigonometric interpolation

On the rate of convergence of a lacunary trigonometric interpolation process

“…For m odd, the estimate shows that the process is uniformly convergent for all f(x)CC2~. This fact was established in [1]. As for error estimates, the result of P. V6rtesi [2, Theorem 2.5] cannot even give O(n-" log n), while (3)…”

“…The latter is equivalent to ahQf, h)=o(h) which, as PROOF. It follows from formulae (5) and (6) of [1] and the unique existence of R. …”

“…A. Sharma--A. K. Varma [1]) that this problem has a unique solution when m is odd and n is arbitrary, or m is even and n is odd. We shall investigate the rate of convergence of this process in these cases.…”

On the divergence of trigonometric lacunary interpolation

“…Moreover, if ni < rj, then 5ni <= rj (cf. (5) By virtue of the estimate IIR~II < 5n (see [8, p. 246]) and (6,8,9) it follows at last that This implies (11), provided that both q > 0 is small and No (and thus rj) is large enough. To establish (12) one may proceed analogously.…”

On the convergence of the trigonometric (O, M) interpolation

“…For sake of brevity sometimes we omit the double indices. We shall use the following estimations (see [2]):…”