Approximation of functions by Whittaker’s cardinal series. (English) Zbl 0582.42002

Stens, R. L.

General inequalities 4, Mem. E. F. Beckenbach, 4th Int. Conf., Oberwolfach/Ger. 1983, ISNM 71, 137-149 (1984).

[For the entire collection see Zbl 0573.00004.]
According to the classical cardinal series theorem any integrable, entire function of exponential type \(\leq \sigma\) can be represented by its cardinal series with nodes \(k\pi\) /\(\sigma\). It is shown that for continuous functions having an absolute integrable Fourier transform or satisfying certain smoothness conditions this representation holds in the limit for \(\sigma\) \(\to \infty\). Similarly, the derivatives of a function as well as its Hilbert transform can be approximated by the derivatives and the Hilbert transform of the cardinal series, respectively. Estimates for the approximation error are given; these are shown to be best possible.

MSC:

42A10

Trigonometric approximation

Keywords:

Whittaker’s cardinal series; Hilbert transform

Citations:

Zbl 0573.00004