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

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

Zbl 0573.00004