T. K. Pogány

Local Growth of Weierstrass s-Function and Whittaker-Type Derivative Sampling

abstract:
Two explicit guard functions $K_j = K_j(\delta_z)$, $j=1,2$, are obtained, which depend on the distance $\delta_z$ between $z$ and the nearest point of the integer lattice in the complex plane, such that $\delta_z K_1(\delta_z) \leq |\sigma(z)|e^{-\pi |z|^2/2} \leq \delta_z K_2(\delta_z),\; z\in {\mathbb C}$, where $\sigma(z)$ stands for the Weierstraß $\sigma$-function. This result is used to improve the circular truncation error upper bound in the $q$-th order Whittaker-type derivative sampling for the Leont'ev functions space $[2, \frac{\pi q}{2})$, $q\geq 1$.