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

Abstract

Abstract Two explicit guard functions 𝐾𝑗= 𝐾𝑗(δ 𝑧), 𝑗 = 1, 2, are obtained, which depend on the distance δ 𝑧 between 𝑧 and the nearest point of the integer lattice in the complex plane, such that δ 𝑧𝐾1(δ 𝑧) ≤ |σ(𝑧)|𝑒–π|𝑧|2/2 ≤ δ 𝑧𝐾2(δ 𝑧), 𝑧 ∈ ℂ, where σ(𝑧) stands for the Weierstraß σ-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 .