ABSTRACT
In the research monograph about metaharmonic lattice point theory, W. Freeden proposed two kinds of summation formulas, namely lattice point–generated Euler summation formulas and lattice ball–generated Euler summation formulas. The first kind of formulas are based on the constituting properties of the Euler–Green functions with respect to (iterated) (Δ + λ)-operators, while the second kind uses (ball-)averaged integral variants of the Euler–Green functions with respect to (Δ + λ)-operators. This chapter extends the lattice ball concept to the operator in order to derive lattice ball Shannon-type sampling variants. The considerations are based on the so-called Λ-Euler–Green τ-ball mean function with respect to the operator L + λ, which is an average of the Λ-Euler–Green function with respect to the operator L + λ over balls around lattice points. In addition to ball-generated means, Gauss–Weierstrass and Abel–Poisson means of the Euler–Green functions can be used to formulate associated Euler-type summation formulas and to derive summability variants of Shannon-type sampling formulas.
