Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in L 2(ℝ3)

We study Jackson’s inequality between the best approximation of a function f ∈ L2(ℝ3) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson’s inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson’s inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue–Morse modulus of continuity of order r ∈ ℕ. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.