Cardinal-Type Approximations of a Function and Its Derivatives

Whittaker’s cardinal function is used to approximate certain analytic functions in Sobolev norm. $L^\infty $ is of primary interest, although attention is also given to $L^2 ( - \infty ,\infty )$. Results are given for functions defined on a general contour in the complex plane, and special treatment is given to the important real domains $( - \infty ,\infty )( - 1,1)$ and $(0,\infty )$. In all cases, it is shpwn that the approximations converge to the function at the rate $C\exp ( - cn^{{1/2}} )$, where n is the number of points of interpolation and C and c are positive constants.