A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval
- Saratov State University, Saratov (Russian Federation)
Classes of functions in the space of continuous functions f defined on the interval [0,{pi}] and vanishing at its end-points are described for which there is pointwise and approximate uniform convergence of Lagrange-type operators S{sub {lambda}}(f,x)={sigma}{sub k=0}{sup n} (y(x,{lambda}))/(y'(x{sub k,{lambda}})(x-x{sub k,{lambda}})) f(x{sub k,{lambda}}). These operators involve the solutions y(x,{lambda}) of the Cauchy problem for the equation y''+({lambda}-q{sub {lambda}}(x))y=0 where q{sub {lambda}} element of V{sub {rho}{sub {lambda}}}[0,{pi}] (here V{sub {rho}{sub {lambda}}}[0,{pi}] is the ball of radius {rho}{sub {lambda}}=o({radical}{lambda}/ln {lambda}) in the space of functions of bounded variation vanishing at the origin, and y(x{sub k,{lambda}})=0). Several modifications of this operator are proposed, which allow an arbitrary continuous function on [0,{pi}] to be approximated uniformly. Bibliography: 40 titles.
- OSTI ID:
- 21301322
- Journal Information:
- Sbornik. Mathematics, Vol. 200, Issue 11; Other Information: DOI: 10.1070/SM2009v200n11ABEH004054; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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