Alternative Derivation and Numerical Analysis of an Optimal Interpolation Formula in the Hilbert Space

Abstract

Background: The problem of optimal interpolation from function values prescribed at the nodes of a uniform grid is considered in the space ????(1,0)2 (0, 1). The coefficients of this formula are known as the ???? = 1 special case of a general result obtained for the space ????(????,????−1)2 (0, 1). In the previous work, they were found by the Sobolev method using an extremal function and a discrete analogue of a differential operator.

Method: Here the same result is derived by a different approach. The integrating factor ????(????) = ????????????(????) and the monotone coordinate transformation ????(????) = (????2???? − 1)/2 reduce the original seminorm to a simple first-derivative energy.

Results: Direct minimization of the Peano-type kernel of the transformed problem proves the locality of the optimal formula. Thus, only the coefficients associated with the two endpoints of the interval containing ???? are nonzero. A separate theorem proves that the resulting local formula involving hyperbolic sines coincides exactly with the signfunction formula for the ???? = 1 case of the previous general result. Exact and uniform estimates of the error-functional norm are obtained.

Convergence analysis: A second-order asymptotic convergence result is established for smooth functions. Numerical experiments for ????2, ????2????, and sin ???? confirm that the maximum error decreases as the number of nodes increases and corroborate the theoretical ????(ℎ2) order.