@MISC{Reif95orthogonalityof, author = {Ulrich Reif}, title = {Orthogonality of Cardinal B-Splines in Weighted Sobolev Spaces}, year = {1995} }

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Abstract

The cardinal B-splines B j;n;h form a basis in the space of splines of order n with knots hZ, which is orthonormal in the Sobolev space H 2;n\Gamma1 (R) provided with the norm kfk 2 !(n;h) := P n\Gamma1 ¯=0 ! ¯ (n; h)k@ ¯ fk 2 for certain positive weights ! ¯ (n; h). These weights are specified explicitly. Further, an application to approximation theory is given. 1 Preliminaries In this section we shall briefly introduce some basic concepts from B-spline theory and functional analysis, see e.g. [Sch81], [dB78], [Ada78] and [SW75] for an introduction to these topics. For m 2 N denote by h\Delta; \Deltai; h\Delta; \Deltai m the inner products and by k \Delta k; k \Delta km the norms of the Hilbert spaces L 2 (R) and H m;2 (R), respectively. Let ! := [! 0 ; : : : ; !m ] be a vector of positive weights and define the weighted Sobolev space H m;2 ! (R) by providing H m;2 (R) with the inner product (f; g) ! := m X ¯=0 ! ¯ h@ ¯ f; @ ¯ gi : (1.1) Supported by BMBF ...