Splinets – efficient orthonormalization of the B-splines

Xijia Liu, Hiba Nassar, Krzysztof Podgorski

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A new efficient orthogonalization of the B-spline basis is proposed and contrasted with some previous orthogonalized methods. The resulting orthogonal basis of splines is best visualized as a net of functions rather than a sequence of them. For this reason, the basis is referred to as a splinet. The splinets feature clear advantages over other spline bases. They efficiently exploit ‘near-orthogonalization’ featured by the B-splines and gains are achieved at two levels: locality that is exhibited through small size of the total support of a splinet and computational efficiency that follows from a small number of orthogonalization procedures needed to be performed on the B-splines to achieve orthogonality. These efficiencies are formally proven by showing the asymptotic rates with respect to the number of elements in a splinet. The natural symmetry of the B-splines in the case of the equally spaced knots is preserved in the splinets, while quasi-symmetrical features are also seen for the case of arbitrarily spaced knots.
Original languageEnglish
JournalBIT Numerical Mathematics
Number of pages59
Publication statusSubmitted - 2024


  • Basis functions
  • B-splines
  • Orthogonalization
  • Splinets


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