A Construction of Bi orthogonal Functions to B splines with Multiple Knots

We present a construction of a re nable compactly supported vector of functions which is bi orthogonal to the vector of B splines of a given degree with multiple knots at the integers with prescribed multiplicity The construction is based on Hermite interpolatory subdivision schemes and on the relation between B splines and divided di erences The bi orthogonal vector of functions is shown to be re nable with a mask related to that of the Hermite scheme For simplicity of presentation the special scalar case corresponding to B splines with simple knots is treated separately Introduction B splines are important for applications and those with integer knots provide ex plicit examples of a compactly supported re nable univariate function Wavelets based on splines have received considerable treatment in the literature see e g In this paper we show how to construct re nable compactly supported functions that are bi orthogonal to B splines with simple or with multiple knots using the re nable scaling functions generated by interpolatory subdivision schemes Compactly supported bi orthogonal func tionals to B splines built on an arbitrary sequence of knots with possible multiplicities are constructed in The case of B splines with simple equidistant knots the scalar case is simpler and is treated here separately for the simplicity of the presentation For this case the bi orthogonal re nable function is generated by a Lagrange type interpolatory subdivision scheme which uses function values at each control point A di erent approach to the construction of bi orthogonal re nable functions to B splines with simple knots from the interpolatory schemes of Deslaurier and Dubuc is mentioned in B splines with multiple equidistant knots of the samemultiplicity constitute a re nable vector of functions Re nable vectors of spline functions are considered in For the case of B splines with multiple knots we generate the bi orthogonal vector of functions by Hermite type interpolatory subdivision schemes which use values of a function and its derivatives at each control point to generate the same type of data in the next level For B splines with knots of multiplicity the corresponding Hermite interpolatory subdivision scheme is of order namely a subdivision scheme which uses and computes the value of the function and the values of its rst derivatives at each control point For information about subdivision schemes the reader is referred to and The construction of the bi orthogonal functions is based on the relation between B splines and divided di erences Information on B spines divided di erences and the relation between them can be found in First we prove the bi orthogonality property and then the re nability property In both proofs we treat separately the case of B splines with simple knots The special case of splines which are not necessarily continuous namely when is maximal exceeds the degree of the B spline by is treated in The Hermite interpo latory schemes in this case are closely related to the moments interpolating schemes Bi orthogonal re nable functions of compact support are important in the construction of bi orthogonal wavelets of compact support B splines with simple knots De ne S n for n S n for n Then the sequence S has the property that