There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints x * i , defined bywhere f belongs to a certain Sobolev space, a i j (·) are piecewise continuous functions over [a, b], b i jk are real numbers, and the points t k belong to [a, b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in P l , denoted by p f , which fulfills the interpolatory data x * i ( p f ) = x * i ( f ), and also the condition that p (n)f best approximates f (n) in P l−n (with respect to the norm induced by the continuous part of the original discrete-continuous bilinear form considered).
We consider the continuous q-Jacobi polynomials {Pn=0 , extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the q-Sobolev character of the non-standard orthogonality for the corresponding family.
Keywords:Continuous q-ultraspherical polynomials Discrete q-ultraspherical polynomials Non-standard orthogonality Favard's theorem In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big qJacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.
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