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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.

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