Linear interpolation and Sobolev orthogonality

Abstract

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).

q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0∞FOR POSITIVE INTEGERS N

“…In fact, the reading of this relation gives us the key to adapt our technique in [11] to the case of q-Laguerre polynomials.…”

“…By considering a suitable modification of our previous result [11,Theorem 3], adapted to the case of generalized q-Laguerre polynomials, we are going to give an orthogonality result for the family {L…”

Non-classical orthogonality relations for big and little q-Jacobi polynomials

“…We will use a suitable modification of our previous result [18,Theorem 3], changing the derivative and integral operators by the q-derivative and q-integral operators, to establish two (not essentially different) non-standard orthogonality results for the family of generalized monic big q-Jacobi polynomials. .…”

“…Again, as in Section 3, we will consider a suitable modification of our previous result [18,Theorem 3] to establish a nonstandard orthogonality condition for the system of generalized monic little q-Jacobi polynomials. .…”

Non-standard orthogonality for the little q-Laguerre polynomials

“…We will fill up this gap by considering a suitable modification of our previous result [8,Theorem 3], using as starting point in our considerations one of the kind suggestions of the referees of that paper. …”