Sequentially ordered Sobolevinner product and Laguerre β Sobolev polynomials
Sequentially ordered Sobolevinner product and Laguerre β Sobolev polynomials
Date
2023-04-20
Authors
Juan HernΓ‘ndez
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Abstract
We study the sequence of polynomials {ππ}πβ₯0{ππ}πβ₯0 that are orthogonal with respect
to the general discrete Sobolev-type inner product β¨π, πβ©{π} = β« π(π₯)π(π₯)ππ(π₯) + β ππ=1 β
πππ=0 ππ,ππ(π)(ππ)π(π)(ππ), β¨π, πβ©{π } = β« π(π₯)π(π₯)ππ(π₯) + β π=1 π β π=0 ππππ,ππ(π)(ππ)π(π)
(ππ) where ππ is a finite Borel measure whose support supp(π) is an infinite set of the real
line, ππ,πβ₯0, and the mass points ππ, π=1,β¦,π are real values outside the interior of the convex
hull of supp(π)suppπ (ππβR\πh(supp(π))β)ππβπ
\πΆh(supp(π))β). Under some restriction of
order in the discrete part of β¨Β·,Β·β©πβ¨Β·,Β·β©π , we prove that ππππ has at least πβπβπβπ* zeros on
πh(supp(π))βπΆh(suppπ)β, being πβπ* the number of terms in the discrete part of β¨Β·,Β·β©πβ¨Β·,Β·β©π .
Finally, we obtain the outer relative asymptotic for {ππ}{ππ} in the case that the measure ππ
is the classical Laguerre measure, and for each mass point, only one order derivative appears
in the discrete part of β¨Β·,Β·β©πβ¨Β·,Β·β©π .
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DΓaz-GonzΓ‘lez, A., HernΓ‘ndez, J., & Pijeira-Cabrera, H. (2023). Sequentially ordered Sobolev inner product and LaguerreβSobolev polynomials. Mathematics, 11(10), 1956.