Sequentially ordered Sobolevinner product and Laguerre – Sobolev polynomials

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dc.contributor.authorJuan HernΓ‘ndez
dc.date.accessioned2024-08-12T15:31:47Z
dc.date.available2024-08-12T15:31:47Z
dc.date.issued2023-04-20
dc.description.abstractWe study the sequence of polynomials {𝑆𝑛}𝑛β‰₯0{𝑆𝑛}𝑛β‰₯0 that are orthogonal with respect to the general discrete Sobolev-type inner product βŸ¨π‘“, π‘”βŸ©{π—Œ} = ∫ 𝑓(π‘₯)𝑔(π‘₯)π‘‘πœ‡(π‘₯) + βˆ‘ 𝑁𝑗=1 βˆ‘ π‘‘π‘—π‘˜=0 πœ†π‘—,π‘˜π‘“(π‘˜)(𝑐𝑗)𝑔(π‘˜)(𝑐𝑗), βŸ¨π‘“, π‘”βŸ©{𝑠} = ∫ 𝑓(π‘₯)𝑔(π‘₯)π‘‘πœ‡(π‘₯) + βˆ‘ 𝑗=1 𝑁 βˆ‘ π‘˜=0 π‘‘π‘—πœ†π‘—,π‘˜π‘“(π‘˜)(𝑐𝑗)𝑔(π‘˜) (𝑐𝑗) where πœ‡πœ‡ is a finite Borel measure whose support supp(πœ‡) is an infinite set of the real line, πœ†π‘—,π‘˜β‰₯0, and the mass points 𝑐𝑖, 𝑖=1,…,𝑁 are real values outside the interior of the convex hull of supp(πœ‡)suppπœ‡ (π‘π‘–βˆˆR\𝐂h(supp(πœ‡))∘)π‘π‘–βˆˆπ‘…\𝐢h(supp(πœ‡))∘). Under some restriction of order in the discrete part of ⟨·,Β·βŸ©π—ŒβŸ¨Β·,Β·βŸ©π‘ , we prove that 𝑆𝑛𝑆𝑛 has at least π‘›βˆ’π‘‘βˆ—π‘›βˆ’π‘‘* zeros on 𝐂h(supp(πœ‡))∘𝐢h(suppπœ‡)∘, being π‘‘βˆ—π‘‘* the number of terms in the discrete part of ⟨·,Β·βŸ©π—ŒβŸ¨Β·,Β·βŸ©π‘ . Finally, we obtain the outer relative asymptotic for {𝑆𝑛}{𝑆𝑛} in the case that the measure πœ‡πœ‡ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨·,Β·βŸ©π—ŒβŸ¨Β·,Β·βŸ©π‘ .
dc.identifier.citationDΓ­az-GonzΓ‘lez, A., HernΓ‘ndez, J., & Pijeira-Cabrera, H. (2023). Sequentially ordered Sobolev inner product and Laguerre–Sobolev polynomials. Mathematics, 11(10), 1956.
dc.identifier.urihttps://repositoriovip.uasd.edu.do/handle/123456789/266
dc.titleSequentially ordered Sobolevinner product and Laguerre – Sobolev polynomials
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