Sequentially Ordered Sobolev Inner Product and Laguerre –Sobolev Polynomials

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Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials.pdf(1.35 MB)
Date
2023-04-20
Authors
Juan HernΓ‘ndez
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We 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=βˆ«π‘“(π‘₯)𝑔(π‘₯)π‘‘πœ‡(π‘₯)+βˆ‘π‘π‘—=1βˆ‘π‘‘π‘—π‘˜=0πœ†π‘—,π‘˜π‘“(π‘˜)(𝑐𝑗)𝑔(π‘˜)(𝑐𝑗),βŸ¨π‘“,π‘”βŸ©π‘ =βˆ«π‘“(π‘₯)𝑔(π‘₯)π‘‘πœ‡(π‘₯)+βˆ‘π‘—=1π‘βˆ‘π‘˜=0π‘‘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‡)suppπœ‡ is an infinite set of the real line, πœ†π‘—,π‘˜β‰₯0πœ†π‘—,π‘˜β‰₯0, and the mass points 𝑐𝑖𝑐𝑖, 𝑖=1,…,𝑁𝑖=1,…,𝑁 are real values outside the interior of the convex hull of supp(πœ‡)suppπœ‡ (π‘π‘–βˆˆβ„\π‚β„Ž(supp(πœ‡))∘)π‘π‘–βˆˆπ‘…\πΆβ„Ž(supp(πœ‡))∘). Under some restriction of order in the discrete part of βŸ¨β€’,β€’βŸ©π—ŒβŸ¨β€’,β€’βŸ©π‘ , we prove that 𝑆𝑛𝑆𝑛 has at least π‘›βˆ’π‘‘βˆ—π‘›βˆ’π‘‘* zeros on π‚β„Ž(supp(πœ‡))βˆ˜πΆβ„Ž(suppπœ‡)∘, being π‘‘βˆ—π‘‘* the number of terms in the discrete part of βŸ¨β€’,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 βŸ¨β€’,β€’βŸ©π—ŒβŸ¨β€’,β€’βŸ©π‘ .
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HernΓ‘ndez, J. (2023). Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials. Mathematics, 11(8), 1956.
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https://repositoriovip.uasd.edu.do/handle/123456789/1380
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