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000118967 005__ 20240319081012.0
000118967 0247_ $$2doi$$a10.1007/s00365-022-09583-4
000118967 0248_ $$2sideral$$a130146
000118967 037__ $$aART-2022-130146
000118967 041__ $$aeng
000118967 100__ $$0(orcid)0000-0003-3636-276X$$aCantero, M. J.$$uUniversidad de Zaragoza
000118967 245__ $$aWall Polynomials on the Real Line: a classical approach to OPRL Khrushchev’s formula
000118967 260__ $$c2022
000118967 5060_ $$aAccess copy available to the general public$$fUnrestricted
000118967 5203_ $$aThe standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (Grünbaum and Velázquez in Adv Math 326:352–464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchev’s formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchev’s formula is used to rederive the characterization given in Simon (J Approx Theory 126:198–217, 2004) for the weak convergence of pn2dµ, where pn are the orthonormal polynomials with respect to a measure µ supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchev’s formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure µ supported on an unbounded subset of the real line yields a weakly convergent sequence pn2dµ (Corollary 8.10), but there exist instances such that pn2dµ becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of pn2dµ in the unbounded case are obtained via Khrushchev’s formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchev’s formula on the real line which sheds light on its graph theoretical meaning, linked to Pólya’s recurrence theory for classical random walks. © 2022, The Author(s).
000118967 536__ $$9info:eu-repo/grantAgreement/ES/DGA/E26-17R$$9info:eu-repo/grantAgreement/ES/DGA/E48-20R$$9info:eu-repo/grantAgreement/ES/MICINN-AEI-FEDER/MTM2017-89941-P
000118967 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/
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000118967 591__ $$aMATHEMATICS$$b19 / 329 = 0.058$$c2022$$dQ1$$eT1
000118967 593__ $$aMathematics (miscellaneous)$$c2022$$dQ1
000118967 593__ $$aAnalysis$$c2022$$dQ2
000118967 593__ $$aComputational Mathematics$$c2022$$dQ2
000118967 594__ $$a3.2$$b2022
000118967 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion
000118967 700__ $$0(orcid)0000-0001-9248-293X$$aMoral, L.
000118967 700__ $$0(orcid)0000-0002-3050-9540$$aVelázquez, L.$$uUniversidad de Zaragoza
000118967 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada
000118967 773__ $$g57 (2022), 75–124$$pConstr. approx.$$tCONSTRUCTIVE APPROXIMATION$$x0176-4276
000118967 8564_ $$s671931$$uhttps://zaguan.unizar.es/record/118967/files/texto_completo.pdf$$yVersión publicada
000118967 8564_ $$s1420486$$uhttps://zaguan.unizar.es/record/118967/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada
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000118967 951__ $$a2024-03-18-15:14:45
000118967 980__ $$aARTICLE