Díaz-González, Abel; Pijeira-Cabrera, Héctor; Pérez-Yzquierdo, Ignacio Rational approximation and Sobolev-type orthogonality. (English) Zbl 1452.41006 J. Approx. Theory 260, Article ID 105481, 19 p. (2020). In this contribution the authors deal with sequences \(\{S_{n}(x)\}_{n\geq0}\) of polynomials orthogonal with respect to a Sobolev-type inner product \[\langle f, g \rangle = \int_{-1}^{1} f(x) g(x) d\mu(x) + \sum_{j=1}^{N} \sum _{i=0}^{d_{j}} M_{j,i} f^{(i)} (c_{j}) g^{(i)} (c_{j}).\] Here, \(\mu\) is a measure in the Nevai class \(M(0,1)\) (see [P. G. Nevai, Orthogonal polynomials. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0405.33009)]), \(M_{j,i} \geq 0, i=0, 1, \cdots, d_{j}-1, M_{j,d_{j}}>0,\) and \(c_{j}, j=1, 2, \cdots, N\) are real numbers with \(|c_{j}| >1, j=1, 2, \cdots, N.\)If we assume the above Sobolev-type inner product is sequentially-ordered, i. e. \(M_{j, i}=0, i=0, 1, \cdots, d_{j}-1, j= 1, 2, \cdots, N,\) then the polynomial \(S_{n}(z)\) has at least \(n-N\) changes of sign in \((-1,1).\) Moreover, for \(n\) large enough all its zeros are real, simple and each sufficiently small neighborhood of \(c_{j}, j= 1, 2, \cdots,N\) contains exactly one zero of \(S_{n}\) and the remaining zeros lie on \((-1,1).\) This last result is a direct consequence of the outer relative asymptotics for Sobolev-type orthogonal polynomials given in [G. López Lagomasino et al., Constr. Approx. 11, No. 1, 107–137 (1995; Zbl 0840.42017)].Let denote by \(\{Q_{n}(x)\}_{n\geq0}\) the sequence of orthogonal polynomials with respect to the measure \(\rho(x) d \mu(x),\) where \(\rho(x)= \prod_{c_{j} <-1} (z-c_{j})^{d_{j} +1} \prod_{c_{j} >1} (c_{j}-z)^{d_{j} +1}\)is a positive polynomial on \((-1,1).\) The authors introduce the sequence of polynomials \(\{S_{n}^{[k]} (z) \} _{n\geq0}\) defined as \[ S_{n}^{[k]} (z)= \int_{-1}^{1} \frac{S_{n+k}(z)- S_{n+k}(x)}{z-x} Q_{k-1}(x) \rho(x) d\mu(x).\] Notice that they constitute a natural extension of the so called associated polynomials of \(k\)th kind in the theory of standard orthogonal polynomials (see [W. Van Assche, J. Comput. Appl. Math. 37, No. 1–3, 237–249 (1991; Zbl 0744.42012)]).The following extended Markov’s theorem is proved. Let consider a sequentially ordered discrete Sobolev inner product and let \(\mu\) be a measure in the Nevai class \(M(0,1).\) Then, for \(k \in \mathbb{N}\) the sequence of rational functions \(\{\frac{S_{n}^{[k]}(z)} {S_{n+k} (z)}\}_{n\geq0}\) uniformly converges to the function \(\int_{-1}^{1} \frac {Q_{k-1}(x)}{z-x} \rho(x) d\mu(x)\) in every compact subset of the exterior of \([-1,1] \bigcup \{c_{1}, c_{2}, \cdots, c_{N}\}\) in the complex plane.The function on the right hand side is said to be the \(k\)th Markov type function associated with the measure \(\rho(x) d\mu(x).\) On the other hand, an estimate of the degree of convergence for the sequence of rational functions in the left hand side is deduced. Reviewer: Francisco Marcellán (Leganes) Cited in 1 Document MSC: 41A20 Approximation by rational functions 30E10 Approximation in the complex plane 33C47 Other special orthogonal polynomials and functions 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:rational approximation; Sobolev orthogonality; Markov’s theorem; zero location Citations:Zbl 0405.33009; Zbl 0840.42017; Zbl 0744.42012 PDFBibTeX XMLCite \textit{A. Díaz-González} et al., J. Approx. Theory 260, Article ID 105481, 19 p. (2020; Zbl 1452.41006) Full Text: DOI arXiv References: [1] Baratchart, L.; L. Yattselev, M., Meromorphic approximations of complex Cauchy transforms with polar singularities, Mat. Sb.. Mat. Sb., Sb. Math., 200, 1261-1297 (2009), English translation · Zbl 1218.41006 [2] de la Calle Ysern, B.; López Lagomasino, G., Convergence of multipoint Padé-type approximants, J. Approx. Theory, 109, 257-278 (2001) · Zbl 0982.41008 [3] Derevyagin, M. S.; Derkach, V. A., On the convergence of Padé approximants of generalized nevanlinna functions, Trans. Mosk. Mat. Obs.. Trans. Mosk. Mat. Obs., Trans. Moscow Math. Soc., 68, 119-162 (2007), English translation in · Zbl 1161.30025 [4] Gonchar, A. A., On the convergence of Padé approximants for some classes of meromorphic functions, Mat. Sb.. Mat. Sb., Math. USSR Sb., 26, 555-575 (1975), English translation · Zbl 0341.30029 [5] Gonchar, A. A.; López Lagomasino, G.; Rakhmanov, E. A., (Some Old and New Results in Rational Approximation Theory. Some Old and New Results in Rational Approximation Theory, Sem. Mat. García de Galdeano (1989), Univ. de Zaragoza: Univ. de Zaragoza Zaragoza), 1-30 [6] Gonchar, A. A.; Rakhmanov, E. A.; Sorokin, V. N., Hermite-Padé approximants for systems of Markov-type functions, Sb. Math., 188, 33-58 (1997) · Zbl 0889.41011 [7] Hille, E., Analytic Function Theory, Vol. I (1982), Chelsea Pub. Co.: Chelsea Pub. Co. NY [8] López Lagomasino, G., (Survey on Multipoint Padé Approximation to Markov Type Meromorphic Functions and Asymptotic Properties of the Orthogonal Polynomials Generated By Them. Survey on Multipoint Padé Approximation to Markov Type Meromorphic Functions and Asymptotic Properties of the Orthogonal Polynomials Generated By Them, Lecture Notes in Math., vol. 1171 (1985), Springer: Springer Berlin), 309-316 · Zbl 0588.41010 [9] López Lagomasino, G.; Marcellán, F.; Van Assche, W., Relative asymptotics for orthogonal polynomials with respect to a discrete Sobolev inner product, Constr. Approx., 11, 107-137 (1995) · Zbl 0840.42017 [10] López Lagomasino, G.; Pijeira, H.; Pérez, I., Sobolev orthogonal polynomials in the complex plane, J. Comput. Appl. Math., 127, 219-230 (2001) · Zbl 0973.42015 [11] Marcellán, F.; Mendes, A.; Pijeira, H., Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation, J. Difference Equations Appl., 19, 1632-1644 (2013) · Zbl 1278.42035 [12] Marcellán, F.; Xu, Y., On Sobolev orthogonal polynomials, Expo. Math., 33, 308-352 (2015) · Zbl 1351.33011 [13] Martinez-Finkelshtein, A., Analytic properties of Sobolev orthogonal polynomials revisited, J. Comput. Appl. Math., 127, 255-266 (2001) · Zbl 0971.33004 [14] Nevai, P., Orthogonal Polynomials, Vol. 213 (1979), Mem. Amer. Math. Soc.: Mem. Amer. Math. Soc. Providence RI [15] Nikishin, E. M.; Sorokin, V. N., (Rational Approximations and Orthogonality. Rational Approximations and Orthogonality, Transl. Math. Monogr., vol. 92 (1991), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0733.41001 [16] Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials (2002), Oxford Univ. Press: Oxford Univ. Press NY · Zbl 1072.30006 [17] Rakhmanov, E. A., Convergence of diagonal Padé approximants, Mat. Sb.. Mat. Sb., Math. USSR Sb., 33, 243-260 (1977), English translation · Zbl 0398.30032 [18] Rakhmanov, E. A., On the asymptotics of the ratio of orthogonal polynomials, Mat. Sb.. Mat. Sb., Math. USSR Sb., 32, 199-213 (1977), English translation · Zbl 0401.30033 [19] Rakhmanov, E. A., On the asymptotics of the ratio of orthogonal polynomials II, Mat. Sb.. Mat. Sb., Math. USSR Sb., 46, 105-117 (1983), English translation · Zbl 0515.30030 [20] Shohat, J., Théorie Générale Des Polynomes Orthogonaux de Tchebichef (1934), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0009.24604 [21] Szegő, G., (Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Series, vol. 23 (1975), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · JFM 61.0386.03 [22] Van Assche, W., Orthogonal polynomials, associated polynomials and functions of the second kind, J. Comput. Appl. Math., 37, 237-249 (1991) · Zbl 0744.42012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.