## General Orthogonal Polynomials (1992)

Venue: | in “Encyclopedia of Mathematics and its Applications,” 43 |

Citations: | 59 - 6 self |

### BibTeX

@INPROCEEDINGS{Totik92generalorthogonal,

author = {Vilmos Totik},

title = {General Orthogonal Polynomials},

booktitle = {in “Encyclopedia of Mathematics and its Applications,” 43},

year = {1992},

publisher = {University Press}

}

### OpenURL

### Abstract

Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.

### Citations

523 |
An introduction to orthogonal polynomials
- Chihara
- 1978
(Show Context)
Citation Context ...tted works. For further backgound on orthogonal polynomials, the reader can consultsOrthogonal Polynomials 73 the books Szegő [91], Simon [80]-[81], Freud [27], Geronimus [34], Gautschi [28], Chicara =-=[18]-=-, Ismail [39]. This is a largely extended version of the first part of the article Golinskii– Totik [36]. Acknowledgement. Research was supported by NSF grant DMS-040650 and OTKA T049448, TS44782, and... |

308 |
Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, 316
- Saff, Totik
- 1997
(Show Context)
Citation Context ...R. It is a general feature of exponential weights that the behavior of zeros of the polynomials is governed by the solution of a weighted energy problem (weighted equilibrium measures, see Saff–Totik =-=[77]-=-). If κn is the leading coefficient of pn, i.e., pn(z) = κnz n + · · ·, then (Lubinsky–Saff [50]) lim n→∞ κnπ 1/2 2 −n e −n/α n (n+1/2)/α = 1,sand we have lim n→∞ |pn(n 1/α γαz)| 1/n � = exp log |z + ... |

294 |
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. AMS, 2000 [2] Deift P, Kamvissis S, Kriecherbauer T, Zhou X. The Toda rarefaction problem. Comm Pure Appl Math
- Deift
- 1996
(Show Context)
Citation Context ... (seeSection 4) can be expressed in terms of the entries of Y1, where Y1 is the matrixdefined by Y (z) ` zn 0 0 zn ' =: I + z1Y1 + O ` 1 z2 ' . For details on this Riemann-Hilbert approach, see Deift =-=[20]-=-. Orthogonal polynomials with respect to inner products Sometimes one talks about orthogonal polynomials with respect to an innerproduct h* , *i which is defined on some linear space containing all po... |

218 |
The classical moment problem and some related questions
- Akhiezer
- 1965
(Show Context)
Citation Context ...first and second kind orthogonal polynomials pn and qn: A(z) = z � n qn(0)qn(z); B(z) = −1 + z � n qn(0)pn(z); C(z) = 1 + z � n pn(0)qn(z); D(z) = z � n pn(0)pn(z). For all these results see Akhiezer =-=[2]-=-, and for an operator theoretic approach to the moment problem see Simon [78] (in particular, Theorems 3 and 4.14). Jacobi matrices and spectral theory of self-adjoint operators Tridiagonal, so called... |

200 |
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
- Deift, Kriecherbauer, et al.
- 1999
(Show Context)
Citation Context ... while the former is strong asymptotics. Strong asymptotics for pn(z) on different parts of the complex plane was given using the Riemann–Hilbert approach, see Deift [20] and Kriecherbauer–McLaughlin =-=[44]-=- and the references there. On the real line we have a Plancherel–Rotach type formula � 2 − π n 1/2α pn(wα; n 1/α γαx)exp(−nγ α α|x| α )− 1 4√ 1 − x2 cos � 1 2 arccosx + nπµw([x, 1]) − π � → 0 4 unifor... |

133 | Krawtchouk polynomials of several variables
- Xu, Hahn
(Show Context)
Citation Context ...nal polynomials of several variables lies also close to this algebraic part of the theory. To discuss them would take us too far from our main direction; rather we refer the reader to the recent book =-=[24]-=- by C. F. Dunkl and Y. Xu. The other part is the analytical aspect of the theory. Its methods are analytical, and it deals with questions that are typical in analysis, or questions that have emerged i... |

105 |
Classical and quantum orthogonal polynomials in one variable
- Ismail
- 2009
(Show Context)
Citation Context ...ynomials. All the discrete polynomials and the qanalogues of classical ones belong to this theory. We will not treat this part; the interested reader can consult the three recent excellent monographs =-=[39]-=- by M. E. H. Ismail, [28] by W. Gautschi and [6] by G. E. Andrews, R. Askey and R. Roy. Much of the present state of the theory of orthogonal polynomials of several variables lies also close to this a... |

100 | Sum rules for Jacobi matrices and their applications to spectral theory
- Killip, Simon
- 2003
(Show Context)
Citation Context ...one in several ways. After numerous works in the subject by Szegő, Shohat, Geronimus, Krein, Kolmogorov and others, a complete characterization for J − J0 being a HilbertSchmidt operator was given in =-=[43]-=- by R. Killip and B. Simon (note that µ is assumed to have total mass 1): � (an − 1/2) 2 + � < ∞ (11.1) if and only if the following conditions hold: n (i) the support of µ is [−1, 1] plus some additi... |

97 |
The isomonodromy approach to matrix models in 2D quantum gravity
- Fokas, Its, et al.
- 1992
(Show Context)
Citation Context ...rm dµ(t) = w(t)dt with some smooth function w. A new approach to generating orthogonal polynomials that has turned out to be of great importance was given in the early 1990’s by Fokas, Its and Kitaev =-=[26]-=-. Consider 2 × 2 matrices � � Y11(z) Y12(z) Y (z) = Y21(z) Y22(z)sOrthogonal Polynomials 75 where the Yij are analytic functions on C \ R, and solve for such matrices the following matrix-valued Riema... |

88 | The classical moment problem as a self-adjoint finite difference operator
- Simon
- 1998
(Show Context)
Citation Context ...(z); B(z) = −1 + z � n qn(0)pn(z); C(z) = 1 + z � n pn(0)qn(z); D(z) = z � n pn(0)pn(z). For all these results see Akhiezer [2], and for an operator theoretic approach to the moment problem see Simon =-=[78]-=- (in particular, Theorems 3 and 4.14). Jacobi matrices and spectral theory of self-adjoint operators Tridiagonal, so called Jacobi matrices ⎛ b0 ⎜ a0 ⎜ J = ⎜ 0 ⎜ ⎝ 0 . a0 b1 a1 0 . 0 a1 b2 a2 . 0 0 a2... |

76 |
Orthogonal Polynomials: Computation and Approximation
- Gautschi
- 2004
(Show Context)
Citation Context ...e polynomials and the qanalogues of classical ones belong to this theory. We will not treat this part; the interested reader can consult the three recent excellent monographs [39] by M. E. H. Ismail, =-=[28]-=- by W. Gautschi and [6] by G. E. Andrews, R. Askey and R. Roy. Much of the present state of the theory of orthogonal polynomials of several variables lies also close to this algebraic part of the theo... |

47 | Orthogonal Polynomials for Exponential Weights
- Levin, Lubinsky
- 2001
(Show Context)
Citation Context ...ogonal polynomials since the early 1980’s. In the last 20 years D. LubinskysOrthogonal Polynomials 105 with coauthors have conducted systematic studies on exponential weights, see e.g. Levin–Lubinsky =-=[46, 45]-=-, Lubinsky [49], Lubinsky–Saff [50], Van Assche [94]; we should mention the names E. Levin, E. B. Saff, W. Van Assche, E. A. Rahmanov and H. N. Mhaskar. In the mid 1990’s a new stimulus came from the ... |

40 |
Assche, Asymptotics for Orthogonal Polynomials
- van
- 1987
(Show Context)
Citation Context ...20 years D. LubinskysOrthogonal Polynomials 105 with coauthors have conducted systematic studies on exponential weights, seee.g. Levin-Lubinsky [46, 45], Lubinsky [49], Lubinsky-Saff [50], Van Assche =-=[94]-=-; we should mention the names E. Levin, E. B. Saff, W. Van Assche, E. A.Rahmanov and H. N. Mhaskar. In the mid 1990's a new stimulus came from the Riemann-Hilbert approach that was used together with ... |

39 |
Über Sturm–Liouvillesche polynomsysteme
- Bochner
- 1929
(Show Context)
Citation Context ...s required that this have a polynomial solution of exact degree n for all n = 0, 1, . . ., for which the corresponding λ and y(x) will be denoted by λn and yn(x), respectively. Bochner’s theorem from =-=[16]-=- states that, except for some trivial solutions of the form y(x) = ax n + bx m and for some polynomials related to Bessel functions, the only solutions are (in all of them we can take h(x) = 0) • Jaco... |

38 |
Orthogonal polynomials
- Nevai
- 1979
(Show Context)
Citation Context ... is equivalent to (10.1). Ratio asymptotics If one assumes weaker conditions then necessarily weaker results will follow. A large and important class of measures is the Nevai class M(b, a) (see Nevai =-=[62]-=-), for which the coefficients in the three-term recurrence xpn(x) = anpn+1(x) + bnpn(x) + an−1pn−1(x) satisfy an → a, bn → b. This is equivalent to ratio asymptotics pn+1(z) lim n→∞ pn(z) = z − b + � ... |

38 |
On the asymptotics of the ratio of orthogonal polynomials
- Rakhmanov
- 1983
(Show Context)
Citation Context ...on-real z, then µ ∈ M(b, a) for some a, b (Simon [79]). The classes M(b, a) are scaled versions of each other, and the most important condition ensuring M(0, 1/2) is given in Rakhmanov’s theorem from =-=[75]-=-: if µ is supported in [−1, 1] and µ ′ > 0 almost everywhere on [−1, 1], then µ ∈ M(0, 1/2). Conversely, Blumenthal’s theorem from [15] states that µ ∈ M(0, 1/2) implies that the support of µ is [−1, ... |

36 |
Extremal polynomials associated with a system of curves
- Widom
- 1969
(Show Context)
Citation Context ...amma n changewith n, and hence so does Fn, and this is the reason that a single asymptoticformula like (10.4) or (10.3) does not hold. The fundamentals of the theory were laid out in H. Widom's paper =-=[97]-=-; and since then many results have beenobtained by F. Peherstorfer and his collaborators, as well as A. I. Aptekarev, J. Geronimo, S. P. Suetin and W. Van Assche. The theory has deep connectionswith f... |

33 |
A.: Spectra of Random and Almost-Periodic Operators. Grundlehren der Mathematischen Wissenschaften
- Pastur, Figotin
- 1992
(Show Context)
Citation Context ... the n-th (weighted) Christoffel function associated with the weight w n , while the limit of the left hand side (as n → ∞) is known as the density of states. See, e.g., Mehta [60] and Pastur–Figotin =-=[68]-=-. 7 Some questions leading to classical orthogonal polynomials There are almost an infinite number of problems where classical orthogonal polynomials emerge. Let us just mention a few. Electrostatics ... |

32 | Sum rules and the Szegő condition for orthogonal polynomials on the real line - Simon, Zlatos |

31 |
The scattering problem for a discrete Sturm-Liouville operator
- Aptekarev, Nikishin
- 1983
(Show Context)
Citation Context ... . ⎥ ⎦s16 Matrix orthogonal polynomials V. Totik 114 In the last 20 years the fundamentals of matrix orthogonal polynomials have been developed mainly by A. Durán and his coauthors (see also the work =-=[9]-=- by A. I. Aptekarev and E. M. Nikishin). The theory shows many similarities with the scalar case, but there is an unexpected richness which is still to be explored. For all the results in this section... |

31 | Riemann-Hilbert problems for multiple orthogonal polynomials
- Assche, Geronimo, et al.
- 2001
(Show Context)
Citation Context ...depends only on m = mn and not on the particularchoice of the sequence mj leading to it.sOrthogonal Polynomials 113 The Riemann-Hilbert problem There is an approach (see Van Assche-Geronimo-Kuijlaars =-=[96]-=-) to both typesof multiple orthogonality in terms of matrix-valued Riemann-Hilbert problem for (r + 1) * (r + 1) matrices Y = (Yij (z))ri,j=0.If duj(x) = wj dx, then one requires that * Y is analytic ... |

28 |
On Rakhmanov’s theorem for Jacobi matrices
- Denisov
(Show Context)
Citation Context ...rom [15] states that µ ∈ M(0, 1/2) implies that the support of µ is [−1, 1] plus at most countably many points that converge to ±1. Thus, in this respect the extension of Rakhmanov’s theorem given in =-=[22]-=- by Denisov is of importance: if µ ′ > 0 almost everywhere on [−1, 1] and outside [−1, 1] the measure µ has at most countably many mass points converging to ±1, then µ ∈ M(0, 1/2). However, M(0, 1/2)s... |

28 |
On the trigonometric moment problem
- Geronimus
- 1946
(Show Context)
Citation Context ...ortance or quality of the omitted works. For further backgound on orthogonal polynomials, the reader can consultsOrthogonal Polynomials 73 the books Szegő [91], Simon [80]-[81], Freud [27], Geronimus =-=[34]-=-, Gautschi [28], Chicara [18], Ismail [39]. This is a largely extended version of the first part of the article Golinskii– Totik [36]. Acknowledgement. Research was supported by NSF grant DMS-040650 a... |

27 |
Souillard,From power pure point to continuous spectrum in disordered systems
- Delyon, Simon, et al.
- 1985
(Show Context)
Citation Context ...countably many mass points converging to ±1, then µ ∈ M(0, 1/2). However, M(0, 1/2)sV. Totik 100 contains many other measures not just those that are in these theorems, e.g. in Delyon–Simon–Souillard =-=[21]-=- a continuous singular measure in the Nevai class was exhibited, and the result in Totik [92] shows that the Nevai class contains practically all types of measures allowed by Blumenthal’s theorem. Wea... |

26 |
Orthogonal polynomials and their zeros
- Nevai, Totik
- 1989
(Show Context)
Citation Context ...). Conversely, if the support of µ is [−1, 1] plus some additional mass points converging to ±1 and µ ′ (x) > 0 for almost all x ∈ [−1, 1], then µ ∈ M(a, b) (Denisov [22], Rakhmanov [75], Nevai–Totik =-=[63]-=-). No spectral characterization of µ ∈ M(0, 1/2) is known; this important class seems to contain all sorts of measures. For example, if ν is any measure with support [−1, 1] then there is a µ ∈ M(0, 1... |

25 | On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur - Geronimus - 1944 |

24 |
Aptekarev, Multiple orthogonal polynomials
- I
- 1998
(Show Context)
Citation Context ... Asymptotic behavior of multiple orthogonal polynomials is not fully understood yet due to the interaction of the different measures. For the existingsOrthogonal Polynomials 111 results see Aptekarev =-=[8]-=-, Van Assche [95], Van Assche’s Chapter 23 in [39] and the references therein. Types and normality On R let there be given r measures µ1, . . .,µr with finite moments and infinite support, and conside... |

22 |
Characterization theorems for orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials: Theory and Practice
- Al-Salam
- 1990
(Show Context)
Citation Context ...of these properties has a converse, namely if a system of orthogonal polynomials possesses any of these properties, then it is (up to a change of variables) one of the classical systems, see Al-Salam =-=[3]-=-. See also Bochner’s result in the next section claiming that the classical orthogonal polynomials are essentially the only polynomial (not just orthogonal polynomial) systems that satisfy a certain s... |

18 |
Asymptotics for Christoffel functions for general measures on the real line
- Totik
(Show Context)
Citation Context ...ngs to the Reg class there (see Section 9) and log µ ′ is integrable over an interval I ⊂ [−1, 1], then for almost all x ∈ I k=0 lim n→∞ nλn(µ, x) = π � 1 − x2 µ ′ (x). This result is true (see Totik =-=[93]-=-) in the form lim n→∞ nλn(µ, x) = dµ(x) , a.e. x ∈ I dωsupp(µ)(x) when the support is a general compact subset of R, µ ∈ Reg and log µ ′ ∈ L 1 (I). Often only a rough estimate is needed for Christoffe... |

16 |
Orthogonal matrix polynomials satisfying second-order differential equations
- Durán, Grünbaum
(Show Context)
Citation Context ...shows many similarities with the scalar case, but there is an unexpected richness which is still to be explored. For all the results in this section see López-Rodriguez–Durán [48] and Durán– Grünbaum =-=[25]-=- and the numerous references there. Matrix orthogonal polynomials An N × N matrix ⎛ p11(t) ⎜ P(t) = ⎝ . · · · . .. ⎞ p1N(t) ⎟ . ⎠ pN1(t) · · · pNN(t) with polynomial entries pij(t) of degree at most n... |

16 |
Orthogonal Polynomials, Akadémiai Kiadó, Budapest and Pergamon
- Freud
- 1971
(Show Context)
Citation Context ...o way on the importance or quality of the omitted works. For further backgound on orthogonal polynomials, the reader can consultsOrthogonal Polynomials 73 the books Szegő [91], Simon [80]-[81], Freud =-=[27]-=-, Geronimus [34], Gautschi [28], Chicara [18], Ismail [39]. This is a largely extended version of the first part of the article Golinskii– Totik [36]. Acknowledgement. Research was supported by NSF gr... |

16 |
Christoffel functions and orthogonal polynomials for Erdős weights
- Levin, Lubinsky, et al.
- 1994
(Show Context)
Citation Context ...ogonal polynomials since the early 1980’s. In the last 20 years D. LubinskysOrthogonal Polynomials 105 with coauthors have conducted systematic studies on exponential weights, see e.g. Levin–Lubinsky =-=[46, 45]-=-, Lubinsky [49], Lubinsky–Saff [50], Van Assche [94]; we should mention the names E. Levin, E. B. Saff, W. Van Assche, E. A. Rahmanov and H. N. Mhaskar. In the mid 1990’s a new stimulus came from the ... |

16 |
Strong asymptotics for extremal polynomials associated with weights on
- Lubinsky, Saff
- 1988
(Show Context)
Citation Context ...’s. In the last 20 years D. LubinskysOrthogonal Polynomials 105 with coauthors have conducted systematic studies on exponential weights, see e.g. Levin–Lubinsky [46, 45], Lubinsky [49], Lubinsky–Saff =-=[50]-=-, Van Assche [94]; we should mention the names E. Levin, E. B. Saff, W. Van Assche, E. A. Rahmanov and H. N. Mhaskar. In the mid 1990’s a new stimulus came from the Riemann–Hilbert approach that was u... |

16 | Orthogonal polynomials with complex-valued weight function - Stahl - 1986 |

15 |
Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains
- I
- 1986
(Show Context)
Citation Context ...S. P. Suetin and W. Van Assche. The theory has deep connections with function theory, the theory of Abelian integrals and the theory of elliptic functions. We refer the reader to the papers Aptekarev =-=[7]-=-, Geronimo–Van Assche [31], Peherstorfer [69]–[72] and Suetin [88]–[89].sAsymptotics for Christoffel functions The Christoffel functions λn(µ, x) −1 = V. Totik 102 n� pk(µ, x) 2 behave somewhat more r... |

15 | A Christoffel-Darboux formula for multiple orthogonal polynomials, preprint math.CA/0402031
- Daems, Kuijlaars
(Show Context)
Citation Context ...he corresponding component of m j . Set Pj = Pm j , Qj = Qm j+1 and with m = m n h (j) m := � Pm(x)x (m)j dµj(x), where (m)j denotes the j-th component of the multiindex m. Then (see Daems– Kuijlaars =-=[19]-=-), again with m = m n , n−1 � (x − y) Pk(x)Qk(y) = Pm(x)Qm(y) − k=0 r� j=1 h (j) m h (j) Pm−e (x)Qm+e (y). j j m−ej Thus, the left hand side depends only on m = m n and not on the particular choice of... |

15 |
Extensions of Szegő’s theory of orthogonal polynomials
- Maté, Nevai, et al.
- 1987
(Show Context)
Citation Context ... Blumenthal’s theorem. Weak and relative asymptotics Under Rahmanov’s condition supp(µ) = [−1, 1], µ ′ > 0 a.e., some parts of Szegő’s theory can be proven in a weaker form (see e.g. Máté–Nevai–Totik =-=[57, 58]-=-). In these the Turán determinants Tn(x) := p 2 n (x) − pn−1(x)pn+1(x) play a significant role. In fact, then given any interval D ⊂ (−1, 1) the Turán determinant Tn is positive on D for all large n, ... |

15 |
Strong and weak convergence of orthogonal polynomials
- Máté, Nevai, et al.
- 1987
(Show Context)
Citation Context ... Blumenthal’s theorem. Weak and relative asymptotics Under Rahmanov’s condition supp(µ) = [−1, 1], µ ′ > 0 a.e., some parts of Szegő’s theory can be proven in a weaker form (see e.g. Máté–Nevai–Totik =-=[57, 58]-=-). In these the Turán determinants Tn(x) := p 2 n (x) − pn−1(x)pn+1(x) play a significant role. In fact, then given any interval D ⊂ (−1, 1) the Turán determinant Tn is positive on D for all large n, ... |

15 |
Szegő’s extremum problem on the unit circle
- Máté, Nevai, et al.
- 1991
(Show Context)
Citation Context ...88]–[89].sAsymptotics for Christoffel functions The Christoffel functions λn(µ, x) −1 = V. Totik 102 n� pk(µ, x) 2 behave somewhat more regularly than the orthogonal polynomials. In Máté– Nevai–Totik =-=[59]-=- it was shown that if µ is supported on [−1, 1], it belongs to the Reg class there (see Section 9) and log µ ′ is integrable over an interval I ⊂ [−1, 1], then for almost all x ∈ I k=0 lim n→∞ nλn(µ, ... |

15 | On the distribution of zeros of polynomials orthogonal on the unit circle - Mhaskar, Saff - 1990 |

14 | Lagomasino, Ratio and relative asymptotics of polynomials on an arc of the unit circle - Hernández, López - 1998 |

12 |
Zeros and critical points of Sobolev orthogonal polynomials
- Gautschi, Kuijlaars
- 1997
(Show Context)
Citation Context .... ., r. It is not even known if the zeros are bounded if all the measures µk have compact support. Nonetheless, for the case r = 1, and µ0, µ1 ∈ Reg (see Section 9) it was shown in Gautschi–Kuijlaars =-=[29]-=- that the asymptotic distribution of the zeros of the derivative Q ′ n is the equilibrium measure ωE0∪E1, where Ei is the support of µi, i = 0, 1 (which also have to be assumed to be regular). Further... |

12 |
Weighted polynomial inequalities with doubling and
- Mastroianni, Totik
(Show Context)
Citation Context ...)(x) when the support is a general compact subset of R, µ ∈ Reg and log µ ′ ∈ L 1 (I). Often only a rough estimate is needed for Christoffel functions, and such a one is provided in Mastroianni–Totik =-=[56]-=-: if µ is supported on [−1, 1] and it is a doubling measure, i.e., µ(2I) ≤ Lµ(I) for all I ⊂ [−1, 1], where 2I is the twice enlarged I, then uniformly on [−1, 1] √ 1 − x2 λn(µ, x) ∼ µ (∆n(x)) ; ∆n(x) ... |

11 | Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight
- Aptekarev, Assche
(Show Context)
Citation Context ...1 log µ exp 2π −1 ′ (t) √ 1 − t2 dt � (with Dµ the Szegő function (10.2)) was proved by J. Nuttall [66], [67], A. A. Gonchar and S. P. Suetin [38]. For a recent Riemann–Hilbert approach see the paper =-=[10]-=- by A. I. Aptekarev and W. Van Assche. A similar result holds on the support of the measure, as well as for the case of varying weights, see Aptekarev–Van Assche [10]. 15 Multiple orthogonality Multip... |

11 |
Orthogonal polynomials on Sobolev spaces: old and new directions
- Marcellán, Alfaro, et al.
- 1993
(Show Context)
Citation Context ...ne is smooth data fitting. The Spanish school around F. Marcellán, G. Lopez and A. Martinez-Finkelshtein has been particularly active in developing this area (see the surveys Marcellán– Alfaro–Rezola =-=[51]-=- and Martinez-Finkelshtein [53, 52] and the references therein). In this section let Qn(z) = zn + · · · denote the monic orthogonal polynomial with respect to the Sobolev inner product (13.1), and qn(... |

10 |
Ueber die Entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für
- Blumenthal
- 1898
(Show Context)
Citation Context ...ondition ensuring M(0, 1/2) is given in Rakhmanov’s theorem from [75]: if µ is supported in [−1, 1] and µ ′ > 0 almost everywhere on [−1, 1], then µ ∈ M(0, 1/2). Conversely, Blumenthal’s theorem from =-=[15]-=- states that µ ∈ M(0, 1/2) implies that the support of µ is [−1, 1] plus at most countably many points that converge to ±1. Thus, in this respect the extension of Rakhmanov’s theorem given in [22] by ... |

10 | Zeros of orthogonal polynomials on the real line
- Denisov, Simon
(Show Context)
Citation Context ...ine. For example, if µ is supported on the real line, then Pc(S(µ)) = S(µ), and if K is a closed interval disjoint from the support, then there is at most one zero in K. It was shown in Denison–Simon =-=[23]-=- that if x0 ∈ R is not in the support, then for some δ > 0 and all n either pn or pn+1 has no zero in (x0 − δ, x0 + δ). Note that if µ is a symmetric measure on [−1, −1/2] ∪ [1/2, 1], then p2n+1(0) = ... |

10 |
Assche, Orthogonal polynomials on several intervals via a polynomial mapping
- Geronimo, Van
- 1988
(Show Context)
Citation Context ...ssche. The theory has deep connections with function theory, the theory of Abelian integrals and the theory of elliptic functions. We refer the reader to the papers Aptekarev [7], Geronimo–Van Assche =-=[31]-=-, Peherstorfer [69]–[72] and Suetin [88]–[89].sAsymptotics for Christoffel functions The Christoffel functions λn(µ, x) −1 = V. Totik 102 n� pk(µ, x) 2 behave somewhat more regularly than the orthogon... |

10 | Denisov’s theorem on recurrence coefficients - Nevai, Totik |

10 |
An introduction to orthogonal polynomials, Mathematics and its
- Chihara
- 1978
(Show Context)
Citation Context ...tted works.For further backgound on orthogonal polynomials, the reader can consultsOrthogonal Polynomials 73 the books Szeg""o [91], Simon [80]-[81], Freud [27], Geronimus [34], Gautschi [28],Chicara =-=[18]-=-, Ismail [39]. This is a largely extended version of the first part of the article Golinskii-Totik [36]. Acknowledgement. Research was supported by NSF grant DMS-040650 andOTKA T049448, TS44782, and w... |