A. I. APTEKAREV, Multiple orthogonal polynomials, J. Comp. Appl. Math. 99 (1998), pp. 423--447.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a vector polynomial Q n n will be called an nth left orthogonal polynomial iff n ; P = 0 for all P 2 C [x] deg P n. Example 7. If r = s = 1, we find the scalar case. If r = 1, s 1, we recover the case of multiple orthogonal polynomials or simultaneous orthogonal polynomials [1]. Definition 8. n; Q n ; Q n Gamma1 ) Similarly, we define Q n; Definition 9. Let M k the moments M k = W (x)dx 2 C ; k = 0; 1 : We gather these moments in an infinite matrix H: H = B B B B B M 1 M 2 M 3 : C C C C ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447.

....are closed subsets of R. We also define # j = the smallest interval that contains supp( j ) Finally we introduce a multi index #n = n 1 , n 2 , n r ) # N r and its length #n = n 1 n 2 . n r . Multiple orthogonal polynomials can now be defined as follows [8, Chapter 4. 3] [2] [10] Definition 2.1 (Type I) An r vector of type I multiple orthogonal polynomials (A #n,1 , A #n,r ) corresponding to the multiindex #n # N r , is such that each A #n,j is a polynomial of degree # n j 1 and the following orthogonality conditions hold: Z x k r X j=1 A ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-- 447.

....Definitions 1 and 2, that the monic polynomial p #,c n,m is unique and that the vector (A #,c n,m , B #,c n,m ) is unique up to a multiplicative factor. In order to achieve this, we will prove that we are dealing here with a special kind of AT system, namely a Nikishin system (see, e.g. [2] [4] 10] 13] 5 Theorem 1 For x 0 and # 1 the following holds: # # 1,c (x) # #,c (x) x # X n=1 1 x j 2 #,n 4 J # 1 (j #,n ) J # # (j #,n ) 12) x Z 0 # d# 1 (t) x t , 13) # #,c (x) # # 1,c (x) # 1 x # X n=1 1 x j 2 # 1,n 4 J # 2 (j ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423--447.

....In this paper we will show that an extension of this Riemann Hilbert approach to (r 1) r 1) matrix valued functions describes certain polynomials which obey orthogonality conditions with respect to r 1 weights on the real line. These polynomials are known as multiple orthogonal polynomials [2] [19] 21] 22] 26] In this introduction we will explain the notion of multiple orthogonal polynomials. In Section 2 we describe the Riemann Hilbert problem for type I multiple orthogonal polynomials and Section 3 gives the Riemann Hilbert problem for type II multiple orthogonal polynomials. In ....

....f j (x i ) exist, so that the Sokhotsky Plemelj formula holds f j (x) f j (x) 2 iw j (x) x 2 R: 1.1) 1.1 Type I multiple orthogonal polynomials Multiple orthogonal polynomials are also known as poly orthogonal polynomials or HermitePad e polynomials. Good references are Aptekarev [2], Mahler [19] the book by Nikishin and Sorokin [21] Nuttall [22] and [26] Here we brie y explain what we mean by type I and type II multiple orthogonal polynomials, how they arise from a problem of simultaneous rational approximation, and state the orthogonality conditions. Let n = n 1 ; n 2 ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447.

....n## d n ( # n) 3 = 1 8 if n # 0 (mod 2) 1 8 if n # 1 (mod 2) 4 Open research problems In the previous sections we gave a short description of multiple orthogonal polynomials and a few examples. For a more detailed account of multiple orthogonal polynomials we refer to Aptekarev [4] and Chapter 4 of the book of Nikishin and Sorokin [34] Multiple orthogonal polynomials arise naturally in Hermite Pade approximation of a system of (Markov) functions. For this kind of simultaneous rational approximation we refer to Mahler [28] and de Bruin [9] 10] Hermite Pade approximation ....

....of indices, etc. The more detailed analytic investigation of the zero distribution, the nth root asymptotics, and the strong asymptotics is more recent and mostly done by researchers from the schools around Nikishin [32] 33] and Gonchar [18] 19] See in particular the work of Aptekarev [4], Kalyagin [20] 25] Bustamante and Lopez [11] but also the work by Driver and Stahl [15] 16] and Nuttall [35] First one needs to understand the analysis of ordinary orthogonal polynomials, and then one has a good basis for studying this extension, for which there are quite a few ....

[Article contains additional citation context not shown here]

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423--447.

.... operators is of interest in all these applications, let us mention the Bogoyavlenskii discrete dynamical system [7] Ruhe s block version of the Lanczos method in Numerical Linear Algebra, or the problem of HermitePad e and Matrix Pad e approximation (for the latter see, e.g. the surveys [4, 5]) In the present paper we will restrict ourselves to the less involved case of three diagonals. To start with, a linear functional c acting on the space of polynomials with complex coefficients is called regular iff det(c(x j k ) j;k=0; n 6= 0 for all n 0. Given a regular c (with c(1) ....

....of the recurrence (1. 2) has been given already by Wall [56, Sections 59 61] Starting with a paper of Aptekarev, Kaliaguine and Van Assche [6] the problem of characterizing the spectrum has received much attention in the last years, see [13, 14, 18, 19] for Jacobi matrices and the survey papers [4, 5] and the references therein for higher order difference operators. A typical example of characterizing the spectrum in terms of only one solution of (1.2) is the following. Theorem 2.9 (see [18, Theorem 2.3] Let A be bounded. Then z 2 Omega Gamma A) if and only if sup n0 P n j=0 jq j (z)j ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998) 423-447.

....deg Q L n n will be called an nth left orthogonal polynomial iff D Q L n ; P E W = 0 for all P 2 C s Theta1 [x] deg P n. Example 7. If r = s = 1, we find the scalar case. If r = 1, s 1, we recover the case of multiple orthogonal polynomials or simultaneous orthogonal polynomials [1]. Definition 8. Q R n; Q R n ; Q R n Gamma1 ) Similarly, we define Q L n; Definition 9. Let M k the moments M k = Z x k W (x)dx 2 C r Thetas ; k = 0; 1 : We gather these moments in an infinite matrix H: H = 0 B B B B B M 0 M 1 M 2 : M 1 M 2 M 3 : M 2 M ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447.

....fixing the leading coefficient to be 1 and leaving n m coefficients to be determined from (2.2) 2. 3) Multiple orthogonal polynomials are related to Hermite Pad e simultaneous rational approximation of a system of Markov functions near infinity, see, e.g. Nikishin and Sorokin [6] and Aptekarev [1]) In the present situation the functions to be approximated are f 1 (z) Z 1 0 x ff ae (x) z Gamma x dx; f 2 (z) Z 1 0 x ff ae 1 (x) z Gamma x dx: Type 1 Hermite Pad e approximation (Latin type) consists of finding polynomials A n;m , B n;m , and C n;m such that A n;m (z)f 1 ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. (to appear)

....refer to Szego [Sz, Chapter 5] MULTIPLE ORTHOGONAL POLYNOMIALS, IRRATIONALITY AND TRANSCENDENCE 5 2. Multiple orthogonal polynomials During the past few years there has been increased interest in multiple orthogonal polynomials, particularly in Eastern Europe. See, e.g. surveys by Aptekarev [A2], de Bruin [dB] and Chapter 4 in the book of Nikishin and Sorokin [NS] For multiple orthogonal polynomials we will need multi indices consisting of r positive integers, for which we use the notation n = n 1 ; n 2 ; n r ) 2 N r , where r 2 N. Furthermore we will use the notation j nj ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423--447.

.... polynomials (see Szego [25] or [27] for a more thorough treatment) and in the next section we will see how some of these facts have an extension to multiple orthogonal polynomials, but that the new setting is richer and still needs further study (see Nikishin and Sorokin [20] and Aptekarev [2] for more information on multiple orthogonal polynomials) Let be a positive measure on the real line for which all the moments exist and for which the support contains infinitely many points. Without loss of generality we will normalize so that it is a probability measure. The monic ....

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423--447.

No context found.

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-- 447.

....polynomial coe#cients # and #. Many of these polynomials have already been studied and applied to diophantine number theory, rational approximation in the complex plane, spectral and scattering problem for higher order di#erence equations and corresponding dynamical systems (see the survey papers [1] [4] 12] 13] 16] 18] In [4] a classification was given of multiple orthogonal polynomials with respect to semiclassical weights of class s # 1, where s = max deg # 2 , deg # 1 , and the weights for the orthogonality relation in (1.1) are the same, but restricted to di#erent contours # j ....

.... satisfy the multiple orthogonality relations Z # Q #n (z)z # w k (z)dz = 0, # = 0, n k 1, k = 1, p, MULTIPLE ORTHOGONAL POLYNOMIALS 9 with respect to the corresponding classical weights from the above tables and the classical paths of integration, i.e. the interval [0, 1] for Jacobi Pineiro polynomials, the interval [0, #) for multiple Laguerre (I and II) polynomials, the interval ( #, #) for multiple Hermite polynomials, and a circle around the origin for multiple Bessel polynomials. 3. Explicit expressions and recurrence relations In this section we present ....

[Article contains additional citation context not shown here]

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-- 447.

No context found.

A. I. APTEKAREV, Multiple orthogonal polynomials, J. Comp. Appl. Math. 99 (1998), pp. 423--447.

No context found.

A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423--447.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC