Multiple Orthogonal Polynomials

Transcription

1 Summer school on OPSF, University of Kent June, 2017

2 Introduction For this course I assume everybody is familiar with the basic theory of orthogonal polynomials:

3 Introduction For this course I assume everybody is familiar with the basic theory of orthogonal polynomials: Definition p n (x)p m (x) dµ(x) = δ m,n, m, n N.

4 Introduction For this course I assume everybody is familiar with the basic theory of orthogonal polynomials: Definition Location of zeros p n (x)p m (x) dµ(x) = δ m,n, m, n N.

5 Introduction For this course I assume everybody is familiar with the basic theory of orthogonal polynomials: Definition Location of zeros p n (x)p m (x) dµ(x) = δ m,n, m, n N. Three term recurrence relation: xp n (x) = a n+1 p n+1 (x) + b n p n (x) + a n p n 1 (x), n 1,

6 Introduction For this course I assume everybody is familiar with the basic theory of orthogonal polynomials: Definition Location of zeros p n (x)p m (x) dµ(x) = δ m,n, m, n N. Three term recurrence relation: xp n (x) = a n+1 p n+1 (x) + b n p n (x) + a n p n 1 (x), n 1, Classical orthogonal polynomials: Jacobi - Laguerre - Hermite

7 Plan of the course

8 Plan of the course lecture 1: Definitions + basic properties

9 Plan of the course lecture 1: Definitions + basic properties lecture 2: Hermite-Padé, Multiple Hermite polynomials

10 Plan of the course lecture 1: Definitions + basic properties lecture 2: Hermite-Padé, Multiple Hermite polynomials lecture 3: Multiple Laguerre polynomials (first and second kind)

11 Plan of the course lecture 1: Definitions + basic properties lecture 2: Hermite-Padé, Multiple Hermite polynomials lecture 3: Multiple Laguerre polynomials (first and second kind) lecture 4: Multiple Jacobi polynomials: Jacobi-Angelesco + Jacobi-Piñeiro

12 Plan of the course lecture 1: Definitions + basic properties lecture 2: Hermite-Padé, Multiple Hermite polynomials lecture 3: Multiple Laguerre polynomials (first and second kind) lecture 4: Multiple Jacobi polynomials: Jacobi-Angelesco + Jacobi-Piñeiro lecture 5: Riemann-Hilbert problem

13 Definition: type I MOPS Let r N and let µ 1,..., µ r be positive measures on the real line, for which all the moments exist.

14 Definition: type I MOPS Let r N and let µ 1,..., µ r be positive measures on the real line, for which all the moments exist. We use multi-indices n = (n 1, n 2,..., n r ) N r and denote their length by n = n 1 + n n r.

15 Definition: type I MOPS Let r N and let µ 1,..., µ r be positive measures on the real line, for which all the moments exist. We use multi-indices n = (n 1, n 2,..., n r ) N r and denote their length by n = n 1 + n n r. Definition (type I) Type I multiple orthogonal polynomials for n consist of the vector (A n,1,..., A n,r ) of r polynomials, with deg A n,j n j 1, for which x k r A n,j (x) dµ j (x) = 0, 0 k n 2, j=1 with normalization x n 1 r A n,j (x) dµ j (x) = 1. j=1

16 Definition: type II MOPS Definition (type II) The type II multiple orthogonal polynomial for n is the monic polynomial P n of degree n for which x k P n (x) dµ j (x) = 0, 0 k n j 1, for 1 j r.

17 Normal indices a multi-index n is normal if the type I vector (A n,1,... A n,r ) exists and is unique

18 Normal indices a multi-index n is normal if the type I vector (A n,1,... A n,r ) exists and is unique if the monic type II multiple orthogonal polynomial P n exists and is unique

19 Normal indices a multi-index n is normal if the type I vector (A n,1,... A n,r ) exists and is unique if the monic type II multiple orthogonal polynomial P n exists and is unique M n (1) 1 M (2) n 2 det.. 0, M n (r) r M(j) n j = m (j) k = m (j) 0 m (j) 1 m (j) n 1 m (j) 1 m (j) 2 m (j).. m (j) n j 1 x k dµ j (x)... n.. m (j) n j m (j) n +n j 2,

20 Special systems: Angelesco systems Definition (Angelesco system) The measures (µ 1,..., µ r ) are an Angelesco system if the supports of the measures are subsets of disjoint intervals j, i.e., supp(µ j ) j and i j = whenever i j.

21 Special systems: Angelesco systems Definition (Angelesco system) The measures (µ 1,..., µ r ) are an Angelesco system if the supports of the measures are subsets of disjoint intervals j, i.e., supp(µ j ) j and i j = whenever i j. Usually one allows that the intervals are touching, i.e., i j = whenever i j.

22 Special systems: Angelesco systems Theorem (Angelesco, Nikishin) The type II multiple orthogonal polynomial P n has exactly n j distinct zeros on j for 1 j r.

23 Special systems: Angelesco systems Theorem (Angelesco, Nikishin) The type II multiple orthogonal polynomial P n has exactly n j distinct zeros on j for 1 j r. Corollary Every multi-index n is normal (an Angelesco system is perfect).

24 Special systems: Angelesco systems Theorem (Angelesco, Nikishin) The type II multiple orthogonal polynomial P n has exactly n j distinct zeros on j for 1 j r. Corollary Every multi-index n is normal (an Angelesco system is perfect). Exercise Show that every A n,j has n j 1 zeros on j.

25 Special systems: AT systems Definition The functions ϕ 1,..., ϕ n are a Chebyshev system on [a, b] if every linear combination n i=1 a iϕ i with (a 1,..., a n ) (0,..., 0) has at most n 1 zeros on [a, b].

26 Special systems: AT systems Definition The functions ϕ 1,..., ϕ n are a Chebyshev system on [a, b] if every linear combination n i=1 a iϕ i with (a 1,..., a n ) (0,..., 0) has at most n 1 zeros on [a, b]. Definition (AT-system) The measures (µ 1,..., µ r ) are an AT-system on the interval [a, b] if the measures are all absolutely continuous with respect to a positive measure µ on [a, b], i.e., dµ j (x) = w j (x) dµ(x) (1 j r), and for every n the functions w 1 (x), xw 1 (x),..., x n 1 1 w 1 (x), w 2 (x), xw 2 (x),..., x n 2 1 w 2 (x), are a Chebyshev system on [a, b]...., w r (x), xw r (x),..., x nr 1 w r (x)

27 Special systems: AT-systems Theorem For an AT-system the function Q n (x) = r A n,j (x)w j (x) j=1 has exactly n 1 sign changes on (a, b).

28 Special systems: AT-systems Theorem For an AT-system the function Q n (x) = r A n,j (x)w j (x) j=1 has exactly n 1 sign changes on (a, b). Theorem If (µ 1,..., µ r ) is an AT-system, then the type II multiple orthogonal polynomial P n has exactly n distinct zeros on (a, b).

29 Special systems: AT-systems Theorem For an AT-system the function Q n (x) = r A n,j (x)w j (x) j=1 has exactly n 1 sign changes on (a, b). Theorem If (µ 1,..., µ r ) is an AT-system, then the type II multiple orthogonal polynomial P n has exactly n distinct zeros on (a, b). Corollary Every multi-index in an AT-system is normal (an AT-system is perfect).

30 Special systems: Nikishin systems Introduced by E.M. Nikishin in 1980.

31 Special systems: Nikishin systems Introduced by E.M. Nikishin in Definition (Nikishin system for r = 2) A Nikishin system of order r = 2 consists of two measures (µ 1, µ 2 ), both supported on an interval 2, and such that dµ 2 (x) dµ 1 (x) = dσ(t) 1 x t, where σ is a positive measure on an interval 1 and 1 2 =.

32 Special systems: Nikishin systems Introduced by E.M. Nikishin in Definition (Nikishin system for r = 2) A Nikishin system of order r = 2 consists of two measures (µ 1, µ 2 ), both supported on an interval 2, and such that dµ 2 (x) dµ 1 (x) = dσ(t) 1 x t, where σ is a positive measure on an interval 1 and 1 2 =. Theorem (Nikishin, Driver-Stahl) A Nikishin system of order two is perfect.

33 Special systems: Nikishin systems Notation: σ 1, σ 2 is a measure which is absolutely continuous with respect to σ 1 and for which the Radon-Nikodym derivative is a Stieltjes transform of σ 2 : ( ) dσ2 (t) d σ 1, σ 2 (x) = dσ 1 (x). x t

34 Special systems: Nikishin systems Notation: σ 1, σ 2 is a measure which is absolutely continuous with respect to σ 1 and for which the Radon-Nikodym derivative is a Stieltjes transform of σ 2 : ( ) dσ2 (t) d σ 1, σ 2 (x) = dσ 1 (x). x t Definition (Nikishin system for general r) A Nikishin system of order r on an interval r is a system of r measures (µ 1, µ 2,..., µ r ) supported on r such that µ j = µ 1, σ j (2 j r), where (σ 2,..., σ r ) is a Nikishin system of order r 1 on an interval r 1 and r r 1 =.

35 Special systems: Nikishin systems Theorem (Fidalgo Prieto and López Lagomasino) Every Nikishin system is perfect.

36 Special systems: Nikishin systems Theorem (Fidalgo Prieto and López Lagomasino) Every Nikishin system is perfect. Proof. Ask Guillermo

37 Special systems: Nikishin systems Theorem (Fidalgo Prieto and López Lagomasino) Every Nikishin system is perfect. Proof. Ask Guillermo or Ulises.

38 Biorthogonality In most cases the measures (µ 1,..., µ r ) are absolutely continuous with respect to one fixed measure µ: dµ j (x) = w j (x) dµ(x), 1 j r. We then define the type I function Q n (x) = r A n,j (x)w j (x). j=1

39 Biorthogonality In most cases the measures (µ 1,..., µ r ) are absolutely continuous with respect to one fixed measure µ: dµ j (x) = w j (x) dµ(x), 1 j r. We then define the type I function Q n (x) = r A n,j (x)w j (x). j=1 Property (biorthogonality) 0, if m n, P n (x)q m (x) dµ(x) = 0, if n m 2, 1, if n = m 1.

40 Recurrence relations Nearest neighbor recurrence relations for type II MOPS xp n (x) = P n+ e1 (x) + b n,1 P n (x) +. xp n (x) = P n+ er (x) + b n,r P n (x) + r a n,j P n ej (x), j=1 r a n,j P n ej (x). j=1 e j = (0,..., 0, j {}}{ 1, 0,..., 0)

41 Recurrence relations Nearest neighbor recurrence relations for type I MOPS xq n (x) = Q n e1 (x) + b n e1,1q n (x) +. xq n (x) = Q n er (x) + b n er,r Q n (x) + r a n,j Q n+ ej (x), j=1 r a n,j Q n+ ej (x). j=1

42 Recurrence relations Theorem (Van Assche) The recurrence coefficients (a n,1,..., a n,r ) and (b n,1,..., b n,r ) satisfy the partial difference equations b n+ ei,j b n,j = b n+ ej,i b n,i r r ( ) b n+ ej,i b a n+ ej,k a n+ ei,k = det n,i, k=1 for all 1 i j r. k=1 a n,i a n+ ej,i b n+ ei,j b n,j = b n e i,j b n ei,i b n,j b n,i

43 Recurrence relations Let ( n k ) k 0 be a path in N r starting from n 0 = 0, such that n k+1 n k = e i for some 1 i r. Then xp nk (x) = P nk+1 (x) + r β nk,jp nk j (x). j=0

44 Recurrence relations Let ( n k ) k 0 be a path in N r starting from n 0 = 0, such that n k+1 n k = e i for some 1 i r. Then xp nk (x) = P nk+1 (x) + r β nk,jp nk j (x). j=0 An important case is the stepline: j {}}{ n k = ( i + 1,..., i + 1, i,... i) k = ri + j, 0 j r 1. }{{} r j

45 Christoffel-Darboux formula Theorem (Daems and Kuijlaars) Let ( n k ) 0 k N be a path in N r starting from n 0 = 0 and ending in n N = n (where N = n ), such that n k+1 n k = e i for some 1 i r. Then N 1 (x y) P nk (x)q nk+1 (y) = P n (x)q n (y) k=0 r a n,j P n ej (x)q n+ ej (y). j=1

46 Christoffel-Darboux formula Theorem (Daems and Kuijlaars) Let ( n k ) 0 k N be a path in N r starting from n 0 = 0 and ending in n N = n (where N = n ), such that n k+1 n k = e i for some 1 i r. Then N 1 (x y) P nk (x)q nk+1 (y) = P n (x)q n (y) k=0 r a n,j P n ej (x)q n+ ej (y). j=1 The sum depends only on the endpoints of the path in N r and not on the path between these points.

47 References M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, 2005 (paperback edition 2009) E.M. Nikishin, V.N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs 92, Amer. Math. Soc., Providence RI, A.I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), A. Martínez-Finkelshtein, W. Van Assche, What is... a multiple orthogonal polynomial?, Notices Amer. Math. Soc. 63 (2016), no. 9, W. Van Assche, Nearest neighbor recurrence relations for multiple orthogonal polynomials, J. Approx. Theory 163 (2011),