Multiple Orthogonal Polynomials

Transcription

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

2 Plan of the course

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

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

5 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)

6 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

7 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

8 Riemann-Hilbert problem for MOPS Fokas, Its and Kitaev 1 formulated a Riemann-Hilbert problem (for 2 2 matrices) that characterizes orthogonal polynomials. 1 A.S. Fokas, A. Its, A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2,

9 Riemann-Hilbert problem for MOPS Fokas, Its and Kitaev 1 formulated a Riemann-Hilbert problem (for 2 2 matrices) that characterizes orthogonal polynomials. There is a Riemann-Hilbert problem (for (r + 1) (r + 1) matrices) that characterizes multiple orthogonal polynomials (W. Van Assche, J.S. Geronimo, A.B.J. Kuijlaars, 2001). Suppose that dµ j (x) = w j (x) dx (1 j r) 1 A.S. Fokas, A. Its, A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2,

10 Riemann-Hilbert problem for MOPS Fokas, Its and Kitaev 1 formulated a Riemann-Hilbert problem (for 2 2 matrices) that characterizes orthogonal polynomials. There is a Riemann-Hilbert problem (for (r + 1) (r + 1) matrices) that characterizes multiple orthogonal polynomials (W. Van Assche, J.S. Geronimo, A.B.J. Kuijlaars, 2001). Suppose that dµ j (x) = w j (x) dx (1 j r) Find Y = C C (r+1) (r+1) for which 1 Y is analytic on C \ R. 1 A.S. Fokas, A. Its, A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2,

11 Riemann-Hilbert problem for MOPS 2 The boundary values Y ± (x) = lim ɛ 0+ Y (x ± iɛ) exist for x R and satisfy 1 w 1 (x) w 2 (x) w r (x) Y + (x) = Y (x) , x R

12 Riemann-Hilbert problem for MOPS 3 Y behaves near infinity as z n 0 ( ) z n 1 Y (z) = I +O(1/z) z n z nr, z.

13 Riemann-Hilbert problem for MOPS 3 Y behaves near infinity as z n 0 ( ) z n 1 Y (z) = I +O(1/z) z n z nr 4 [some condition near the endpoints of the supports of µ 1,..., µ r ], z.

14 Riemann-Hilbert problem for MOPS Theorem (VA, Geronimo, Kuijlaars) If n and n e j (1 j r) are normal indices, then Y (z) = P n (z) 2πiγ 1 P n e1 (z). 2πiγ r P n er (z) 1 P n (x)w 1 (x) dx 2πi x z P n e1 (x)w 1 (x) γ 1 1 P n (x)w r (x) dx 2πi x z P n e1 (x)w r (x) dx γ x z 1 x z. P n er (x)w γ 1 (x) P n er (x)w r (x) r dx γ x z r dx x z. dx where 1 = 1 γ j γ j ( n) = x n j 1 P n ej w j (x) dx, 1 j r.

15 Riemann-Hilbert problem for MOPS Theorem (VA, Geronimo, Kuijlaars) If n and n e j (1 j r) are normal indices, then Q n (x) dx 2πiA z x n,1(z) 2πiA n,r (z) c 1 Q n+ e1 (x) Y T dx c 2πi z x 1A n+ e1,1(z) c 1A n+ e1,r (z) (z) =... c r Q n+ er (x) dx c 2πi z x r A n+ er,1(z) c r A n+ er,r (z) where A n+ ej (z) = zn j + lower order terms. c j ( n)

16 Nikishin system Introduced by E.M. Nikishin in 1980.

17 Nikishin system Introduced by E.M. Nikishin in µ 2 is absolutely continuous with respect to µ 1

18 Nikishin system Introduced by E.M. Nikishin in µ 2 is absolutely continuous with respect to µ 1 The Radon-Nikodym derivative is dµ d 2 dσ(t) = w(x) = dµ 1 c x t,

19 Nikishin system Introduced by E.M. Nikishin in µ 2 is absolutely continuous with respect to µ 1 The Radon-Nikodym derivative is [c, d] is disjoint from [a, b]. dµ d 2 dσ(t) = w(x) = dµ 1 c x t,

20 Nikishin system Introduced by E.M. Nikishin in µ 2 is absolutely continuous with respect to µ 1 The Radon-Nikodym derivative is [c, d] is disjoint from [a, b]. dµ d 2 dσ(t) = w(x) = dµ 1 c x t, We will take c < d < a < b.

21 Multiple orthogonal polynomials for a Nikishin system We consider the following case: dµ 1 (x) = (x a) α (b x) β h 1 (x) dx = w 1 (x) dx, x [a, b], dσ(t) = (t c) γ (d t) δ h 2 (t) dt = w 2 (t) dt t [c, d], where h 1 is analytic in a neighborhood of [a, b] and h 2 is analytic in a neighborhood of [c, d].

22 Multiple orthogonal polynomials for a Nikishin system We consider the following case: dµ 1 (x) = (x a) α (b x) β h 1 (x) dx = w 1 (x) dx, x [a, b], dσ(t) = (t c) γ (d t) δ h 2 (t) dt = w 2 (t) dt t [c, d], where h 1 is analytic in a neighborhood of [a, b] and h 2 is analytic in a neighborhood of [c, d]. b a x k( ) A n,m (x)+w(x)b n,m (x) w 1 (x) dx = 0, 0 k n+m 2, w(x) = d c w 2 (t) x t dt.

23 Riemann-Hilbert problem We use the Riemann-Hilbert problem for X = Y T

24 Riemann-Hilbert problem We use the Riemann-Hilbert problem for X = Y T Find X : C C 3 3 such that

25 Riemann-Hilbert problem We use the Riemann-Hilbert problem for X = Y T Find X : C C 3 3 such that 1 X is analytic in C \ [a, b].

26 Riemann-Hilbert problem We use the Riemann-Hilbert problem for X = Y T Find X : C C 3 3 such that 1 X is analytic in C \ [a, b]. 2 lim ɛ 0+ X (x ± iɛ) = X ± (x) exists for x (a, b) and X + (x) = X (x) 2πiw 1 (x) 1 0, 2πiw(x)w 1 (x) 0 1 x (a, b).

27 Riemann-Hilbert problem 3 Near infinity one has ( X (z) = I + O(1/z)) z n m z n 0, z. 0 0 z m

28 Riemann-Hilbert problem 3 Near infinity one has ( X (z) = I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Near a the behavior is O(r a (z)) O(1) O(1) X (z) = O(r a (z)) O(1) O(1), z a, O(r a (z)) O(1) O(1) where z a α, 1 < α < 0, r a (z) = log z a, α = 0, 1, α > 0.

29 Riemann-Hilbert problem 3 Near infinity one has ( X (z) = I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Near b the behavior is O(r b (z)) O(1) O(1) X (z) = O(r b (z)) O(1) O(1), z b, O(r b (z)) O(1) O(1) where z b β, 1 < β < 0, r b (z) = log z b, β = 0, 1, β > 0.

30 Riemann-Hilbert problem The solution is b A n,m (x) + w(x)b n,m (x) w 1 (x) dx A n,m (z) B n,m (z) a z x b X (z) = A n+1,m (x) + w(x)b n+1,m (x) c 1 w 1 (x) dx c 1 A n+1,m (z) c 1 B n+1,m (z) a z x b A n,m+1 (x) + w(x)b n,m+1 (x) c 2 w 1 (x) dx c 2 A n,m+1 (z) c 2 B n,m+1 (z) z x a with c 1 = c 1 (n, m) and c 2 = c 2 (n, m) such that c 1 A n+1,m (z) = z n + lower order terms, c 2 B n,m+1 (z) = z m + lower order terms.

31 Riemann-Hilbert analysis The idea is to transform this Riemann-Hilbert problem for X to a Riemann-Hilbert problem for R that behaves nicely, uniformly in C, lim R(z) = I. n,m Then undo the transformations for the asymptotic behavior of X.

32 Riemann-Hilbert analysis The idea is to transform this Riemann-Hilbert problem for X to a Riemann-Hilbert problem for R that behaves nicely, uniformly in C, lim R(z) = I. n,m Then undo the transformations for the asymptotic behavior of X. First transformation: U(z) = X (z) d w 2 (t) 0 z t dt 1 c

33 Riemann-Hilbert analysis The idea is to transform this Riemann-Hilbert problem for X to a Riemann-Hilbert problem for R that behaves nicely, uniformly in C, lim R(z) = I. n,m Then undo the transformations for the asymptotic behavior of X. First transformation: U(z) = X (z) d w 2 (t) 0 z t dt 1 This brings in the measure dσ(t) = w 2 (t) dt on the interval [c, d] c

34 Riemann-Hilbert problem for U 1 U is analytic in C \ ([a, b] [c, d])

35 Riemann-Hilbert problem for U 1 U is analytic in C \ ([a, b] [c, d]) 2 Jumps U + (x) = U (x) 2πiw 1 (x) 1 0, U + (x) = U (x) 0 1 0, 0 2πiw 2 (x) 1 x (a, b), x (c, d).

36 Riemann-Hilbert problem for U 3 Asymptotic behavior (here one needs m n) U(z) = ( I + O(1/z)) z n m z n 0, z. 0 0 z m

37 Riemann-Hilbert problem for U 3 Asymptotic behavior (here one needs m n) U(z) = ( I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Behavior near a O(r a (z)) O(1) O(1) U(z) = O(r a (z)) O(1) O(1), z a, O(r a (z)) O(1) O(1)

38 Riemann-Hilbert problem for U 3 Asymptotic behavior (here one needs m n) U(z) = ( I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Behavior near b O(r b (z)) O(1) O(1) U(z) = O(r b (z)) O(1) O(1), z b, O(r b (z)) O(1) O(1)

39 Riemann-Hilbert problem for U 3 Asymptotic behavior (here one needs m n) U(z) = ( I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Behavior near c O(1) O(r c (z)) O(1) U(z) = O(1) O(r c (z)) O(1), z c, O(1) O(r c (z)) O(1)

40 Riemann-Hilbert problem for U 3 Asymptotic behavior (here one needs m n) U(z) = ( I + O(1/z)) z n m z n 0, z. 0 0 z m 4 Behavior near d O(1) O(r d (z)) O(1) U(z) = O(1) O(r d (z)) O(1), z d, O(1) O(r d (z)) O(1)

41 Vector equilibrium problem Equilibrium problem: ( ) 2I (ν 1 )+2q1I 2 (ν 2 ) 2q 1 I (ν 1, ν 2 ) = inf 2I (µ 1 )+2q 2 µ 1,µ 2 1I (µ 2 ) 2q 1 I (µ 1, µ 2 ), µ 1, µ 2 are probability measures, supp(µ 1 ) [a, b], supp(µ 2 ) [c, d] and q 1 = I (µ 1, µ 2 ) = d b c m n+m a log 1 x y dµ 1(x) dµ 2 (x).

42 Vector equilibrium problem Equilibrium problem: ( ) 2I (ν 1 )+2q1I 2 (ν 2 ) 2q 1 I (ν 1, ν 2 ) = inf 2I (µ 1 )+2q 2 µ 1,µ 2 1I (µ 2 ) 2q 1 I (µ 1, µ 2 ), µ 1, µ 2 are probability measures, supp(µ 1 ) [a, b], supp(µ 2 ) [c, d] and q 1 = I (µ 1, µ 2 ) = d b c m n+m a log 1 x y dµ 1(x) dµ 2 (x). ν 1 gives the asymptotic distribution of the zeros of A n,m (x) + w(x)b n,m (x) on [a, b]

43 Vector equilibrium problem Equilibrium problem: ( ) 2I (ν 1 )+2q1I 2 (ν 2 ) 2q 1 I (ν 1, ν 2 ) = inf 2I (µ 1 )+2q 2 µ 1,µ 2 1I (µ 2 ) 2q 1 I (µ 1, µ 2 ), µ 1, µ 2 are probability measures, supp(µ 1 ) [a, b], supp(µ 2 ) [c, d] and q 1 = I (µ 1, µ 2 ) = d b c m n+m a log 1 x y dµ 1(x) dµ 2 (x). ν 1 gives the asymptotic distribution of the zeros of A n,m (x) + w(x)b n,m (x) on [a, b] ν 2 gives the asymptotic distribution of the zeros of B n,m on [c, d].

44 Vector equilibrium problem Variational conditions: 2U(x; ν 1 ) q 1 U(x; ν 2 ) = l 1, 2q 1 U(x; ν 2 ) U(x; ν 1 ) = l 2, x [a, b], x [c, d], where U(x; µ) = log 1 x y dµ(y).

45 Riemann-Hilbert problem: g-functions Introduce g 1 (z) = b a log(z x) dν 1 (x), g 2 (z) = d c log(z t) dν 2 (t).

46 Riemann-Hilbert problem: g-functions Introduce g 1 (z) = b a log(z x) dν 1 (x), g 2 (z) = d c log(z t) dν 2 (t). U(x; ν 1 ), x > b, g 1 ± (x) = U(x, ν 1 ) ± iπ, x < a, U(x; ν 1 ) ± iπϕ 1 (x), a < x < b, U(x; ν 2 ), x > d, g 2 ± (x) = U(x, ν 2 ) ± iπ, x < c, U(x; ν 2 ) ± iπϕ 2 (x), c < x < d, ϕ 1 (x) = ϕ 2 (x) = b x d x dν 1 (t), dν 2 (t),

47 Riemann-Hilbert problem: second transformation Normalizing the Riemann-Hilbert problem: e (n+m)g 1(z) 0 0 V (z) = LU(z) 0 e (n+m)g 1(z)+mg 2 (z) 0 L 1, 0 0 e mg 2(z)

48 Riemann-Hilbert problem: second transformation Normalizing the Riemann-Hilbert problem: e (n+m)g 1(z) 0 0 V (z) = LU(z) 0 e (n+m)g 1(z)+mg 2 (z) 0 L 1, 0 0 e mg 2(z) where L = L(n, m) = e n+m 3 (2l 1+l 2 ) e n+m 3 (l 1 l 2 ) e n+m 3 (l 1+2l 2 )

49 Riemann-Hilbert problem for V 1 V is analytic on C \ ([a, b] [c, d]).

50 Riemann-Hilbert problem for V 1 V is analytic on C \ ([a, b] [c, d]). 2 Asymptotic behavior in normalized V (z) = I + O(1/z), z.

51 Riemann-Hilbert problem for V 1 V is analytic on C \ ([a, b] [c, d]). 2 Asymptotic behavior in normalized V (z) = I + O(1/z), z. 3 Oscillatory jumps on (a, b) and (c, d) V + (x) = V (x) e2πi(n+m)ϕ1(x) 0 0 2πiw 1 (x) e 2πi(n+m)ϕ1(x) 0, V + (x) = V (x) 0 e 2πimϕ2(x) 0, 0 2πiw 2 (x) e 2πimϕ2(x) x (a, b), x (c, d).

52 Riemann-Hilbert problem for V 1 V is analytic on C \ ([a, b] [c, d]). 2 Asymptotic behavior in normalized V (z) = I + O(1/z), z. 3 Oscillatory jumps on (a, b) and (c, d) V + (x) = V (x) e2πi(n+m)ϕ1(x) 0 0 2πiw 1 (x) e 2πi(n+m)ϕ1(x) 0, V + (x) = V (x) 0 e 2πimϕ2(x) 0, 0 2πiw 2 (x) e 2πimϕ2(x) 4 Behavior near a, b, c, d under control. x (a, b), x (c, d).

53 Steepest descent for oscillatory Riemann-Hilbert problems Deift and Zhou (1993) developed a method for obtaining the asymptotic behavior (in our case for n, m ) for oscillatory Riemann-Hilbert problems. It starts with factoring the oscillatory jump.

54 Steepest descent for oscillatory Riemann-Hilbert problems e 2πi(n+m)ϕ 1(x) 0 0 Φ n+m 2πiw 1 (x) e 2πi(n+m)ϕ (x) 0 = v 1 Φ n+m Φ n+m 1 /v /v Φ n+m 1 /v 1 0 = v

55 Steepest descent for oscillatory Riemann-Hilbert problems e 2πi(n+m)ϕ 1(x) 0 0 Φ n+m 2πiw 1 (x) e 2πi(n+m)ϕ (x) 0 = v 1 Φ n+m Φ n+m 1 /v /v Φ n+m 1 /v 1 0 = v e 2πimϕ 2(x) 0 = 0 Φ m 0 2πiw 2 (x) e 2πimϕ 2 0 2(x) 0 v 2 Φ m = 0 1 Φ m 2 /v /v Φ m /v v

56 Opening the lenses c d a b

57 Opening the lenses c d a b Σ c,d + Σ a,b + c Σ c,d d a Σ a,b b

58 Riemann-Hilbert problem: third transformation V (z), 1 Φ (n+m) 1 /v 1 0 V (z) 0 1 0, Φ (n+m) 1 /v 1 0 V (z) 0 1 0, S(z) = V (z) 0 1 Φ m 2 /v 2, V (z) 0 1 Φ m 2 /v 2, outside the lenses inside the [a, b]-lens, upper part inside the [a, b]-lens, lower part inside the [c, d]-lens, upper part inside the [c, d]-lens, lower part

59 Riemann-Hilbert problem for S 1 S is analytic in C \ ([a, b] [c, d] Σ a,b Σ c,d ).

60 Riemann-Hilbert problem for S 1 S is analytic in C \ ([a, b] [c, d] Σ a,b Σ c,d ). 2 S is normalized near infinity: S(z) = I + O(1/z).

61 Riemann-Hilbert problem for S 1 S is analytic in C \ ([a, b] [c, d] Σ a,b Σ c,d ). 2 S is normalized near infinity: S(z) = I + O(1/z). 3 Jumps 1 Φ (n+m) 1 /v 1 0 S (z) 0 1 0, z Σ a,b, /v 1 0 S (z) v 1 0 0, z (a, b), S + (z) = S (z) 0 1 Φ m 2 /v 2, z Σ c,d, S (z) 0 0 1/v 2, z (c, d). 0 v 2 0

62 Riemann surface R We need to extend Φ 1 (defined originally on [a, b]) and Φ 2 (defined originally on [c, d]) to the complex plane.

63 Riemann surface R We need to extend Φ 1 (defined originally on [a, b]) and Φ 2 (defined originally on [c, d]) to the complex plane. These functions live more naturally on a Riemann surface with three sheets. a b R 0 R 1 c d R 2

64 Riemann surface R We need to extend Φ 1 (defined originally on [a, b]) and Φ 2 (defined originally on [c, d]) to the complex plane. These functions live more naturally on a Riemann surface with three sheets. a b R 0 R 1 c d R 2 Φ ± 1 (x) = e±2πiϕ 1(x), Φ ± 2 (x) = e±2πiϕ 2(x)

65 Oscillatory jumps exponentially small jumps Claim: { Φ 1 (z) > 1, z Σ a,b, Φ 2 (z) > 1, z Σ c,d.

66 Oscillatory jumps exponentially small jumps Claim: { Φ 1 (z) > 1, z Σ a,b, Φ 2 (z) > 1, z Σ c,d. As n, m the jumps on Σ a,b and Σ c,d tend to I exponentially fast.

67 Oscillatory jumps exponentially small jumps Claim: { Φ 1 (z) > 1, z Σ a,b, Φ 2 (z) > 1, z Σ c,d. As n, m the jumps on Σ a,b and Σ c,d tend to I exponentially fast. The main contribution to the Riemann-Hilbert matrix S will be the jumps on [a, b] and [c, d].

68 Global parametrix We ignore the jumps that tend to I.

69 Global parametrix We ignore the jumps that tend to I. The main contribution to S is the matrix N with satisfies the Riemann-Hilbert problem 1 N is analytic in C \ ([a, b] [c, d]).

70 Global parametrix We ignore the jumps that tend to I. The main contribution to S is the matrix N with satisfies the Riemann-Hilbert problem 1 N is analytic in C \ ([a, b] [c, d]). 2 N is normalized near infinity: N(z) = I + O(1/z).

71 Global parametrix We ignore the jumps that tend to I. The main contribution to S is the matrix N with satisfies the Riemann-Hilbert problem 1 N is analytic in C \ ([a, b] [c, d]). 2 N is normalized near infinity: N(z) = I + O(1/z). 3 Jumps 0 1/v 1 0 N + (x) = N (x) v 1 0 0, N + (x) = N (x) 0 0 1/v 2, 0 v 2 0 x (a, b), x (c, d).

72 Global parametrix We ignore the jumps that tend to I. The main contribution to S is the matrix N with satisfies the Riemann-Hilbert problem 1 N is analytic in C \ ([a, b] [c, d]). 2 N is normalized near infinity: N(z) = I + O(1/z). 3 Jumps 0 1/v 1 0 N + (x) = N (x) v 1 0 0, N + (x) = N (x) 0 0 1/v 2, 0 v Convenient behavior near a, b, c, d. x (a, b), x (c, d).

73 Szegő functions We write this global parametrix as D 1 ( ) 0 0 D 1 (z) 0 0 N(z) = 0 D 2 ( ) 0 N 0 (z) 0 D 2 (z) D 3 ( ) 0 0 D 3 (z) where (D 1, D 2, D 3 ) correspond to the Szegő functions of (v 1, v 2 ) for the Riemann surface R: 1

74 Szegő functions We write this global parametrix as D 1 ( ) 0 0 D 1 (z) 0 0 N(z) = 0 D 2 ( ) 0 N 0 (z) 0 D 2 (z) D 3 ( ) 0 0 D 3 (z) where (D 1, D 2, D 3 ) correspond to the Szegő functions of (v 1, v 2 ) for the Riemann surface R: D 1, D 2, D 3 are analytic in C \ ([a, b] [c, d]), with D k ( ) 0 and D 2 + (x) = v 1(x)D1 (x), D2 (x) = v 1(x)D 1 + (x),, x [a, b], D 3 + (x) = D 3 (x), D 1 + (x) = D 1 (x), D 3 + (x) = v 2(x)D2 (x),, x [c, d]. D3 (x) = v 2(x)D 2 + (x), 1

75 Standardized global parametrix The remaining matrix N 0 satisfies the Riemann-Hilbert problem 1 N 0 is analytic in C \ ([a, b] [c, d]).

76 Standardized global parametrix The remaining matrix N 0 satisfies the Riemann-Hilbert problem 1 N 0 is analytic in C \ ([a, b] [c, d]). 2 N 0 is normalized near infinity: N 0 (z) = I + O(1/z).

77 Standardized global parametrix The remaining matrix N 0 satisfies the Riemann-Hilbert problem 1 N 0 is analytic in C \ ([a, b] [c, d]). 2 N 0 is normalized near infinity: N 0 (z) = I + O(1/z). 3 Jumps N 0 + (x) = N 0 (x) 1 0 0, N 0 + (x) = N 0 (x) 0 0 1, x (a, b), x (c, d).

78 Standardized global parametrix The remaining matrix N 0 satisfies the Riemann-Hilbert problem 1 N 0 is analytic in C \ ([a, b] [c, d]). 2 N 0 is normalized near infinity: N 0 (z) = I + O(1/z). 3 Jumps N 0 + (x) = N 0 (x) 1 0 0, N 0 + (x) = N 0 (x) 0 0 1, Convenient behavior near a, b, c, d. x (a, b), x (c, d).

79 Final transformation? We now consider the matrix R = SN 1.

80 Final transformation? We now consider the matrix R = SN 1. Σ c,d + Σ a,b + c Σ c,d d a Σ a,b b

81 Final transformation? We now consider the matrix R = SN 1. Σ c,d + Σ a,b + c Σ c,d d a Σ a,b b The jumps on these curves converge to the identity matrix I.

82 Perturbation result for Riemann-Hilbert problems Theorem Suppose R n satisfies a normalized Riemann-Hilbert problem on a system Σ = k i=1 Σ i of contours. Suppose that the jumps R + n (x) = R n (x)j i x Σ i, converge uniformly to the identity matrix lim J i I = 0. n Then R n converges uniformly to the identity matrix lim R n(z) I = 0. n

83 Final asymptotic result? lim n SN 1 = I lim n S = N.

84 Final asymptotic result? lim n SN 1 = I lim n S = N. For z on compact sets of C \ ([a, b] [c, d]): outside the lenses S = V lim n V = N.

85 Final asymptotic result? lim n SN 1 = I lim n S = N. For z on compact sets of C \ ([a, b] [c, d]): outside the lenses S = V lim n V = N. For z on [a, b] or [c, d]: inside the lenses S + = V + J + lim n V +J + = N +

86 Final asymptotic result? lim n SN 1 = I lim n S = N. For z on compact sets of C \ ([a, b] [c, d]): outside the lenses S = V lim n V = N. For z on [a, b] or [c, d]: inside the lenses S + = V + J + lim n V +J + = N + There is a problem: the jumps for SN 1 on Σ a,b and Σ c,d do not converge uniformly to I. Uniformity is lost near a, b, c, d.

87 Final asymptotic result? lim n SN 1 = I lim n S = N. For z on compact sets of C \ ([a, b] [c, d]): outside the lenses S = V lim n V = N. For z on [a, b] or [c, d]: inside the lenses S + = V + J + lim n V +J + = N + There is a problem: the jumps for SN 1 on Σ a,b and Σ c,d do not converge uniformly to I. Uniformity is lost near a, b, c, d. A local analysis around a, b, c, d is needed.

88 Local parametrices In the neighborhood of a the Riemann-Hilbert problem for S looks like Γ a a

89 Local parametrices In the neighborhood of a the Riemann-Hilbert problem for S looks like Γ a a We approximate it by a model RHP P a which matches the global parametrix N on the boundary Γ a with an error O(1/n).

90 Local parametrices In the neighborhood of a the Riemann-Hilbert problem for S looks like Γ a a We approximate it by a model RHP P a which matches the global parametrix N on the boundary Γ a with an error O(1/n). This model Riemann-Hilbert problem uses the Bessel function J α.

91 Local parametrices In the neighborhood of a the Riemann-Hilbert problem for S looks like Γ a a We approximate it by a model RHP P a which matches the global parametrix N on the boundary Γ a with an error O(1/n). This model Riemann-Hilbert problem uses the Bessel function J α. Around b we need the Bessel function J β. Around c we need the Bessel function J γ. Around d we need the Bessel function J δ.

92 The final transformation! R(z) = { SN 1, z outside Γ a, Γ b, Γ c, Γ d, SP 1 e, z inside Γ e, e {a, b, c, d}.

93 The final transformation! R(z) = { SN 1, z outside Γ a, Γ b, Γ c, Γ d, SP 1 e, z inside Γ e, e {a, b, c, d}. c d a b

94 The final transformation! R(z) = { SN 1, z outside Γ a, Γ b, Γ c, Γ d, SP 1 e, z inside Γ e, e {a, b, c, d}. c d a b Then R n I = O(1/n).

95 Final asymptotic results

96 Final asymptotic results Theorem Let (n, m) be multi-indices that tend to infinity but for which m/(n + m) = q 1 remains constant, with 0 < q 1 1/2. Then uniformly on compact subsets of C \ (]a, b] [c, d]) A n,m (z) = [N 1 (ψ 1 (z)) + O(1/n)] D 0( ) D 1 (z) e(n+m)g 1(z) mg 2 (z)+(n+m)l 1 [N 1 (ψ 2 (z)) + O(1/n)] D 0( ) D 2 (z) emg 2(z)+(n+m)(l 1 +l 2 ) and d c dσ(t) z t, B n,m (z) = [N 1 (ψ 2 (z)) + O(1/n)] D 0( ) D 2 (z) emg 2(z)+(n+m)(l 1 +l 2 ).

97 Final asymptotic results Theorem Furthermore d dσ(t) A n,m (z) + B n,m (z) c z t = [N 1 (ψ 1 (z)) + O(1/n)] D 0( ) D 1 (z) e(n+m)g 1(z) mg 2 (z)+(n+m)l 1.

98 Final asymptotic results: on (c, d) Theorem Uniformly on closed subintervals of (c, d) one has B n,m (x) = 2[N 1 (ψ 1 + (x)) + O(1/n)] D 0( ) 2) D 2 + (x) e mu(x;ν ) cos (mπϕ 2 (x) arg D 2 + (x).

99 Final asymptotic results: on (a, b) Theorem Uniformly on closed subintervals of (a, b) one has d dσ(t) A n,m (x) + B n,m (x) c x t = 2[N 1(ψ 1 + (x)) + O(1/n)] D ) 0( ) 1) D 1 + cos ((n + m)ϕ 1 (x) arg D + (x) e(n+m)u(x;ν 1 (x).

100 Type II multiple orthogonal polynomials Take the inverse-transpose of the Riemann-Hilbert matrix: X T = P b P n,m(t)w 1 (t) b n,m(z) a t z dt a γ 1 P n 1,m (z) γ 2 P n,m 1 (z) where P n,m(t)w(t)w 1 (t) t z dt 1 γ 1 = 1 γ 2 = b a b a t n 1 P n 1,m (t)w 1 (t) dt, t m 1 P n,m 1 (t)w(t)w 1 (t) dt.

101 Asymptotics for type II polynomials Theorem Let (n, m) be multi-indices that tend to infinity but for which m/(n + m) = q 1 remains constant, with 0 < q 1 1/2. Then uniformly on compact subsets of C \ [a, b] P n,m (z) = D 0(z) D 0 ( ) N 1(ψ 0 (z))e (n+m)g 1(z).

102 Asymptotics for type II polynomials Theorem Let (n, m) be multi-indices that tend to infinity but for which m/(n + m) = q 1 remains constant, with 0 < q 1 1/2. Then uniformly on compact subsets of C \ [a, b] P n,m (z) = D 0(z) D 0 ( ) N 1(ψ 0 (z))e (n+m)g 1(z). For x on closed intervals of (a, b) one has P n,m (x) = 2i D+ 0 (x) D 0 ( ) [N 1(ψ 0 + (x)) + O(1/n)] ) sin ((n + m)ϕ 1 (x) + arg D 0 + (x).

103 References P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 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) ; in particular Chapter 23. Guillermo López Lagomasino,, Riemann-Hilbert analysis for a Nikishin system, Math. Sbornik (submitted), arxiv: [math.ca] W. Van Assche, J.S. Geronimo, A.B.J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, in Special Functions 2000: Current Perspective and Future Directions, (J. Bustoz, M.E.H. Ismail, S.K. Suslov, eds.), NATO Science Series II. Mathematics, Physics and Chemistry, Vol. 30, Kluwer, Dordrecht, 2001, pp