Problems Session. Nikos Stylianopoulos University of Cyprus

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1 [ 1 ] University of Cyprus Problems Session Nikos Stylianopoulos University of Cyprus Hausdorff Geometry Of Polynomials And Polynomial Sequences Institut Mittag-Leffler Djursholm, Sweden May-June 2018 Approximation Theory Operator Theory

2 [ 2 ] University of Cyprus Lebesgue spaces and Orthonormal Polynomials Let µ be a finite positive Borel measure having compact and infinite support S µ := supp(µ) in the complex plane C. Then, the measure yields the Lebesgue spaces L 2 (µ) with inner product f, g µ := f (z)g(z)dµ(z) and norm f L2 (µ) := f, f 1/2 µ. Let {p n (µ, z)} n=0 denote the sequence of orthonormal polynomials associated with µ. That is, the unique sequence of the form p n (µ, z) = γ n (µ)z n +, γ n (µ) > 0, n = 0, 1, 2,..., satisfying p m (µ, ), p n (µ, ) µ = δ m,n. Approximation Theory Operator Theory Tools Bergman

3 [ 3 ] University of Cyprus Distribution of zeros: The tools For any polynomial q n (z), of degree n, we denote by ν qn the normalized counting measure for the zeros of q n (z); that is, ν qn := 1 n q n(z)=0 where δ z is the unit point mass (Dirac delta) at the point z. For any measure µ with compact support in C, U µ 1 (z) := log dµ(t), z C. z t δ z, denotes the logarithmic potential on µ. Then U νqn (z) = 1 n log 1 q n (z), z C. With µ E we denote the equilibrium measure of a compact set E of positive logarithmic capacity. Approximation Theory Operator Theory Tools Bergman

4 [ 4 ] University of Cyprus Bergman polynomials {p n } on an Jordan domain G Γ Ω Φ Ψ Δ D G 0 Γ := G Ω := C \ G f, g := f (z)g(z)da(z), f L2 (G) := f, f 1/2. G The Bergman polynomials {p n } n=0 of G are the orthonormal polynomials w.r.t. the area measure on G: p m, p n = p m (z)p n (z)da(z) = δ m,n, G with p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Approximation Theory Operator Theory Tools Bergman

5 [ 5 ] University of Cyprus Shift Operator Let L 2 a(g) denote the Bergman space of square integrable and analytic functions in G and consider the Bergman shift operator on L 2 a(g). That is, S z : L 2 a(g) L 2 a(g) with S z f = zf. Properties of S z (i) S z defines a subnormal operator on L 2 a(g). (ii) σ(s z ) = G and σ ess (S z ) = G (Axler, Conway & McDonald, Can. J. Math.,1982). (iii) S z (f ) = P G (zf ), where P G denotes the orthogonal projection from L 2 (G) to L 2 a(g). Proof of (iii): For any f, g L 2 a(g) it holds that S z f, g = f, S z g = f, zg = zf, g = P G (zf ), g.

6 [ 6 ] University of Cyprus Recurrences for Bergman polynomials {p n } In general it holds that n+1 zp n (z) = b k,n p k (z), where b k,n := zp n, p k. k=0

7 [ 7 ] University of Cyprus Matrix representation for S z The Bergman operator S z has the following upper Hessenberg matrix representation with respect to the Bergman polynomials {p n } n=0 of G: b 00 b 01 b 02 b 03 b 04 b 05 b 10 b 11 b 12 b 13 b 14 b 15 0 b 21 b 22 b 23 b 24 b 25 M = 0 0 b 32 b 33 b 34 b 35, b 43 b 44 b b 54 b where b k,n = zp n, p k are the Fourier coefficients of S z p n = zp n. Note The eigenvalues of the n n principal submatrix M n of M coincide with the zeros of p n.

8 [ 8 ] University of Cyprus Example: G D This example shows why modern text books on Functional Analysis or Operators Theory do not refer to matrices: Indeed, in this case we have: n + 1 p n (z) = π zn, n = 0, 1,.... Therefore, in the matrix representation M of S z the only non-zero diagonals are the main subdiagonal, and hence for any n N, M n is a nilpotent matrix. As a result, the Caley-Hamilton theorem implies: This is in sharp contrast to: σ(m n ) = {0}. σ ess (M) = σ ess (S z ) = {w : w = 1} and σ(m) = σ(s z ) = {w : w 1}.

9 [ 9 ] University of Cyprus The inverse conformal map Ψ Γ Ω Φ Ψ Δ D G 0 Recall that Φ(z) = γz + γ 0 + γ 1 z + γ 2 z 2 +, and let Ψ := Φ 1 : {w : w > 1} Ω, denote the inverse conformal map. Then, where Ψ(w) = bw + b 0 + b 1 w + b 2 +, w < 1, w 2 b = cap(γ) = 1/γ.

10 [ 10 ] University of Cyprus The Toeplitz matrix with (continuous) symbol Ψ T ψ = b 0 b 1 b 2 b 3 b 4 b 5 b 6 b b 0 b 1 b 2 b 3 b 4 b 5 0 b b 0 b 1 b 2 b 3 b b b 0 b 1 b 2 b b b 0 b 1 b b b 0 b b b

11 [ 11 ] University of Cyprus Spectral properties Theorem (St, Constr, Approx., 2013) If Γ is piecewise analytic without cusps, then 1 b n c 1 (Γ), n N, (1) n1+ω where ωπ (0 < ω < 2) is the smallest exterior angle of Γ. Therefore, in this case, the symbol Ψ of the Toeplitz matrix T Ψ belongs to the Wiener algebra. Thus, T Ψ defines a bounded linear operator on the Hilbert space l 2 (N) and σ ess (T Ψ ) = Γ ; (2) see e.g., Bottcher & Grudsky, Toeplitz book, 2005.

12 [ 12 ] University of Cyprus Faber polynomials of G The Faber polynomial of the 2nd kind G n (z), is the polynomial part of the expansion of the Laurent series expansion of Φ n (z)φ (z) at : ( ) 1 G n (z) = Φ n (z)φ (z) + O, z. z These polynomials satisfy the recurrence relation: zg n (z) = bg n+1 (z) + n+1 Recall: zp n (z) = b k,n p k (z). Note k=0 n b k G n k (z), n = 0, 1,..., k=0 The eigenvalues of the n n principal submatrix T n of T ψ coincide with the zeros of G n.

13 [ 13 ] University of Cyprus M T ψ diagonally The next series of theorems show that the connection between the two matrices M and T Ψ is much more substantial. Theorem (Saff & St., CAOT, 2012 and Beckemann & St., Constr. Approx., 2018) Assume that Γ is piecewise analytic without cusps. Then, it holds as n, ( ) n n + 1 b n+1,n = b + O, (3) n and for k 0, n k + 1 n + 1 b n k,n = b k + O where O depends on k but not on n. ( ) 1, (4) n

14 [ 14 ] University of Cyprus M T ψ diagonally: Smooth curve Improvements in the order of convergence occur in cases when Γ is smooth. Theorem (Saff & St., CAOT, 2012 and Beckemann & St., Constr. Approx., 2018) Assume that Γ C(p + 1, α), with p + α > 1/2. Then, it holds as n, ( ) n n + 1 b n+1,n = b + O, (5) n 2(p+α) and for k 0, ( ) n k n + 1 b n k,n = b k + O, (6) n 2(p+α) where O depends on k but not on n.

15 [ 15 ] University of Cyprus M T ψ diagonally: Analytic curve For the case of an analytic boundary Γ further improved asymptotic results can be obtained. Theorem (Saff & St., CAOT, 2012 and Beckemann & St., Constr. Approx., 2018) Assume that the boundary Γ is analytic and let ρ < 1 be the smallest index for which Φ is conformal in the exterior of L ρ. Then, it holds as n, n + 2 n + 1 b n+1,n = b + O(ρ 2n ), (7) and for k 0, n k + 1 n + 1 b n k,n = b k + O(ρ 2n ), (8) where O depends on k but not on n.

16 [ 16 ] University of Cyprus Is M T ψ compact? Corollary If the upper Hessenberg matrix M is banded, with constant bandwidth, then M T ψ defines a compact operator on l 2 (N). Theorem (Putinar & St, CAOT, 2007) If the Bergman polynomials {p n } satisfy a 3-term recurrence relation, then Γ = G is an ellipse. Theorem (Khavinson & St, Springer, 2009 (St, CRAS, 2010)) Assume that: (i) Γ = G is C 2 continuous (piecewise analytic without cusps). (ii) The Bergman polynomials {p n } n=0 satisfy an m + 1-term recurrence relation, with some m 2. Then m = 2 and Γ is an ellipse.

17 [ 17 ] University of Cyprus Example: G is a 3-cusped hypocycloid 1 0,5-0, ,5 1 1,5-0,5-1 Note that supp(µ Γ ) = Γ and recall σ ess (M) = Γ = σ ess (T Ψ ). Levin, Saff & St., Constr. Approx. (2003): ν(p n ) He & Saff, JAT (1994): µ Γ, n, n N, N N. σ(t n ) [0, 1.5] [0, 1.5e i2π/3 ] [0, 1.5e i4π/3 ].

18 [ 18 ] University of Cyprus Example: G is the square 0,6 0,4 0,2 0-0,6-0,4-0,2 0 0,2 0,4 0,6-0,2-0,4-0,6 σ ess (M) = Γ = σ ess (T Ψ ). Maymeskul & Saff, JAT (2003): σ(m n ) the two diagonals. Kuijlaars & Saff, Math. Proc. Cambrigde Phil. Soc. (1995): ν(g n ) µ Γ, n, n N, N N

19 [ 19 ] University of Cyprus Example: G is the canonical pentagon 1 Regular 5 gon: zeros of orthogonal polynomials up to degree 50. σ ess (M) = Γ = σ ess (T Ψ ). Levin, Saff & St., Constr. Approx. (2003): ν(p n ) µ Γ, n, n N, N N Kuijlaars & Saff, Math. Proc. Cambrigde Phil. Soc. (1995): ν(g n ) µ Γ, n, n N, N N

20 [ 20 ] University of Cyprus The challenge Problem Describe the three distinct behaviours in the spectral properties of M n and T n, by using the two infinite matrices M and T ψ ONLY! Note that each of the matrix alone, carries all the information of the domain G, because it contains, either as limits, or explicitly, all the coefficients of the inverse conformal mapping ψ : Ω.