Estimates for Bergman polynomials in domains with corners

Transcription

1 [ 1 ] University of Cyprus Estimates for Bergman polynomials in domains with corners Nikos Stylianopoulos University of Cyprus The Real World is Complex 2015 in honor of Christian Berg Copenhagen August 2015 Asymptotics Estimates Conjecture

2 [ 2 ] University of Cyprus Bergman polynomials Let Γ be a Jordan curve in C let G := int(γ) and consider the Lebesgue space L 2 (G) with inner product and norm: f, g G := f (z)g(z)da(z), f L 2 (G) := f, f 1/2 G G The Bergman polynomials {p n } n=0 of G are the unique orthonormal polynomials w.r.t. the area measure on G: p m, p n G = p m (z)p n (z)da(z) = δ m,n, with G p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... We will investigate the asymptotic behaviour of p n (z) on Γ, in cases when Γ has corners. Asymptotics Estimates Conjecture Theory Analytic Smooth Corners

3 [ 3 ] University of Cyprus Associated conformal maps Γ Ω Φ Ψ Δ D G 0 Ω := C \ G Φ(z) = γz + γ 0 + γ 1 z + γ 2 +. cap(γ) = 1/γ z2 Ψ(w) = bw + b 0 + b 1 w + b 2 w 2 +. The Bergman polynomials of G: cap(γ) = b p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Asymptotics Estimates Conjecture Theory Analytic Smooth Corners

4 [ 4 ] University of Cyprus Strong asymptotics when Γ is analytic Ω Φ Δ G D 0 ρ 1 Lρ Γ Theorem (Carleman, Ark. Mat. Astr. Fys., 1922) If ρ < 1 is the smallest index for which Φ is conformal in ext(l ρ ), then n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c 1 (Γ) ρ 2n, n where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, n N, A n (z) c 2 (Γ) n ρ n, z Ω. Asymptotics Estimates Conjecture Theory Analytic Smooth Corners

5 [ 5 ] University of Cyprus Strong asymptotics when Γ is smooth We say that Γ C(p, α), for some p N and 0 < α < 1, if Γ is given by z = g(s), where s is the arclength, with g (p) Lipα. Then both Φ and Ψ := Φ 1 are p times continuously differentiable in Ω \ { } and \ { } respectively, with Φ (p) and Ψ (p) Lipα. Theorem (Suetin, Proc. Steklov Inst. Math. AMS, 1974) Assume that Γ C(p + 1, α), with p + α > 1/2. Then, for n N, n + 1 γ 2(n+1) 1 π λ 2 = 1 α n, where 0 α n c 1 (Γ) n n, 2(p+α) where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, A n (z) c 2 (Γ) log n, z Ω. np+α Asymptotics Estimates Conjecture Theory Analytic Smooth Corners

6 [ 6 ] University of Cyprus Strong asymptotics for Γ non-smooth Theorem (St, C. R. Acad. Sci. Paris, 2010 & Constr. Approx., 2013) Assume that Γ is piecewise analytic without cusps.then, for n N, and for any z Ω, n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c(γ) 1 n n, p n (z) = n + 1 π Φn (z)φ (z) {1 + A n (z)}, where A n (z) c 1 (Γ) dist(z, Γ) Φ (z) 1 n + c 2 (Γ) 1 n. Asymptotics Estimates Conjecture Theory Analytic Smooth Corners

7 [ 7 ] University of Cyprus More on the geometry of Γ By Γ being piecewise analytic without cusps we mean that Γ consists of N analytic arcs that meet at points z j, where they form exterior angles ω j π, with 0 < ω j < 2, j = 1,..., N. Our results below will be given in terms of ω := max{1, ω 1,..., ω N }. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

8 [ 8 ] University of Cyprus A standard argument Consider L 1/n := {z Ω : Φ(z) = 1 + 1/n}, n N, and use: (i) The well-known estimate for the distance of L 1/n from Γ, dist(z, Γ) c(γ) 1 n ω, z L 1/n. (ii) The double inequality, which is a simple consequence of Koebe s 1/4-theorem, 1 Φ(z) 1 4 dist(z, Γ) Φ (z) 4 Φ(z) 1 dist(z, Γ), z Ω \ { }. The above, in conjunction with the estimate for A n (z), lead easily to p n L (G) p n L1/n c(γ)n ω. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

9 [ 9 ] University of Cyprus Other known estimates The previous estimate agrees with the one obtained by Abdullaev, under somewhat weaker assumptions: Theorem (Abdullaev, J. Ukrain. Mat., 2000) If Γ is a quasiconformal curve such that Φ Lipα, 1/2 < α 1. Then, p n L (G) c(γ)n 1/α. To see the relevance, note that our assumptions above, regarding the geometry of Γ, imply the relation ω = 1/α. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

10 [ 10 ] University of Cyprus Other known estimates The "standard argument" result is also in agreement with the following norm-comparison estimate, which is valid for any polynomial P n of degree n. Theorem (Pritsker, J. Math. Anal. Appl., 1997) Assume that Γ consists of N Dini-smooth arcs that meet at points z j, where they form exterior angles ω j π, with 0 < ω j 2, j = 1,..., N. Then, P n L (G) c(γ)n ω P n L2 (G), where ω has the same meaning as above. This estimate was shown to be sharp by Totik and Varga in Proc. London Math. Soc. (2015), under the slightly stronger assumption that the arcs constituting Γ are C 1+ smooth. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

11 [ 11 ] University of Cyprus Uniform estimate on Γ However, as our first result shows, the standard argument has led to a non-optimal exponent of n in the upper bound for p n L (G). Theorem (St, Contemp. Math., 2015) Assume that Γ is piecewise analytic without cusps and recall the notation ω := max{1, ω 1,..., ω N }. Then, p n L (G) c(γ)n ω 1/2. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

12 [ 12 ] University of Cyprus Pointwise estimate on Γ The next theorem gives a pointwise estimate for p n (z), z Γ. Theorem (St, Contemp. Math., 2015) Assume that Γ is piecewise analytic without cusps. Then, for any z Γ away from corners, p n (z) c(γ, z)n 1/2. If z j is a corner of Γ with exterior angle ω j π, 0 < ω j < 2, then p n (z j ) c(γ, z)n ω j 1/2 log n. It is interesting to note that the above yields the following limit provided 0 < ω j < 1/2. lim p n(z j ) = 0, n Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

13 [ 13 ] University of Cyprus On sharpness The following result settles, in a certain sense, the question of sharpness of the pointwise estimate, and hence that of the uniform estimate. Theorem (Totik & Varga, Proc. London Math. Soc., 2015) Assume that Γ has a C 1+ smooth corner of exterior angle ωπ, with 1 ω < 2 at the point z. Then, for infinitely many n, p n (z) c(γ, z)n ω 1/2. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

14 [ 14 ] University of Cyprus An estimate inside Γ Theorem (Maymeskul, Saff & St, Numer. Math., 2002) Assume, as above, that Γ is piecewise analytic without cusps and let s := min j=1,...,n {ω j /(2 ω j )}. Then, for any z B, where B is a compact set in G, it holds that p n (z) C(Γ, B) 1 n s, n N. Furthermore, if 1/(2 ω j ) / N, then for each ε > 0, there exists a subsequence Λ ε such that 1 p n (z) C(Γ, B) n, n Λ ε. s ε We note that a well-known result of Fejer implies that all the zeros of p n (z), n N, are contained in the convex hull of G. This result was refined by Saff to the interior of the convex hull. Asymptotics Estimates Conjecture Uniform Pointwise Sharpness Inside

15 [ 15 ] University of Cyprus A result of Lehman For the statement of a conjecture regarding the behaviour of p n (z) on Γ, we need a result of Lehman, for the asymptotics of both Φ and Φ. Assume that ωπ, 0 < ω < 2, is the opening of the exterior angle at a point z Γ. Then, for any ζ near z: Φ(ζ) = Φ(z) + a 1 (ζ z) 1/ω + o( ζ z 1/ω ), and Φ (ζ) = 1 ω a 1(ζ z) 1/ω 1 + o( ζ z 1/ω 1 ), with a 1 0. Asymptotics Estimates Conjecture Statement Numerical Theoretical

16 [ 16 ] University of Cyprus A Conjecture for the asymptotics of p n on Γ Conjecture Assume that Γ is piecewise analytic without cusps. Then, at any point z of Γ with exterior angle ωπ, 0 < ω < 2, it holds that p n (z) = ω(n + 1)ω 1/2 a1 ωφn+1 ω (z) {1 + β n (z)}, π Γ(ω + 1) with lim n β n (z) = 0. Asymptotics Estimates Conjecture Statement Numerical Theoretical

17 [ 17 ] University of Cyprus The two intersecting circles Consider the case where G is defined by the two intersecting circles z 1 = 2 and z + 1 = 2. Then, Φ(z) = 1 ( z 1 ). 2 z We test the conjecture numerically for z = i (corner) and z = (non-corner). Asymptotics Estimates Conjecture Statement Numerical Theoretical

18 [ 18 ] University of Cyprus The two intersecting circles Zeros of the Bergman polynomials p n (z), with n = 80, 100, 120. Theorem (Saff & St, JAT 2015) Let ν n denote the normalised counting measure of zeros of p n. Then ν n µ Γ, n, n N, where µ Γ denotes the equilibrium measure on Γ. The reluctance of the zeros to approach the points ±i, is due to the fact that dµ Γ (z) = Φ (z) ds, where s denotes the arclength on Γ. Asymptotics Estimates Conjecture Statement Numerical Theoretical

19 [ 19 ] University of Cyprus The two intersecting circles: z = i Then, Φ(i) = i, ω = 1/2, Γ(3/2) = π/2 and a 1 = 1/(2i). Thus, in this case the conjecture takes the form p n (i) = (i n / 2π){1 + β n }. n β n n β n Asymptotics Estimates Conjecture Statement Numerical Theoretical

20 [ 20 ] University of Cyprus The two intersecting circles: z = Now ω = 1, Φ(z) = 1, β n R, and the conjecture takes the form p n (1 + n ) = π (1 + 2) {1 + β n}. 2 n β n n β n Asymptotics Estimates Conjecture Statement Numerical Theoretical

21 [ 21 ] University of Cyprus Faber polynomials of the second kind We consider the polynomial part of Φ n (z)φ (z) and denote the resulting series by {G n } n=0. Thus, with Φ n (z)φ (z) = G n (z) H n (z), z Ω, G n (z) = γ n+1 z n + and H n (z) = O ( 1/ z 2), z. G n (z) is the so-called Faber polynomial of the 2nd kind (of degree n). We also consider the auxiliary polynomial q n 1 (z) := G n (z) γn+1 p n (z). λ n Observe that q n 1 (z) has degree at most n 1, but it can be identical to zero, as the special case G = {z : z < 1} shows. Asymptotics Estimates Conjecture Statement Numerical Theoretical

22 [ 22 ] University of Cyprus Related asymptotics Theorem (Pritsker, JAT 2002) Assume that Γ is rectifiable and let z Γ be formed by two analytic arcs meeting with exterior angle ωπ, 0 < ω 2. Then, G n (z) = ω(n + 1)ω 1 a1 ωφn+1 ω (z) {1 + o(1)}, Γ(ω + 1) For polygonal Γ the above result was established by Szegő in Math. Z., Theorem (St, Constr. Approx., 2013) If Γ is piecewise analytic without cusps, then it holds { ( )} λ n n γ n+1 = 1 + O and q n 1 π n L2 (G) = O ( ) 1. n Asymptotics Estimates Conjecture Statement Numerical Theoretical

23 [ 23 ] University of Cyprus Theoretical support of conjecture The above two theorems suggest the conjecture, because p n (z) = λ n γ n+1 {G n(z) q n 1 (z)}. Asymptotics Estimates Conjecture Statement Numerical Theoretical

24 [ 24 ] University of Cyprus A useful Lemma Under the assumption Γ is piecewise analytic without cusps, the following three inequalities hold for any polynomial P n of degree n: (i) The uniform estimate (ii) The pointwise estimate P n L (G) c(γ)n ω P n L2 (G). P n (z) c(γ, z)n ω log n P n L 2 (G), valid for any z Γ, formed from arcs meeting with exterior angle ωπ. (iii) The pointwise estimate P n (z) c(γ, z)n P n L 2 (G), valid for z Γ, not a corner point (hence ω = 1). Asymptotics Estimates Conjecture Statement Numerical Theoretical