Avram Parter and Szegő limit theorems turned inside out
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1 Avram Parter and Szegő limit theorems turned inside out Egor A. Maximenko joint work with Johan M. Bogoya Ramírez, Albrecht Böttcher, and Sergei M. Grudsky Instituto Politécnico Nacional, ESFM, México Norte-Sur CINVESTAV, México 20 de noviembre de 204 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur 204 / 32
2 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
3 Toeplitz matrices Szegő s limit theorem Quantile function Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
4 Toeplitz matrices Szegő s limit theorem Quantile function Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
5 Toeplitz matrices T 5 (a) = a 0 a a 2 a 3 a 4 a a 0 a a 2 a 3 a 2 a a 0 a a 2 a 3 a 2 a a 0 a a 4 a 3 a 2 a a 0. It is convenient to think that a j are the Fourier coefficients of a certain function a defined on [0, 2π]: a j = 2π a(θ) e jiθ dθ. 2π 0 The function a is called the generating symbol of the matrices T n (a) = [ a j k ] n j,k=. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
6 Hermitian Toeplitz matrices, real bounded symbols We suppose that the generating symbol is bounded and real: a L ([0, 2π], R). In this case the matrices are Hermitian: a k = a k, a 0 R. T 5 (a) = a 0 a a 2 a 3 a 4 a a 0 a a 2 a 3 a 2 a a 0 a a 2 a 3 a 2 a a 0 a a 4 a 3 a 2 a a 0. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
7 stealed from loveiscomix.com Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
8 [ a0 ] a a 2 a 3 a a 0 a a 2 a 2 a a 0 a a 3 a 2 a a 0 To understand an Hermitian matrix is... Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
![[ a0 ] a a 2 a 3 a a 0 a a 2 a 2 a a 0 a a 3 a 2 a a 0 [ λ ] 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4 To understand an Hermitian matrix is.](/docs-images/82/85140650/images/9-0.jpg "..... to know the behavior of its eigenvalues and eigenvectors.")
9 [ a0 ] a a 2 a 3 a a 0 a a 2 a 2 a a 0 a a 3 a 2 a a 0 [ λ ] λ λ λ 4 To understand an Hermitian matrix is to know the behavior of its eigenvalues and eigenvectors. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
10 Eigenvalues of Hermitian Toeplitz matrices For a in L ([0, 2π], R), sp T n (a) [ess inf(a), ess sup(a)]. Widom (994): sp T n (a) in the Hausdorff distance [ess inf(a), ess sup(a)]. λ (n),..., λ(n) n := the eigenvalues of T n (a) written in the ascending order: λ (n)... λ (n) n. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
11 Behavior of the eigenvalues of Hermitian Toeplitz matrices Graph of a Eigenvalues of T 8 (a) 3/4 /4 0 π/2 π 3π/2 2π 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
12 Behavior of the eigenvalues of Hermitian Toeplitz matrices Graph of a Eigenvalues of T 6 (a) 3/4 /4 0 π/2 π 3π/2 2π 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
13 Behavior of the eigenvalues of Hermitian Toeplitz matrices Graph of a Eigenvalues of T 32 (a) 3/4 /4 0 π/2 π 3π/2 2π 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
14 Behavior of the eigenvalues of Hermitian Toeplitz matrices Graph of a Eigenvalues of T 32 (a) 3/4 β=0.4 /4 how many? 0 π/2 π 3π/2 2π α=0. 0 First question: How many eigenvalues are in [α, β]? (Szegő, 920). Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
15 Behavior of the eigenvalues of Hermitian Toeplitz matrices Graph of a Eigenvalues of T 32 (a) 3/4 β=0.4 /4 0 π/2 π 3π/2 2π how many? α=0. 0 λ (32)? 7 First question: How many eigenvalues are in [α, β]? (Szegő, 920). Second question: λ (n) j? Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
16 Toeplitz matrices Szegő s limit theorem Quantile function Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
17 Szegő s limit theorem (920) generating symbol a L ([0, 2π], R) test function ϕ C(R) n n j= ϕ(λ (n) j ) 2π ϕ(a(θ)) dθ 2π 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur 204 / 32
18 Avram Parter limit theorem (988) an analogue of Szegő limit theorem for the singular values generating symbol a L ([0, 2π], C) test function ϕ C(R) n n j= ϕ(s (n) j ) 2π ϕ( a(θ) ) dθ 2π 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
19 Corollary from Szegő s limit theorem distribution of the eigenvalues of Hermitian Toepltiz matrices generating symbol a C([0, 2π], R) segment [α, β] α < β #{j : α λ (n) j β} n µ {θ [0, 2π]: α a(θ) β} 2π Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
20 Example to illustrate the corollary Graph of a Eigenvalues of T 32 (a) π 0. 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
21 Example to illustrate the corollary Graph of a Eigenvalues of T 32 (a) 0.4 eigenvalues 0 2π Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
22 Example to illustrate the corollary Graph of a Eigenvalues of T 32 (a) 0.4 eigenvalues 0 2π Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
23 Example to illustrate the corollary Graph of a Eigenvalues of T 32 (a) 0.4 eigenvalues 0 2π Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
24 Example to illustrate the corollary Graph of a Eigenvalues of T 32 (a) 0.4 eigenvalues 0 2π 0. 0 µ {θ : 0. a(θ) 0.4} 2π = Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
25 Szegő found an approximate answer to the first question (the number of the eigenvalues on any segment [α, β]). The second question is open: λ (n) j? Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
26 Corollary by Trench (202): the reordering idea a C([0, 2π], R) n n λ (n) j j= v (n) j 0 where v (n),..., v (n) ( ) ( ) ( ) n are the values a 2π n, a 4π n,..., a 2πn n written in the ascending order: v (n) v n (n) Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
27 Toeplitz matrices Szegő s limit theorem Quantile function Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
28 Quantile function of a list of numbers The same numbers in the ascending order: QuantileFunction(/3) = 8 because 8 is the minimal number v such that at least /3 of the elements are less or equal to v. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
29 Quantile function associated to a L ([0, 2π], R) F a := the cumulative distribution function of a : F a (v) := µ {θ [0, 2π]: a(θ) v}, v R. 2π Q a := the corresponding quantile function : Q a (p) := inf{v R: F a (v) p}, p (0, ]. Q a increases and has the same distribution as a. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
30 Construction of the quantile function associated to a piecewise-linear generating symbol Graph of a Graph of Q a π 2 π 3π 2 2π Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
31 Construction of the quantile function associated to a piecewise-linear generating symbol Graph of a Graph of Q a 3 4 v v 4 0 π 2 π 3π 2 2π a and Q a are identically distributed: 2π µ{θ [0, 2π]: a(θ) v} = µ{p (0, ]: Q a(p) v} Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
32 Construction of the quantile function associated to a piecewise-linear generating symbol Graph of a Graph of Q a π 2 π 3π 2 2π a reordering in the Lebesgue-style Q a Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
33 Q a may be continuous even when a is not 0 π 2π Graph of a Graph of Q a Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
34 Criterion of continuity of the quantile function Let a : [0, 2π] R be a measurable function. R(a) := the essential range of a. Then the following conditions are equivalent. R(a) is connected F a is strictly increasing Q a is continuous Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
35 Toeplitz matrices Szegő s limit theorem Quantile function Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
36 Main result: uniform convergence of the eigenvalues a L ([0, 2π], R) R(a) is connected max j n λ (n) j Q a ( j/n) 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
37 Idea of the proof Szegő s limit theorem convergence in distribution portmanteau lemma convergence of the quantile functions Uniform continuity of Q a Monotonicity: Q a λ (n)... λ (n) n Uniform convergence Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
38 Uniform convergence of the singular values (our corollary from Avram Parter theorem) a L ([0, 2π], C) R( a ) = [0, a ] max j n s (n) j Q a ( j/n) 0 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
39 First example continuous piecewise-linear generating symbol Graph of a Graph of Q a Eigenvalues of T 64 (a) 3/4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 9/48 /6 64 Every eigenvalue λ (n) j is shown as a point The third picture mimics the second one. ( ) j n, λ(n) j. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
40 First example continuous piecewise-linear generating symbol Graph of a Graph of Q a 3/4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 9/48 /6 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
41 First example continuous piecewise-linear generating symbol Graph of a Graph of Q a 3/4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 9/48 /6 and the points ( j/n, λ (n) j ) Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
42 First example continuous piecewise-linear generating symbol Graph of a 3/4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 9/48 /6 ( ) 3 64, λ(64) 3 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
43 First example continuous piecewise-linear generating symbol Graph of a ( 3 64, Q a ( )) /4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 9/48 /6 ( ) 3 64, λ(64) 3 Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
44 Second example trigonometric polynomial 8/9 Graph of a a(θ) = cos(θ) 2 9 cos(2θ) 8/9 Graph of Q a Eigenvalues of T 64 (a) 0 2π 3 π 4π 3 2π 0 /2 64 If the generating symbol is a trigonometric polynomial, then the maximal error is of the order O(/n): C > 0 max λ (n) j n j Q a ( j/n) C n Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
45 Third example a is not continuous, but R(a) is connected Graph of a Graph of Q a Eigenvalues of T 64 (a) 3/4 3/4 /4 /4 0 π/2 π 3π/2 2π 0 /6 5/6 64 In this example, λ (n) j is also uniformly approximated by Q a ( j/n) as n. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
46 Fourth example If R(a) is not connected, then the uniform convergence fails Graph of a Graph of Q a Eigenvalues of T 64 (a) 3/4 3/4 /4 /4 0 π 2π 0 /2 64 In this example, λ (n) n/2 can not be approximated by values of Q a. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
47 Summary The Szegő limit theorem combined with the notion of quantile function yields the main term of the individual asymptotics of the eigenvalues: λ (n) j Q a ( j n ) assuming that a L ([0, 2π], R) and R(a) is connected. Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
48 Summary The Szegő limit theorem combined with the notion of quantile function yields the main term of the individual asymptotics of the eigenvalues: λ (n) j Q a ( j n ) assuming that a L ([0, 2π], R) and R(a) is connected. Thanks for attention! Egor Maximenko (IPN-ESFM, Mexico) Avram Parter and Szegő theorems Norte-Sur / 32
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