AvramParter and Szeg o limit theorems: from weak convergence to - - PowerPoint PPT Presentation

Avram–Parter and Szeg˝

• limit theorems:

from weak convergence to uniform approximation

Egor A. Maximenko based on joint works with Johan Manuel Bogoya, Albrecht B¨

• ttcher, and Sergei M. Grudsky

Instituto Polit´ ecnico Nacional, ESFM, M´ exico

International Workshop

• n Operator Theory and its Applications

Chemnitz, TU August, 2017

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 1 / 30

Szeg˝

• ’s first limit theorem

and other theorems about asymptotic distribution can be turned inside out.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 2 / 30

Szeg˝

• ’s first

limit theorem Quantile function Uniform convergence

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 3 / 30

Szeg˝

• ’s first

limit theorem Quantile function Uniform convergence

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 4 / 30

Toeplitz matrices

T5(a) = a0 a−1 a−2 a−3 a−4 a1 a0 a−1 a−2 a−3 a2 a1 a0 a−1 a−2 a3 a2 a1 a0 a−1 a4 a3 a2 a1 a0 . It is convenient to think that ak are the Fourier coefficients

• f a certain function a defined on [0, 2π]:

ak = 1 2π

2π

a(θ) e−kiθ dθ. The function a is called the generating symbol of the matrices Tn(a) =

aj−k n

j,k=1.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 5 / 30

Hermitian Toeplitz matrices, real bounded symbols

We suppose that the generating symbol is bounded and real: a ∈ L∞([0, 2π], R). The corresponding Toeplitz matrices are Hermitian: a−k = ak, a0 ∈ R. T5(a) =

      

a0 a1 a2 a3 a4 a1 a0 a1 a2 a3 a2 a1 a0 a1 a2 a3 a2 a1 a0 a1 a4 a3 a2 a1 a0

      

. The spectra of Tn(a) “asymptotically fill” [ess inf(a), ess sup(a)]: ess inf(a) ≤ λ(n)

1

≤ . . . ≤ λ(n)

n

≤ ess sup(a), sp(Tn(a)) → [ess inf(a), ess sup(a)] (in Hausdorff distance).

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 6 / 30

Behavior of the eigenvalues of Hermitian Toeplitz matrices

1 π 2π

Graph of a

1

Eigenvalues of T8(a)

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 7 / 30

Behavior of the eigenvalues of Hermitian Toeplitz matrices

1 π 2π

Graph of a

1

Eigenvalues of T16(a)

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 7 / 30

Behavior of the eigenvalues of Hermitian Toeplitz matrices

1 π 2π

Graph of a

1

Eigenvalues of T32(a)

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 7 / 30

Behavior of the eigenvalues of Hermitian Toeplitz matrices

1 π 2π

Graph of a

α=0.1 β=0.4 how many? 1

Eigenvalues of T32(a)

First question: How many eigenvalues are in [α, β] ?

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 7 / 30

Behavior of the eigenvalues of Hermitian Toeplitz matrices

1 π 2π

Graph of a

λ(32) 20 ≈ ? 1

Eigenvalues of T32(a)

First question: How many eigenvalues are in [α, β] ? Second question: λ(n)

j

≈ ?

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 7 / 30

Szeg˝

• ’s first limit theorem (1920)

generating symbol a ∈ L∞([0, 2π], R) test function ϕ ∈ C(R) 1 n

n

• j=1

ϕ(λ(n)

j

) − − − → 1 2π

2π

ϕ(a(θ)) dθ

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 8 / 30

Another form of the Szeg˝

• ’s first limit theorem

generating symbol a ∈ L∞([0, 2π], R) α < β a(θ) = α, β a.e. #{j : α ≤ λ(n)

j

≤ β} n − − − → µR(a−1([α, β])) 2π

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 9 / 30

Another form of the Szeg˝

• ’s first limit theorem

distribution of the eigenvalues of Hermitian Toeplitz matrices

generating symbol a ∈ L∞([0, 2π], R) v ∈ R a(θ) = v a.e. #{j : λ(n)

j

≤ v} n − − − → µR({θ ∈ [0, 2π]: a(θ) ≤ v}) 2π

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 10 / 30

Example to illustrate Szeg˝

• ’s first limit theorem

2π 1

Graph of a

0.4 1

Eigenvalues of T32(a)

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 11 / 30

Example to illustrate Szeg˝

• ’s first limit theorem

2π 1

Graph of a

0.4 1

Eigenvalues of T32(a)

15 eigenvalues

15 32 ≈ 0.469

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 11 / 30

Example to illustrate Szeg˝

• ’s first limit theorem

2π 1

Graph of a

0.4 1

Eigenvalues of T32(a)

15 eigenvalues

15 32 ≈ 0.469

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 11 / 30

Example to illustrate Szeg˝

• ’s first limit theorem

2π 1

Graph of a

0.4 1

Eigenvalues of T32(a)

15 eigenvalues

15 32 ≈ 0.469

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 11 / 30

Example to illustrate Szeg˝

• ’s first limit theorem

2π 1

Graph of a

µR {θ: a(θ)≤ 0.4} 2π

= 0.483

0.4 1

Eigenvalues of T32(a)

15 eigenvalues

15 32 ≈ 0.469

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 11 / 30

Szeg˝

• found an approximate answer to the first question:

how many eigenvalues belong to a given interval? The second question was still open:

λ(n)

j

≈ ?

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 12 / 30

Szeg˝

• ’s first

limit theorem Quantile function Uniform convergence

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 13 / 30

Definition of the quantile function

BPM(R) := Borel probability measures over R. Given µ ∈ BPM(R), one defines: the cumulative distribution function Fµ : R → [0, 1], Fµ(v) := µ(−∞, v],

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 14 / 30

Definition of the quantile function

BPM(R) := Borel probability measures over R. Given µ ∈ BPM(R), one defines: the cumulative distribution function Fµ : R → [0, 1], Fµ(v) := µ(−∞, v], the quantile function Qµ : (0, 1) → R, Qµ(p) := inf{v ∈ R: Fµ(v) ≥ p},

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 14 / 30

Definition of the quantile function

BPM(R) := Borel probability measures over R. Given µ ∈ BPM(R), one defines: the cumulative distribution function Fµ : R → [0, 1], Fµ(v) := µ(−∞, v], the quantile function Qµ : (0, 1) → R, Qµ(p) := inf{v ∈ R: Fµ(v) ≥ p}, and the support of µ: supp(µ) := {v ∈ R: ∀ε > 0 µ(v − ε, v + ε) > 0}.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 14 / 30

Quantile function associated to a list of real numbers

X = (−1, 0.6, 1.1). Associate the weight 1/3 to each one of the elements of X: µ({−1}) = µ({0.6}) = µ({1.1}) = 1 3. The corresponding cdf and the quantile function:

1/3 2/3 1 −1 0.6 1.1 −1 0.6 1.1 1 3 2 3 1 Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 15 / 30

Quantile function associated to a list of real numbers

118 100 195 166 164 123 102 172 164 117

The same numbers in the ascending order (α1 ≤ α2 ≤ . . . ≤ α10):

100 102 117 118 123 164 164 166 172 195

Qα(1/3) = α⌈10/3⌉ = α4 = 118. Qα(p) = α⌈np⌉.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 16 / 30

Quantile function associated to a function

Let a ∈ L∞([0, 2π], R). Pushforward measure µ ∈ BPM(R): µ(B) := 1 2πµR(a−1(B)). Fa := the cumulative distribution function of a : Fa(v) := 1 2π µR {θ ∈ [0, 2π]: a(θ) ≤ v}, v ∈ R. Qa := the corresponding quantile function : Qa(p) := inf{v ∈ R: Fa(v) ≥ p}, p ∈ (0, 1).

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 17 / 30

Construction of the quantile function associated to a piecewise-linear real function

2π 1 π 2 π 3π 2 3 4 1 4

Graph of a

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Construction of the quantile function associated to a piecewise-linear real function

2π 1 π 2 π 3π 2 3 4 1 4

Graph of a

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Construction of the quantile function associated to a piecewise-linear real function

2π 1 π 2 π 3π 2 3 4 1 4

Graph of a

3 4 1 1 4 19 48 11 16 1

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Construction of the quantile function associated to a piecewise-linear real function

2π 1 π 2 π 3π 2 3 4 1 4

Graph of a

3 4 1 1 4 19 48 11 16 1

1 1

Graph of Qa

3 4 1 4 19 48 11 16 Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Construction of the quantile function associated to a piecewise-linear real function

v

2π 1

Graph of a

v

1 1

Graph of Qa a and Qa are identically distributed:

1 2πµR{θ ∈ [0, 2π]: a(θ) ≤ v} = µR{p ∈ [0, 1]: Qa(p) ≤ v}

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Construction of the quantile function associated to a piecewise-linear real function

2π 1

Graph of a

1 1

Graph of Qa a

reordering in Lebesgue-style

− − − − − − − − − − − − − − − − − − − − → Qa

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 18 / 30

Criterion for continuity of the quantile function

Let µ ∈ BPM(R) with compact supp(µ). Then the following conditions are equivalent: supp(µ) is connected Fµ is strictly increasing Qµ is continuous

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 19 / 30

Convergence in distribution (µn Λ)

(convergence in law, weak convergence)

Let Λ ∈ BPM(R) and let (µn)∞

n=1 be a sequence in BPM(R).

Then the following conditions are equivalent. ∀ϕ ∈ Cb(R) lim

n→∞

• R

ϕ dµn =

• R

ϕ dΛ ∀v ∈ C(FΛ) lim

n→∞ Fµn(v) = FΛ(v)

∀p ∈ C(QΛ) lim

n→∞ Qµn(p) = QΛ(p)

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 20 / 30

Szeg˝

• ’s first

limit theorem Quantile function Uniform convergence

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 21 / 30

Main result

µn ∈ BPM(R) supp(µn) ⊆ [α, β] Λ ∈ BPM(R) supp(Λ) = [α, β] µn Λ Qµn

[0,1]

= = = ⇒ QΛ

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 22 / 30

Application to Toeplitz matrices: uniform approximation of the eigenvalues

a ∈ L∞([0, 2π], R) R(a) = [α, β] max

1≤j≤n

• λ(n)

j

− Qa( j/n)

• −

− − → 0

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 23 / 30

First example

continuous piecewise-linear generating symbol

π/2 π 3π/2 2π 1/4 3/4 1

Graph of a

19/48 11/16 1 1/4 3/4 1

Graph of Qa

1 64 1

Eigenvalues of T64(a) Every eigenvalue λ(n)

j

is shown as a point

• j

n, λ(n) j

• .

The third picture mimics the second one.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 24 / 30

Second example

a is not continuous, but R(a) is connected

π/2 π 3π/2 2π 1/4 3/4 1

Graph of a

1/6 5/6 1 1/4 3/4 1

Graph of Qa

1 64 1

Eigenvalues of T64(a) In this example, λ(n)

j

is also uniformly approximated by Qa( j/n) as n → ∞.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 25 / 30

Third example

If R(a) is not connected, then the uniform convergence fails

π 2π 1/4 3/4 1

Graph of a

1/2 1 1/4 3/4 1

Graph of Qa

1 64 1

Eigenvalues of T64(a) In this example, λ(n)

⌊n/2⌋ can not be approximated by values of Qa.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 26 / 30

Summary

Szeg˝

• ’s first

limit theorem concept of quantile function

λ(n)

j

≈ Qa

j

n

• Egor Maximenko (IPN, Mexico)

weak → uniform IWOTA 2017 27 / 30

Summary

Szeg˝

• ’s first

limit theorem concept of quantile function

λ(n)

j

≈ Qa

j

n

• Similar results, without using the terminology of quantile function:

Di Benedetto, Serra, Fiorentino (1993): pointwise convergence. Trench (2012): convergence in L1 sence.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 27 / 30

More applications

There are other results about asymptotic distribution: Avram–Parter theorem, Szeg˝

• type theorems for locally Toeplitz matrices,

L´ evy’s arcsine law for random walks, Weyl’s theorem about uniformly distributed sequences. Applying the concept of quantile function

• ne easily deduces corollaries about uniform approximation.

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 28 / 30

Uniform approximation of the singular values (quantile version of Avram–Parter theorem)

a ∈ L∞([0, 2π], C) R(|a|) = [0, a∞] max

1≤j≤n

• s(n)

j

− Q|a|( j/n)

• −

− − → 0

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 29 / 30

Interactive visualization

http://www.egormaximenko.com Thanks for attention!

Egor Maximenko (IPN, Mexico) weak → uniform IWOTA 2017 30 / 30