Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols
A.A. Batalshchikov, S.M. Grudsky, I.S. Malisheva, S.S. Mihalkovich, E. Ramirez de Arellano, V.A. Stukopin
AAsymptotics of eigenvalues of large symmetric Toeplitz matrices with smoothsimple-loop symbols
A.A. Batalshchikov
ASM Solutions LLC, Nyzhnyaya Pervomayskaya Str. 48/9, Moscow, Russia, e-mail: [email protected]
S.M. Grudsky
CINVESTAV, Departamento de Mathematicas, Ciudad de Mexico 07360, Mexico, e-mail: [email protected], tel.+52 555747 3800 ext.6457
I.S. Malisheva
S.S. Mihalkovich
E. Ram´ırez de Arellano
CINVESTAV, Departamento de Mathematicas, Av. Instituto Polit´ecnico Nacional 2508, Mexico, tel.+52 55 5061 3869
V.A. Stukopin
1) Moscow Institute of Physics and Technology (MIPT)2) SMI of VSC RAS (South Mathematical Institute of Vladikavkaz Scientific Center of Russian Academy of Sciences)3) Interdisciplinary Scientific Center J.-V. Poncelet (CNRS UMI 2615) and Scoltech Center for Advanced Study
Abstract
This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitzmatrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimensionof the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. Theconstructed expansion is uniform with respect to the number of eigenvalues.
Keywords:
Toeplitz matrices, Eigenvalues, Asymptotic expansions
Primary 47B35, Secondary 15A18, 41A25, 65F15
1. Introduction
Given a function a ( t ) in L on the complex unit circle T we denote by a l the l-th Fourier coefficient a l = 12 π (cid:90) π a ( e ix ) e − ilx dx, (1.1) l ∈ Z , and by T n ( a ) the n × n Toeplitz matrix ( a j − k ) nj,k =1 .The object of our study is the behavior of the spectral characteristics (eigenvalues and singular numbers, eigen-vectors, determinants, condition numbers, etc.) of Toeplitz matrices in the case when the dimension of the matricestends to infinity. It has been intensively studied for a century (see [1], [2], [3], [4], and literature cited there).This problem is important for statistical mechanics and other applications ([1], [5], [6], [7], [8]). First of all, wemention the numerous versions of the Szeg¨o theorem on the asymptotic distribution of eigenvalues and theoremsof Abram-Parter type on the asymptotic distribution of singular numbers ([9], [10], [11], [12]). There is a richliterature devoted to the asymptotics of the determinants of Toeplitz matrices. (see monographs [2], [3], papers Preprint submitted to LINEAR ALGEBRA AND ITS APPLICATIONS a r X i v : . [ m a t h . SP ] M a r n tending to infinity is a nontrivial question. This question is resolved in [13]for the case of the Fisher-Hartwig singularities considered in the above-mentioned papers. In this paper it is shownthat the limit set coincides with the image of the symbol in the complex plane.There is another well-known case, when the limit set also coincides with the symbol image. It is the case when thisimage is a “curve without an interior”. Using this fact, in [27] we solved the problem of the asymptotic behaviorof the eigenvalues of Toeplitz symmetric matrices with a polynomial symbol satisfying the following condition.Namely, the symbol passes its own curve-image exactly two times, when the variable t makes one turn on the unitcircle.Note that the asymptotic structure of the eigenvectors in the case of n → ∞ is considered in the papers [28],[29], [30].In this paper we generalize the results of the article [27], extending the class of symbols from polynomials tothe class of smooth functions that have only two continuous derivatives. For this purpose the method used in [27]needed a significant change. The main obstacle here is that the considered symbol is defined only on the unitcircle and does not allow, in general, unlike the case of the polynomial symbol of [27], an analytic continuationto the neighborhood of the unit circle T . At the same time, the eigenvalues are not located on the image-curveof the symbol a ( t ), but they are located in some of its neighborhoods. Therefore, the question arises about thecontinuation of a ( t ) to the complex plane. In this regard, we replace the symbol with a polynomial approximation(first terms of the Laurent series) of a n ( t ) of degree ( n −
1) (see (2.4)) and note that the operator correspondingto the Toeplitz matrix does not change. The function a n ( t ) is considered in an annulus with the width of order1 /n , containing T . We show that all eigenvalues lie in the image of the mapping w = a n ( z ) of this annulus. On theone hand, it is necessary to transfer the methods and results of [20], [21] from the real segment to a region in thecomplex plane. On the other hand, we ensure that all constructions are uniform with respect to the parameters ofthe family of functions { a n } , n ∈ N .The paper is organized as follows. Section 2 contains the main results of the work. We consider in Section 3an example with numerical calculations of all eigenvalues for different values of n . The presented figures bring upseveral questions about the location the eigenvalues. The main results that are formulated in Section 2 allows toanswer these questions. In particular we give the asymptotics of the eigenvalues that are located near the points z = a (1) and z = a ( −
1) (see Lemma 3.2), where the derivative of the symbol vanishes. This result is a generalizationto the complex case of the well-known results about the asymptotics of the smallest and largest eigenvalues of largeToeplitz matrices with real value symbols (see [9], [18]).Section 4 presents the results on the smoothness properties of the functions a ( t ) and a n ( t ) that we need and thefunctions b ( t, λ ), b n ( t, λ ) are constructed on the basis of a ( t ), a n ( t ) (see (2.10), (2.12)). In Section 5, a nonlinearequation is introduced for determining the eigenvalues and then its asymptotic properties are investigated. Section6 is devoted to the analysis of the solvability of the above mentioned nonlinear equation in the complex domainsurrounding the image-curve of the symbol a ( t ). The main results are proved in section 7.
2. The main results
Let α ≥
0. We denote by W α the weighted Wiener algebra of all complex-valued functions a : T → C , whoseFourier coefficients satisfy (cid:107) a (cid:107) α := ∞ (cid:88) j = −∞ | a j | ( | j | + 1) α < ∞ . (2.1)Let m be the integer part of α . It is readily seen that if a ∈ W α then the function g defined by g ( ϕ ) := a ( e iϕ )is a 2 π -periodic C m function on R . In what follows we consider complex-valued symmetric simple-loop functions in2 α . To be more precise, for α ≥
2, we let
CSL α be the set of all a ∈ W α such that g ( ϕ ) = a ( e i ϕ ) has the followingproperties:(1) the function g ( ϕ ) is symmetrical in the following sense: g ( ϕ ) = g (2 π − ϕ ) , ϕ ∈ [0 , π ] . (2.2)(It is equivalent to the conditon a j = a − j , j = 1 , , . . . .)(2) Im( g ) is a simple (without self-intersections) arc with non-coincident end points M , M : g (0) = g (2 π ) = M , g ( π ) = M , M (cid:54) = M , so that g (cid:48) ( ϕ ) (cid:54) = 0 for ϕ ∈ (0 , π ) and g (cid:48)(cid:48) (0) = g (cid:48)(cid:48) (2 π ) (cid:54) = 0, g (cid:48)(cid:48) ( π ) (cid:54) = 0.It should be noted that if we have (2.2) then g (cid:48) (0) = g (cid:48) (2 π ) = g (cid:48) ( π ) = 0 . (2.3)We introduce the following notation. Let f ( t ) = (cid:80) ∞ j = −∞ f j t j ( t ∈ T ) be a function from the space L ( T ) suchthat (cid:80) ∞ j = −∞ | f j | < ∞ . We consider the projectors[ P n f ] ( t ) := n − (cid:88) j =0 f j t j , n = 1 , , . . . . We will also denote the image of the operator P n by L ( n )2 . Note that for the symbol a ∈ L ∞ ( T ) the Toeplitzmatrix T n ( a ) can be identified with the operator T n ( a ) : L ( n )2 → L ( n )2 , defined by T n ( a ) f = P n ( af ) . We introduce then the functions a n ( t ) = n − (cid:88) j = − ( n − a j t j (2.4)and note that T n ( a ) = T n ( a n ) . (2.5)Therefore, we will use the function a n ( t ) instead of the symbol a ( t ), when it will be convenient, and respectivelythe function g n ( ϕ ) := a n ( e i ϕ ) instead of g ( ϕ ).Note that the functions a n ( t ) and g n ( ϕ ) satisfy all conditions of the definition of CSL α for a sufficiently large n .Besides, if a ( t ) ∈ W α then sup t ∈ T | a ( t ) − a n ( t ) | = o ( / n α ) , n → ∞ (2.6)(see Lemma 4.1, i) below).Introduce the sets: R ( a ) := { g ( ϕ ) : ϕ ∈ (0 , π ) } ; (2.7)Π n ( a ) = (cid:8) ψ = ϕ + i δ | ϕ ∈ [ cn − , π − cn − ] , δ ∈ [ − Cn − , Cn − ] (cid:9) , (2.8)where c small enough and C large enough are fixed positive numbers. Let us denote R n ( a ) := { g n ( ψ ) : ψ ∈ Π n ( a ) } . (2.9)It is well known (see for example [2]) that the limit spectrum of the operator family { T n ( a ) } ∞ n =1 coincideswith the curve R ( a ). Thus, for sufficiently large n the spectrum of T n ( a ) is located in the neighborhood of R ( a ).Moreover, we will show that sp T n ( a ) ⊂ R n ( a ).According to the conditions (1)-(2), for any λ ∈ R ( a ) there exists exactly one ϕ ( λ ) ∈ (0 , π ) such that g ( ϕ ( λ )) = λ . The symmetry implies that the function ϕ ( λ ) := (2 π − ϕ ( λ )) ∈ ( π, π ) also has this property: g ( ϕ ( λ )) = λ .For all λ ∈ R ( a ) consider the functionˆ b ( t, λ ) = ( a ( t ) − λ ) e i ϕ ( λ ) ( t − e i ϕ ( λ ) )( t − − e i ϕ ( λ ) ) . (2.10)3Note that e − i ϕ ( λ ) = e i ϕ ( λ ) , therefore the function (cid:0) t − − e i ϕ ( λ ) (cid:1) goes to zero at a single point t = e i ϕ ( λ ) ). In asimilar manner, for all λ ∈ R n ( a ) there exist ϕ ,n ( λ ) ∈ Π n , such that g n ( ϕ ,n ( λ )) = g n (2 π − ϕ ,n ( λ )) = λ. (2.11)It will be shown below that the point ϕ ,n ( λ ), satisfying condition (2.11) (see Lemma 4.3) is unique.Together with (2.10) we consider the functionˆ b n ( t, λ ) = ( a n ( t ) − λ ) e i ϕ ,n ( λ ) ( t − e i ϕ ,n ( λ ) )( t − − e i ϕ ,n ( λ ) ) , λ ∈ R n ( a ) , (2.12)which is a polynomial of powers of t and t − of finite degree and does not vanish in the domain t ∈ T , λ ∈ R n ( a ).We show that this function allows a Wiener-Hopf factorization of the formˆ b n ( t, λ ) = ˆ b n, + ( t, λ )ˆ b n, + ( t − , λ ) , where ˆ b n, + ( t, λ ) is a polynomial of degree ( n −
2) of the variable t . We introduce the function θ n ( λ ) = log ˆ b n, + ( e i ϕ ,n ( λ ) , λ )ˆ b n, + ( e − i ϕ ,n ( λ ) , λ ) , λ ∈ R n ( a ) . (2.13)Note that in the case of λ ∈ R ( a ), ( e i ϕ ( λ ) ∈ T ) (2.13) the function θ n ( λ ) can be represented as θ n ( λ ) = 12 π i (cid:90) T log ˆ b n ( τ, λ ) τ − e i ϕ ,n ( λ ) dτ − π i (cid:90) T log ˆ b n ( τ, λ ) τ − e − i ϕ ,n ( λ ) dτ. (2.14)We also introduce the function θ ( λ ) = 12 π i (cid:90) T log ˆ b ( τ, λ ) τ − e i ϕ ( λ ) dτ − π i (cid:90) T log ˆ b ( τ, λ ) τ − e − i ϕ ( λ ) dτ, λ ∈ R ( a ) , (2.15)where the integrals in (2.14) and (2.15) are understood in the sense of the principal value. It is more convenient forus to consider the introduced functions as functions of the parameter s := ϕ ,n ( λ ) ( λ = g n ( s )): η ( s ) := θ ( g ( s )) , s ∈ (0 , π ) (2.16)and η n ( s ) := θ n ( g n ( s )) , s ∈ Π n ( s ) (2.17)Introduce the values d j,n = πjn + 1 , j = 1 , , . . . , n, (2.18) e j,n = d j,n − η n ( d j,n ) n + 1 , j = 1 , , . . . , n. (2.19)We will also need the following small areas:Π j,n ( a ) := (cid:26) s ∈ Π n ( a ) , | s − e j,n | ≤ c n n + 1 (cid:27) , j = 1 , , . . . , n, (2.20)where the constants c n does not depend on j and decrease to 0 with n → ∞ (see (6.5)). Now we are ready toformulate the main results of this work. Let λ ( n ) j , j = 1 , , . . . , n be a numeration of the eigenvalues of the operator T n ( a ). Theorem 1.
Let a be a symbol such that a ∈ CSL α , α ≥ . Then, for a sufficiently large natural number n , thefollowing statements hold:i) all eigenvalues T n ( a ) are different, and λ ( n ) j ∈ g (Π j,n ) for j = 1 , , . . . , n , i) the values s j,n such that λ ( n ) j = g n ( s j,n ) satisfy the equation ( n + 1) s + η n ( s ) = πj + ∆ n ( s ) , j = 1 , , . . . , n (2.21) with | ∆ n ( s ) | = o (1 /n α − ) where ∆ n ( s ) → as n → ∞ uniformly respect to s ∈ Π n ( a ) .iii) Equation (2.21) has a unique solution in the domain Π j,n . Theorem 2.
Under the conditions of the Theorem 1 s j,n = d j,n + [ α ] − (cid:88) k =1 p k ( d j,n )( n + 1) k + ∆ ( n )2 ( j ) where ∆ ( n )2 ( j ) = (cid:40) o (1 /n ) , α = 2 O (1 /n α − ) , α > as n → ∞ uniformly in j . The functions p k can be calculated explicitly; in particular p ( s ) = − η ( s ) , p ( s ) = η ( s ) η (cid:48) ( s ) . Theorem 3.
Under the conditions of Theorem 1 λ ( n ) j = g ( d j,n ) + [ α ] − (cid:88) k =1 r k ( d j,n )( n + 1) k + ∆ ( n )3 ( j ) (2.22) where ∆ ( n )3 ( j ) = o (cid:16) d j,n ( π − d j,n ) n (cid:17) , α = 2 ,O (cid:16) d j,n ( π − d j,n ) n α − (cid:17) , α > . as n → ∞ uniformly in j . The coefficients r k can be calculated explicitly; in particular r ( ϕ ) = − g (cid:48) ( ϕ ) η ( ϕ ) and r ( ϕ ) = 12 g (cid:48)(cid:48) ( ϕ ) η ( ϕ ) + g (cid:48) ( ϕ ) η ( ϕ ) η (cid:48) ( ϕ ) .
3. Numerical example and the consequences of the main results
Define a symbol by the function g ( ϕ ) (= a ( e i ϕ )): g ( ϕ ) = c sin( c ϕ ) + c ((1 + ϕ ) / + (1 − ϕ ) / ) , ϕ ∈ [ − π ; π ] , (it is more convenient for us to consider the symbol in this section on the segment [ − π ; π ]) where: c = 15 −
16 i , c = (1 − π ) / − ( π + 1) / πc cos( π c ) , c = 120 . The expression for the constants c , are derived from the conditions g (cid:48) ( − π ) = g (cid:48) ( π ). (The equalities g ( − π ) = g ( π )and g (cid:48)(cid:48) ( − π ) = g (cid:48)(cid:48) ( π ) are a consequence of the symmetry of function g ( ϕ ).) It can be verified that the constructedfunction satisfies the conditions (1) and (2) at the begining of Section 2. (The condition g (cid:48) ( ϕ ) (cid:54) = 0 for ϕ ∈ ( − π ; π )can be verified numerically.)We can see that the third derivative of the symbol a ( t ) has singularities at the points t = e i and t = e − i . It iseasy to see that a ( t ) ∈ CSL . − δ for arbitrary small δ > a ( t ) and the eigenvalues of the matrices T n ( a ) for n = 20 and n = 80 are shown in theFigure 3.1 and Figure 3.2 correspondingly. If we look at these Figures then we can make the following observations:1. The limit set of the eigenvalues for T n ( a ) if n → ∞ really coincide with R ( a ).2. The points of concentration for the eigenvalues of T n ( a ) are z = a (1) and z = a ( − z and z is much less than outside of these neighborhoods.5 .25 0.50 0.75 1.00 1.25 1.50 1.75Re z I m z Figure 3.1: Image of the symbol a ( t ) and eigenvalues of the matrix T ( a ) z I m z Figure 3.2: Image of the symbol a ( t ) and eigenvalues of the matrix T ( a )
6. Some eigenvalues are located under the curve R ( a ) and others above the curve.We are going to show that Theorem 3 allows to explain and clarify these observations.Designate the spectrum of T n ( a ) by Sp( T n ( a )) := (cid:110) λ ( n ) j (cid:111) nj =1 . Then the limit set of the eigenvalues of thesequence { T n ( a ) } ∞ n =1 is the following: Λ( a ) := lim sup n →∞ Sp( T n ( a )) . The next Lemma is a direct consequence of the Theorem 3.
Lemma 3.1.
Under the conditions of Theorem 1 Λ( a ) = R ( a ) . In addition sup ≤ j ≤ n ρ (cid:16) λ ( n ) j , R ( a ) (cid:17) ≤ const n . (Here ρ ( z, R ( a )) is the distance between the point z and the curve R ( a ) in the complex plane). Consider now the observation 2. The situation in the neighborhoods of the points z and z clarifies the followingstatement. Lemma 3.2.
Let the conditions of the Theorem 1 be fulfilled. Then:i) if jn +1 → , then the following asymptotic formula is true: λ ( n ) j = g (0) + π g (cid:48)(cid:48) (0)2 j ( n + 1) + ∆ ( n )4 ( j ) , where ∆ ( n )4 ( j ) = o ( j / n ) . (3.1) ii) if n +1 − jn +1 → , then the following asymptotic formula is true: λ ( n ) j = g ( π ) + π g (cid:48)(cid:48) ( π )2 ( n + 1 − j ) ( n + 1) + ∆ ( n )5 ( j ) , where ∆ ( n )5 ( j ) = o (cid:18) ( n + 1 − j ) n (cid:19) . (3.2) Proof.
Since a ( t ) ∈ W then, according to (2.22), we have: λ ( n ) j = g ( d j,n ) − g (cid:48) ( d j,n ) η ( d j,n ) n + 1 + o (cid:18) d j,n ( π − d j,n ) n (cid:19) . Consider i). We have in this case that o (cid:18) d j,n ( π − d j,n ) n (cid:19) = o (cid:18) jn (cid:19) . Taking into account that g (cid:48) ( d j,n ) = g (0) + g (cid:48)(cid:48) (0)2 ( d j,n ) + o ( d j,n ) ,g (cid:48) ( d j,n ) = O ( d j,n ), and η ( d j,n ) = O ( d j,n ), one has | g (cid:48) ( d j,n ) η ( d j,n ) | n + 1 = o (cid:18) j n (cid:19) . Thus we get i). Case ii) is proved analogously. 7hus Lemma 3.2 shows that the distance between consecutive eigenvalues located in a neighborhood of the point z = g (0) ( j/n →
0) is of o ( j/n ). So if j is bounded (“the first eigenvalues”) then this distance is of o ( / n ) . (3.3)The same result is true for neighborhoods of the point z = g ( π ) (“the last eigenvalues”). On the other handfor ε < d j,n < π − ε , where ε > | λ ( n ) j +1 − λ ( n ) j | = o ( / n ) . (3.4)Thus (3.3) and (3.4) confirm and explain of the observation 2 above.Pass to observation 3. Consider “inner eigenvalues” that is ε < d j,n < π − ε . Suppose that a ( t ) ∈ W theformula (2.22) give to us that λ ( n ) j = g ( d j,n ) − g (cid:48) ( d j,n ) η ( d j,n ) n + 1 + O ( / n ) . Let ˜ e j,n := d j,n − Re η ( d j,n ) n + 1 , then λ ( n ) j = g (˜ e j,n ) − i g (cid:48) (˜ e j,n ) Im η ( d j,n ) n + 1 + O ( / n ) . (3.5)Thus λ ( n ) j is located on the normal to the curve R ( a ) at the point z = g (˜ e j,n ) with exactitude O (cid:0) /n (cid:1) .In addition we can see that the point λ ( n ) j is located “above” or “below” of the curve R ( a ) depending the signof the real number Im η ( d j,n ). Moreover, formula (3.5) give us that | g (cid:48) (˜ e j,n ) Im η ( d j,n ) | n + 1 + O (cid:0) /n (cid:1) is the distancebetween λ ( n ) j and R ( a ).It should be noted that Theorem 3 has not only a qualitative sense but also a qualitative (numerical) one. Ifone has numerical values for the function η ( d j,n ) we can calculate all eigenvalues λ ( n ) j , j = 1 , , . . . , n very rapidlyfor different values of n . This idea was applied in the article [31] (in case of real valued symbols) where the function η ( d j,n ) was calculated with the help of the eigenvalues λ ( n ) j for not very large n .In the rest of this section we illustrate the accuracy of asymptotic formulas. Introduce the notation for approx-imated eigenvalues from (2.22): λ ( n )1 ,j = g ( d j,n ) − g (cid:48) ( ϕ ) η ( ϕ ) n + 1 ,λ ( n )2 ,j = g ( d j,n ) − g (cid:48) ( ϕ ) η ( ϕ ) n + 1 + g (cid:48)(cid:48) ( ϕ ) η ( ϕ ) + g (cid:48) ( ϕ ) η ( ϕ ) η (cid:48) ( ϕ )( n + 1) . The relative approximation error will be characterized by the following values:∆ ( n )1 = max j =1 ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( n )1 ,j − λ ( n ) j λ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∆ ( n )2 = max j =1 ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( n )2 ,j − λ ( n ) j λ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The resulting accuracy of the spectrum of T n ( a ) is shown in table 1. n
20 40 80 160 320∆ ( n )1 ( n )2 Table 1: Comparative accuracy of the calculation of the spectrum T n for the symbol a ( t ) We can see for even n = 20 the accuracy good enough for both approximations.8 . Preliminary results Let the function f be f ( t ) = (cid:80) ∞ j = −∞ f j t j ∈ L ( T ). We introduce the operator Q n by the formula( Q n f ) := ∞ (cid:88) j = n f j t j . (4.1)The following Lemma holds. Lemma 4.1. i) If f ∈ W α ( T ) , α ≥ , then sup t ∈ T | ( Q n f ) ( t ) | ≤ (cid:107) Q n f (cid:107) α n α . ii) For a natural number k ≤ α , then f ( k ) ( t ) ∈ W α − k ( T ) and sup t ∈ T (cid:12)(cid:12)(cid:12)(cid:16) Q n f ( k ) (cid:17) ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:107) Q n f (cid:107) α n α − k . iii) For a real number k > α , then n − (cid:88) j = − ( n − | f j || j | k ≤ n k − α (cid:107) f (cid:107) α . Proof.
Items i) and ii) were proved in [20], [21]. Let us prove iii): n − (cid:88) j = − ( n − | f j || j | k = n − (cid:88) j = − ( n − ( | f j | (1 + | j | ) α ) | j | k (1 + | j | ) α ≤ (1 + n ) k − α n (cid:88) j = − n | f j | (1 + | j | ) α ≤ n k − α (cid:107) f (cid:107) α . Let a ∈ W α . Consider the functions g ( ϕ ) = a ( e i ϕ ) = ∞ (cid:88) j = −∞ a j e i jϕ and g n ( ϕ ) = a n ( e i jϕ ) = ( n − (cid:88) j = − ( n − a j e i jϕ . The following Lemma gives an asymptotic representation when n → ∞ of the function g n ( ϕ ) in the complexdomain Π n ( a ) using the function g ( ϕ ) (and its derivatives) defined on [0 , π ]. Lemma 4.2.
Let the point ψ = ϕ + i δ ∈ Π n ( a ) . If a ( t ) ∈ W α , α ≥ , m = [ α ] then g n ( ψ ) = g ( ϕ ) + m (cid:88) k =1 g ( k ) ( ϕ )(i δ ) k + m +1 (cid:88) k =0 α n,k ( ϕ )(i δ ) k , where α n,k ( ϕ ) ∈ W , (cid:107) α n,k (cid:107) = o ( n k − α ) , k = 0 , , . . . , m and (cid:107) α n,m +1 (cid:107) = O ( n m +1 − α ) . Proof.
We can represent g n ( ψ ) in the form: g n ( ψ ) = n − (cid:88) j = − n − a j e i j ( ϕ +i δ ) . | jδ | ≤ C , we can use the following asymptotics e i j (i δ ) = 1 + m (cid:88) k =1 ( − δj ) k k ! + O ( δj ) m +1 . So we obtain g n ( ψ ) = n − (cid:88) j = − ( n − a j e i jϕ (cid:32) m (cid:88) k =1 ( − δj ) k k ! + O ( δj ) m +1 (cid:33) = g ( ϕ ) + α n, ( ϕ ) + m (cid:88) k =1 (i δ ) k ( n − (cid:88) j = − ( n +1) a j e i jϕ ( ij ) k + n +1 (cid:88) j = − ( n +1) a j O ( δj ) m +1 = g ( ϕ ) + m (cid:88) k =1 g ( k ) ( ϕ )(i δ ) k + α n, ( ϕ ) + m (cid:88) k =1 (i δ ) k α n,k ( ϕ ) + δ m +1 α n,m +1 ( ϕ ) . It’s easy to see, due to Lemma 4.1, that (cid:107) α n, ( ϕ ) (cid:107) = (cid:88) | j | > ( n − | a j | ≤ (1 + n ) − α (cid:88) | j | > ( n − | a j | (1 + | j | ) α = o ( n − α ) . Similarly, for k = 1 , , . . . , m (cid:107) α n,k ( ϕ ) (cid:107) = (cid:88) | j | > ( n − | a j || j | k ≤ (1 + n ) k − α (cid:88) | j | > ( n − | a j | (1 + | j | ) α = o ( n k − α ) . Finally, (cid:107) α n,m +1 ( ϕ ) (cid:107) ≤ const n − (cid:88) j = − ( n − | a j | (1 + | j | ) m +1 ≤ const n m +1 − α n − (cid:88) j = − ( n − | a j | (1 + | j | ) α = O ( n m +1 − α ) . Now we can prove the correctness of the introduction of the value ϕ ,n ( λ ) (see (2.11)) Lemma 4.3.
Let a ( t ) ∈ W α , α ≥ . Then mapping g n ( ψ ) : Π n ( a ) → R n ( a ) is a bijection for a sufficiently large natural number n .Proof. The surjectivity of this mapping follows from the definition of the set R n ( a ).We will prove injectivity by contradiction. Suppose that for each n there exists couple of different points ψ ,n , ψ ,n ∈ Π n ( a ) and λ n ∈ R n ( a ) such that g n ( ψ ,n ) = g n ( ψ ,n ) = λ n . (4.2)Without loss of generality, we can assume that the sequences { ψ ,n } ∞ n =1 , { ψ ,n } ∞ n =1 and { λ n } ∞ n =1 have limits ϕ , ϕ ∈ [0 , π ] respectively, and λ ∈ R ( a ). It is obvious that ϕ = ϕ := ϕ because of g ( ϕ ) = g ( ϕ ) = λ , andmapping g ( ϕ ) : [0 , π ] → R ( a ) is injective by condition (2). Suppose that ϕ / ∈ { , π } . Then by (4.2) we have g n ( ψ ,n ) − g n ( ψ ,n ) = 0 . (4.3)Applying Taylor’s formula, we obtain g (cid:48) n ( ψ ,n )( ψ ,n − ψ ,n ) + o ( | ψ ,n − ψ ,n | ) = 0 . The latter is impossible since lim n →∞ g (cid:48) n ( ψ ,n ) = g (cid:48) ( ψ ) (cid:54) = 0 . ϕ = 0 or ϕ = π . Let’s suppose for definiteness, that ϕ = 0. Then for a sufficiently large natural number n | ψ ,n | ≤ σ and | ψ ,n | ≤ σ where σ > g n ( ψ ,n ) − g n ( ψ ,n ) = (cid:90) I n g (cid:48) n ( s ) ds = (cid:90) I n (cid:32)(cid:90) [0 ,s ] g (cid:48)(cid:48) n ( u ) du (cid:33) ds, (4.4)where I n := [ ψ ,n , ψ ,n ] and [0 , s ] are segments of the complex plane connecting these points. We note that, identity(4.4) is true because g (cid:48) n (0) = 0. We rewrite (4.4) as g n ( ψ ,n ) − g n ( ψ ,n ) = g (cid:48)(cid:48) n ( ψ ,n )2 ( ψ ,n − ψ ,n ) + (cid:90) I n (cid:32)(cid:90) [0 ,s ] ( g (cid:48)(cid:48) n ( u ) − g (cid:48)(cid:48) n ( ψ ,n )) du (cid:33) ds. (4.5)Let us estimate the integral term. Since a ( t ) ∈ W α ( T ), α ≥
2, then a (cid:48)(cid:48) ( t ) ∈ W α − ( T ).Therefore, the function g (cid:48)(cid:48) n ( ψ ) is uniformly continuous in Π n ( a ) with respect to n . Thus, the following estimateholds for the difference in the integral expression (4.5)sup | u |≤ σ, | ψ ,n |≤ σ | g (cid:48)(cid:48) n ( u ) − g (cid:48)(cid:48) n ( ψ ,n ) | = o (1) , σ → n . Replacing the variables in (4.5) by u = e i arg s v and s = pψ ,n + (1 − p ) ψ ,n weobtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) I n (cid:32)(cid:90) [0 ,s ] ( g (cid:48)(cid:48) n ( u ) − g (cid:48)(cid:48) n ( ψ ,n )) du (cid:33) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) == | ψ ,n − ψ ,n | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e i arg s (cid:32)(cid:90) | s | (cid:0) g (cid:48)(cid:48) n ( e i arg s v ) − g (cid:48)(cid:48) n ( ψ ,n ) (cid:1) dv (cid:33) dp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ψ ,n − ψ ,n | o (1) (cid:90) (cid:32)(cid:90) | s | dv (cid:33) dp ≤ o ( | ψ ,n − ψ ,n | ) (cid:90) | pψ ,n + (1 − p ) ψ ,n | dp ≤ o ( | ψ ,n − ψ ,n | ) ( | ψ ,n | + | ψ ,n | ) . Thus from (4.5) we get that for a sufficiently large n : | g n ( ψ ,n ) − g n ( ψ ,n ) | ≥ | ψ ,n − ψ ,n | (cid:18) | g (cid:48)(cid:48) (0) | | ψ ,n + ψ ,n | − o ( | ψ ,n | + | ψ ,n | ) (cid:19) . Further, taking in account that ψ ,n = ϕ ,n + i δ ,n , ψ ,n = ϕ ,n + i δ ,n , where ϕ , ,n ≥ cn and | δ , ,n | ≤ Cn , we obtain | g n ( ψ ,n ) − g n ( ψ ,n ) || ψ ,n − ψ ,n | ≥ | g (cid:48)(cid:48) (0) | (cid:113) ( ϕ ,n + ϕ ,n ) + ( δ ,n + δ ,n ) − o (cid:16)(cid:113) ϕ ,n + δ ,n + (cid:113) ϕ ,n + δ ,n (cid:17) > | g (cid:48)(cid:48) (0) | ϕ ,n + ϕ ,n ) − o ( ϕ ,n + | δ ,n | + ϕ ,n + | δ ,n | ) ≥ | g (cid:48)(cid:48) (0) | · cn − o ( / n ) > . Consequently, | g n ( ψ ,n ) − g n ( ψ ,n ) | >
0, which contradicts (4.3). Case ϕ = π is treated similarly. Remark 4.1.
The function g n ( ψ ) is analytic, therefore it is a conformal mapping of the domain Π n ( a ) onto R n ( a ) . Now consider the function ˆ b ( t, λ ) of the form (2.10), where λ ∈ R ( a ). It is convenient to consider it in twoforms. As a function, the second argument of which is s = ϕ ( λ ) (= g − ( λ )) ∈ (0 , π ): b ( t, s ) := ˆ b ( t, g ( s )) = ( a ( t ) − g ( s )) e i s ( t − e i s )( t − − e i s ) (4.6)11nd as a function, the second argument of which is τ = e i s ∈ T :˜ b ( t, τ ) := ( a ( t ) − a ( τ )) τ ( t − τ )( t − − τ ) . (4.7)Similarly, we will consider (2.12) in the form b n ( t, s ) := ( a n ( t ) − g n ( s )) e i s ( t − e i s )( t − − e i s ) , s ∈ Π n ( a ) (4.8)and in the form ˜ b n ( t, τ ) := ( a n ( t ) − a n ( τ )) τ ( t − τ )( t − − τ ) , τ ∈ R n ( a ) . (4.9)The following Lemma gives conditions for the functions introduced above and for their partial derivatives to bein W α . Lemma 4.4.
Let a ∈ CSL α , α ≥ . If s ∈ (0 , π ) , and s ∈ Π n ( a ) , theni) ˜ b ( · , τ ) ∈ W α − , ˜ b ( t, · ) ∈ W α − and (cid:107) ˜ b ( · , τ ) (cid:107) α − ≤ const (cid:107) a (cid:107) α , (cid:107) ˜ b ( t, · ) (cid:107) α − ≤ const (cid:107) a (cid:107) α , (4.10) (cid:107) b n ( · , s ) (cid:107) α − ≤ const (cid:107) a (cid:107) α (4.11) and (cid:107) b ( · , s ) − b n ( · , s ) (cid:107) α − ≤ const n − ( α − ; (4.12) ii) for α ≥ p + (cid:96) , we have that ∂ p + (cid:96) ˜ b ( · , τ ) ∂ p t∂ (cid:96) τ ∈ W α − − p − (cid:96) , ∂ p + (cid:96) ˜ b ( t, · ) ∂ p t∂ (cid:96) τ ∈ W α − − p − (cid:96) , and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) ˜ b ( · , τ ) ∂ p t∂ (cid:96) τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α − − p − (cid:96) ≤ const (cid:107) a (cid:107) α ; (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) ˜ b ( t, · ) ∂ p t∂ (cid:96) τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α − − p − (cid:96) ≤ const (cid:107) a (cid:107) α , (4.13) (cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) b n ( · , s ) ∂ p t∂ (cid:96) s (cid:13)(cid:13)(cid:13)(cid:13) α − − p − (cid:96) ≤ const (cid:107) a (cid:107) α , (4.14) (cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) b ( · , s ) ∂ p t∂ (cid:96) s − ∂ p + (cid:96) b n ( · , s ) ∂ p t∂ (cid:96) s (cid:13)(cid:13)(cid:13)(cid:13) α ≤ const n − ( α − − p − (cid:96) ) ; (4.15) iii) for α < p + (cid:96) , we have (cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) b n ( · , s ) ∂ p t∂ (cid:96) s (cid:13)(cid:13)(cid:13)(cid:13) ≤ const n p + (cid:96) − α (cid:107) a (cid:107) α . (4.16) Here all values of “const” do not depend on τ , t , s , s and n , respectively.Proof. We represent the function b ( t, s ) in the form b ( t, s ) = c ( s ) t ( b ( t, s ) − b ( t, s )) , where b ( t, s ) := a ( t ) − g ( s ) t − e i s , b ( t, s ) := a ( t ) − g ( s ) t − e − i s , c ( s ) = 12i sin s . Since g ( s ) = g ( − s ), we get b ( t, s ) = c ( s ) t ∞ (cid:88) j = −∞ ,j (cid:54) =0 a j (cid:18) t j − e i js t − e i s − t j − e − i js t − e − i s (cid:19) := b (+) ( t, s ) + b ( − ) ( t, s ) , b ( ± ) ( t, s ) are responsible for summation over j > j <
0, respectively. We estimate the first term. b (+) ( t, s ) = c ( s ) t ∞ (cid:88) j =1 a j j − (cid:88) k =0 t k (cid:16) e i( j − − k ) s − e − i( j − − k ) s (cid:17) = 2i c ( s ) t ∞ (cid:88) j =1 a j j − (cid:88) k =0 t k sin( j − − k ) s . Changing the order of summation, we have b (+) ( t, s ) = t ∞ (cid:88) k =0 ∞ (cid:88) j = k +2 a j sin( j − − k ) s sin s t k . (4.17)Let us get the estimate of type (4.10). For this we use the following inequality: (cid:12)(cid:12)(cid:12)(cid:12) sin( L · s )sin s (cid:12)(cid:12)(cid:12)(cid:12) ≤ const L, L > , (4.18)which is true for all real s (in particular for s = s ∈ [0 , π ]), and complex s , such that | L · Im s | ≤ M, (4.19)where M > (cid:107) b (+) ( · , s ) (cid:107) α − ≤ const ∞ (cid:88) k =0 ∞ (cid:88) j = k +2 | a j | · | j − − k | (2 + k ) α − = const ∞ (cid:88) j =2 | a j | (cid:32) j − (cid:88) k =0 | j − − k | · (2 + k ) α − (cid:33) . It is not difficult to show that j − (cid:88) k =0 | j − − k | (2 + k ) α − ≤ const (1 + j ) α , then (cid:107) b (+) ( · , s ) (cid:107) α − ≤ const ∞ (cid:88) j =2 | a j | (1 + j ) α = const (cid:107) a (cid:107) α . (4.20)Similarly, it can be shown that (cid:107) b ( − ) ( · , s ) (cid:107) α − ≤ const · j = − (cid:88) −∞ | a j | (1 + | j | ) α . Obviously, the last two inequalities imply the fulfillment of the first inequality (4.10). The second relation (4.10)is true by symmetry ˜ b ( t, τ ) = ˜ b ( τ, t ). The inequality (4.12) also follows from (4.10) if instead of a ( t ) we take thedifference ( a ( t ) − a n ( t )) and use the statement of the Lemma 4.1, i). The inequality (4.11) is proved in the sameway as in (4.20), it should only be replaced s with s in (4.17), infinite upper limits by j to ( n − s ∈ Π n ( a ) the condition (4.19) is satisfied because | Im s | ≤ Cn , and value ( j − − k ) < n .Let us prove item ii). According to Lemma 4.1 ii), if f ( t ) ∈ W α , then ∂ p f∂ p t ( t ) ∈ W α − p and (cid:13)(cid:13)(cid:13)(cid:13) ∂ p f∂ p t (cid:13)(cid:13)(cid:13)(cid:13) α − p ≤ const (cid:107) f (cid:107) p . (4.21)13herefore, in case (cid:96) = 0, the first of the relations (4.13) is proved. By the symmetry of ˜ b ( t, τ ) we can argue thatthe second of the inequalities is proved. (4.13) in case (cid:96) = 0. Now let p = 0, in case of the first of the relations(4.13), using (4.17) we have ∂ (cid:96) ∂ (cid:96) s b (+) ( t, s ) = ∞ (cid:88) k =0 ∞ (cid:88) j = k +2 a j ∂ (cid:96) ∂ (cid:96) s (cid:18) sin( j − − k ) s sin s (cid:19) t k . (4.22)Then, by analogy with (4.18), we can use the inequality ∂ (cid:96) ∂s (cid:96) (cid:18) sin( L · s ) s (cid:19) ≤ const L (cid:96) +1 (4.23)which is true for all real s = s and complex s because (4.19). Then we get (cid:13)(cid:13)(cid:13)(cid:13) ∂ (cid:96) ∂s (cid:96) b (+) ( t, s ) (cid:13)(cid:13)(cid:13)(cid:13) α − − (cid:96) ≤ const ∞ (cid:88) k =0 ∞ (cid:88) j = k +2 | a j | · | j − − k | (cid:96) +1 (2 + k ) α − − (cid:96) = const ∞ (cid:88) j =2 | a j | (cid:32) j − (cid:88) k =0 | j − − k | (cid:96) +1 (2 + k ) α − − (cid:96) (cid:33) = const ∞ (cid:88) j =2 | a j | (1 + j ) α ≤ const (cid:107) a (cid:107) α . The assertion for b ( − ) ( t, s ) and the other assertions ii) are proved by analogy with i) and by taking into accountthe property (4.21). We turn to the proof of iii). Differentiating (4.22) p times over t , we obtain ∂ p + (cid:96) b n ( t, s ) ∂ p t∂ (cid:96) s = n − (cid:88) k =0 n − (cid:88) j = k +2 a j ∂ (cid:96) ∂ (cid:96) s (cid:18) sin( j − − k ) s sin s (cid:19) (cid:32) p − (cid:89) v =0 ( k − v ) (cid:33) t k − p . Thus, using (4.23), (replacing ∞ with ( n + 1)), we get (cid:13)(cid:13)(cid:13)(cid:13) ∂ p + (cid:96) b n ( t, s ) ∂ p t∂ (cid:96) s (cid:13)(cid:13)(cid:13)(cid:13) ≤ const n − (cid:88) k = p n − (cid:88) j = k +1 | a j | · | j − − k | (cid:96) +1 k p ≤ const n − (cid:88) j =2 | a j | j − (cid:88) k = p ( j − − k ) (cid:96) +1 k p ≤ const n − (cid:88) j =2 | a j | ( j + 1) p + (cid:96) +2 ≤ const n p + (cid:96) +2 − α n − (cid:88) j =2 | a j | ( j + 1) α ≤ const n p + (cid:96) +2 − α (cid:107) a (cid:107) α . The case of the function b ( − ) ( t ) is treated similarly. Lemma 4.5.
Let s = s + i δ ∈ Π n ( a ) . If a ( t ) ∈ W α , p + (cid:96) ≤ α − , m = [ α − − p − (cid:96) ] then ∂ p + (cid:96) ∂ p t∂ (cid:96) s b n ( t, s ) = ∂ p + (cid:96) ∂ p t∂ (cid:96) s b ( t, s ) + m (cid:88) k =1 ∂ p + (cid:96) + k ∂ p t∂ (cid:96) + k s b ( t, s )(i δ ) k + m +1 (cid:88) k =0 β p,ln,k ( t, s )(i δ ) k , where β p,ln,k ( · , s ) ∈ W and besides (cid:107) β p,ln,k ( · , s ) (cid:107) = o (cid:0) n k − α − − p − (cid:96) (cid:1) , k = 0 , , . . . , m. (cid:107) β p,ln,m +1 ( · , s ) (cid:107) = O (cid:16) n ( m +1) − α − − p − (cid:96) (cid:17) The proof of this Lemma is carried out on the basis of the previous Lemma, similarly to the Lemma 4.2.An important role in the theory of Toeplitz operators is played by the concept of the topological index of afunction. 14 efinition 1.
Let the function a ( t ) be continuous on the unit circle T , and a ( t ) (cid:54) = 0 , t ∈ T , then the topologicalindex of the function a ( t ) with respect to the point z = 0 is an integer of the form wind a ( t ) := 12 π arg a ( t ) (cid:12)(cid:12)(cid:12)(cid:12) T where arg a ( t ) | T is the increment of the continuous branch of the argument of the function a ( t ) , when the point t makes a full turn on the curve T in the positive direction. Consider the problem about topological index of the functions b ( t, s ) and b n ( t, s ). Lemma 4.6.
Let a ( t ) ∈ W α , α ≥ . Then for any s ∈ (0 , π ) we have wind b ( t, s ) = 0 and for any s ∈ Π n ( a ) , we have wind b n ( t, s ) = 0 . Proof.
For any s ∈ (0 , π ) the function b ( t, s ) (cid:54) = 0. In addition, due to the symmetry b ( e i ϕ , s ) = b ( e i(2 π − ϕ ) , s ) wecan see that the image of b ( e i ϕ , s ), ϕ ∈ [0 , π ] represents a curve without interior, described twice: once and back,when ϕ describe the segments [0 , π ] and [ π, π ], respectively. Thus, the first relation in the formulation of Lemma4.6 is proved. The second is proved similarly.Let us now consider the sequence of functions η n ( s ) of the type (2.13)- (2.14), (2.17) and compare it with thelimit function (2.15)-(2.16) η ( s ) = 12 π i (cid:90) T log b ( τ, s ) τ − e i s dτ − π i (cid:90) T log b ( τ, − s ) τ − e − i s dτ, s ∈ (0 , π ) . (4.24)It’s obvious that b ( τ, s ) (cid:54) = 0 for τ ∈ T and s ∈ [0 , π ]. Since T × [0 , π ] is a compact set we haveinf s ∈ [0 ,π ] inf t ∈ T | b ( t, s ) | = ∆ > . (4.25)Thus, according to Lemma 4.5, there is such a large enough natural number N , thatinf n ≥ N inf s ∈ Π n ( a ) inf t ∈ T | b n ( t, s ) | ≥ ∆2 . (4.26)To analyze the functions η n ( s ), η ( s ), we need generalized H¨older classes. We say that f ( t ) ∈ H µ ( K ), 0 < µ ≤ K is a compact domain of the complex plane C , if the following condition is satisfied: (cid:107) f (cid:107) H µ := sup t ∈ K | f ( t ) | + sup t , t ∈ K | f ( t ) − f ( t ) || t − t | µ < ∞ . We define the class C m + µ ( K ). Let us say that f ( z ) ∈ C m + µ ( K ), m = 0 , , , . . . , 0 < µ ≤ f ( m ) ( z ) ∈ H µ ( K ).Moreover, the norm of the function f ( t ) in this space is introduced by the formula: (cid:107) f (cid:107) C m + µ ( K ) = m − (cid:88) k =0 sup t ∈ K | f ( k ) ( t ) | + (cid:107) f ( m ) ( t ) (cid:107) H µ . Note that C µ ( K ) = H µ ( K ). We also agree that C := W . In the following, we use the following known result.(see [20], Lemma 3.6). Lemma 4.7.
Let f ( t ) ∈ W α , α > , then f ( t ) ∈ C m + µ ( T ) , where m = [ α ] , µ = α − [ α ] . Introduce the following notation:Λ( t, s ) := 12 π i (cid:90) T log b ( τ, s ) τ − t dτ, s ∈ (0 , π )and Λ n ( t, s ) := 12 π i (cid:90) T log b n ( τ, s ) τ − t dτ, s ∈ Π n ( a ) . Then we get that η ( s ) = Λ( e i s , s ) − Λ( e − i s , s ) , s ∈ (0 , π ) , and η n ( s ) = Λ n ( e i s , s ) − Λ n ( e − i s , s ) , s ∈ Π n ( a ) . emma 4.8. Let the function a ( t ) ∈ CSL α , α ≥ , theni) η ( s ) ∈ C m + µ ([0 , π ]) , where m = [ α − , µ = α − − m ;ii) the point s = s + i δ ∈ Π n ( a ) allows following representation η n ( s ) = η ( s ) + m (cid:88) k =1 η ( k ) ( s )(i δ ) k + m +1 (cid:88) k =1 γ n,k ( s )(i δ ) k where | γ n,k ( s ) | = o ( n k − ( α − ) , k = 0 , , . . . , m, | γ n,m +1 ( s ) | = O ( n m +1 − ( α − ) Proof.
According to Lemma 4.6, for any s ∈ [0 , π ] we can choose a continuous branch of the log b ( τ, s ), moreoverthis function will be continuous in s ∈ [0 , π ]. Further, according to the well-known theorem of the theory of Banachalgebras log b ( · , s ) ∈ W α − since b ( · , s ) ∈ W α − and the relation (4.25) is satisfied, and the norm (cid:107) b ( · , s ) (cid:107) α − is continuous in s . Note that the function Λ( t, s ) also has these properties, since the Cauchy singular integraloperator involved in the definition of the function Λ( t, s ) is bounded in the space W α − .This implies that the function Λ( t, s ) has partial derivatives with respect to t (with a fixed s ), up to the orderof m and ∂ m ∂ m t Λ( · , s ) ∈ H µ ( T ). On the other hand, Λ( t, s ) has continuous partial derivatives with respect to s .Indeed: ∂ p + (cid:96) ∂ p t∂ (cid:96) s Λ( t, s ) = ( p − π i (cid:90) T ∂ (cid:96) ∂ (cid:96) s log b ( τ, s )( τ − t ) p dτ. Applying integration by parts p times to the last integral, we obtain ∂ p + (cid:96) ∂ p t∂ (cid:96) s Λ( t, s ) = 12 π i (cid:90) T ∂ p + (cid:96) ∂ p τ∂ (cid:96) s log b ( τ, s ) τ − t dτ. Thus, we obtain that if p + (cid:96) ≤ m ( m = [ α − ∂ p + (cid:96) ∂ p t∂ (cid:96) s Λ( · , s ) ∈ C α − − p − (cid:96) ( T ) and ∂ p + (cid:96) ∂ p t∂ (cid:96) s Λ( t, · ) ∈ C α − − p − (cid:96) ([0 , π ]) . From the last two relations it follows that for k = 0 , , . . . , m we obtain: ∂ k η ( s ) ∂ k s ∈ C α − − k ([0 , π ]) . Indeed, taking for example, k = 2, we get ∂ η ( s ) ∂ s = (cid:18) ∂ ∂ t Λ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e i s − ∂ ∂ t Λ( t − , s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e − i s (cid:19) + 2 (cid:18) ∂ ∂t∂s Λ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e i s − ∂ ∂t∂s Λ( t − , s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e − i s (cid:19) + (cid:18) ∂ ∂ s Λ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e i s − ∂ ∂ s Λ( t − , s ) (cid:12)(cid:12)(cid:12)(cid:12) t = e − i s (cid:19) . Similarly ∂ k η n ( s ) ∂ k s ∈ C α − − k ([0 , π ]) , moreover (cid:13)(cid:13)(cid:13)(cid:13) ∂ k η ( s ) ∂ k s − ∂ k η n ( s ) ∂ k s (cid:13)(cid:13)(cid:13)(cid:13) H α − − k ([0 ,π ]) ≤ const n − ( α − − k ) , (4.27)where the “const” is independent of n .Let now s = s + i δ , then η n ( s ) = η n ( s ) + m (cid:88) k =1 η ( k ) n ( s ) k ! (i δ ) k + o m +1 ( s , δ ) , (4.28)16here m = [ α −
2] and | o m +1 ( s , δ ) | ≤ const (cid:12)(cid:12)(cid:12) η ( m +1) n ( s ) (cid:12)(cid:12)(cid:12) · | i δ | m +1 . (4.29)According to (4.27) we have η ( k ) n ( s ) = η ( k ) ( s ) + γ n,k ( s ) , where | γ n,k ( s ) | = o ( n k − α +2 ). To estimate o m +1 ( δ ), we use an inequality of the form: (cid:13)(cid:13)(cid:13)(cid:13) d k + (cid:96) (ln b ( · , s )) d k td (cid:96) s (cid:13)(cid:13)(cid:13)(cid:13) ≤ const n ( m +1) − α +2 − p − (cid:96) , where m + 1 = k + (cid:96) > α − n . Then it is easy to understand that | η ( m +1) n ( s ) | ≤ const n ( m +1) − ( α − . Thus, the Lemma 4.8 is proved.
5. Equation for the eigenvalues
In this section we derive an equation for finding the eigenvalues of the considered Toeplitz matrices and subjectthis equation to asymptotic analysis when the parameter n → ∞ . So, we consider the standard equation for findingeigenvalues and eigenvectors: T n ( a n − λ ) X n = 0 , λ ∈ R n ( a ) (5.1)in the space L ( n )2 . Let us present the expression a n ( t ) − λ as a product p ( t, λ )ˆ b n ( · , λ ), where ˆ b n ( · , λ ) is the continuousnon-degenerate zero index function and p ( t, λ ) is a Laurent polynomial with three terms, which inherits the zerosof the original function a n ( t ) − λ . Further, through some transformations we reduce (5.1) to an equation with aninvertible operator T n +2 (ˆ b n ( · , λ )) on the left-hand side. By applying the operator T − n +2 (ˆ b n ( · , λ )) to this equationand considering the zeros of p ( t, λ ) we get a homogeneous system of linear equations, the main determinant of whichgives us the above-mentioned equation for finding eigenvalues.We first prove the following result. Lemma 5.1.
Let a ∈ CSL α , α ≥ , then there is such a natural N , independent of λ ∈ R n ( a ) so that for all λ ∈ R n ( a ) and for all n ≥ N the operator T n +2 (ˆ b n ( · , λ )) is invertible and besides (cid:13)(cid:13)(cid:13) T − n +2 (ˆ b n ( · , λ )) (cid:13)(cid:13)(cid:13) L ≤ M, (5.2) where M do not depend on n and λ ∈ R n ( a ) .Proof. According to Lemma 4.6, wind ˆ b ( · , λ ) = 0 for all λ ∈ R n ( a ). Thus the finite section method (see, forexample [2]) ensures the existence of a natural number N ( λ ) such that for n ≥ N ( λ ) the operator T n (ˆ b ( · , λ ))is invertible. Since the set R ( a ) is compact, then, for all λ from R ( a ), we can choose a single number N =sup λ ∈R ( a ) N ( λ ) < ∞ such that for n > N (cid:13)(cid:13) T − n +2 ( b ( · , λ )) (cid:13)(cid:13) L ≤ M , (5.3)where M does not depend on n and λ .According to the Lemma 4.5 for all small enough ε > N such that for n > N (cid:107) ˆ b ( · , λ ) − ˆ b n ( · , λ ) (cid:107) < ε, (5.4)where λ = g ( s ), λ = g ( s ), s = s + i δ ( ∈ Π n ( a )), and the numbers ε and N does not depend on s and δ .Obviously (5.3) and (5.4) imply (5.2), where N = max( N , N ).We state the main result of this section. Let λ = g n ( s ), s ∈ Π n ( a ). Then the following theorem holds.17 heorem 4. Let a ∈ CSL α , α ≥ , and n > N where N is a rather large natural number. Then λ ∈ R n ( a ) is aneigenvalue of T n ( a ) if and only if e − i( n +1) s Θ n +2 (cid:0) e i s , λ (cid:1) ˆΘ n +2 (cid:0) e − i s , λ (cid:1) − e i( n +1) s Θ n +2 (cid:0) e − i s , λ (cid:1) ˆΘ n +2 (cid:0) e i s , λ (cid:1) = 0 , (5.5) where the functions Θ n +2 and ˆΘ n +2 are defined by the formulas: Θ n +2 ( t, λ ) = T − n +2 (cid:16) ˆ b n ( · , λ ) χ (cid:17) ( t ); ˆΘ n +2 ( t, λ ) = T − n +2 (cid:16) ˜ b n ( · , λ ) χ (cid:17) ( t − ) , and ˜ b n ( t, λ ) = ˆ b n (1 /t, λ ) .Proof. Consider the equation T n ( a − λ ) X n = 0 , X n ∈ L ( n )2 . Rewrite this equation in the following form T n ( a n − λ ) X n = 0 , λ ∈ R n ( a ) . (5.6)By the (2.12) the above equation can be rewritten as follows: P n ˆ b n ( · , λ ) p ( · , λ ) X n = 0 , (5.7)where p ( t, λ ) = (cid:0) t − e i s (cid:1) (cid:0) t − − e i s (cid:1) . (Recall that g n ( s ) = λ .) Multiply equality (5.6) by the base vector χ = t . We obtain( P n +1 − P )ˆ b ( · , λ ) p ( · , λ ) χ X n = 0 . (5.8)Note that P n +1 − P is a finite dimensional orthogonal space projector from L ( T ) to the linear shell of vectors χ , χ , . . . , χ n , where χ j = t j . It is easy to see that p ( · , λ ) χ X n ∈ L ( n +2)2 . We set by definition Y n +2 := P n +2 ( a n − λ ) χ X n = P n +2 ˆ b n ( · , λ )( χ p ( · , λ ) X n ) = P n +2 (ˆ b n ( · , λ ))( χ p ( · , λ ) X n ) . Then equation (5.8) can be rewritten in the following form( P n +1 − P ) Y n +2 = 0 . The last equality means that the values of Y have the following representation Y n +2 = y χ + y n +1 χ n +1 . According to the Lemma 5.1 the operator T n +2 (ˆ b n ( · , λ )) is invertible, so we get that T − n +2 (ˆ b n ( · , λ )) Y n +2 = χ p ( · , λ ) X n , that is y [ T − n +2 (ˆ b n ( · , λ )) χ ]( t ) + y n +1 [ T − n +2 (ˆ b n ( · , λ )) χ n +1 ]( t ) = tp ( t, λ ) X n ( t ) . (5.9)We introduce the reflection operator acting on the space L ( n )2 :( W n f n )( t ) = n − (cid:88) j =0 f n − − j t j , where f n ( t ) = n − (cid:88) j =0 f j t j . From the identity W n +2 T n +2 (ˆ b n ) W n +2 = T n +2 (˜ b n ), which is easy to verify by simple calculation, we get[ T − n +2 (ˆ b n ( · , λ )) χ n +1 ]( t ) = t n +1 T − n +2 (˜ b n ( · , λ ))( t − ) . y Θ n +2 ( t, λ ) + y n +1 t n +1 ˆΘ n +2 ( t, λ ) = tp ( t, λ ) X n ( t ) . (5.10)Considering that the multiplier p ( t, λ ) disappears when t = e i s and when t = e − i s , we conclude that y and y n +1 must satisfy the following homogeneous system of linear algebraic equations:Θ n +2 ( e i s , λ ) y + e i( n +1) s ˆΘ n +2 ( e i s , λ ) y n +1 = 0 , Θ n +2 ( e − i s , λ ) y + e − i( n +1) s ˆΘ n +2 ( e − i s , λ ) y n +1 = 0 . (5.11)Note that if y = y n +1 = 0 then by (5.10) we get X n ≡
0. Therefore, the original equation (5.6) has a non-trivialsolution X n if, and only if, the determinant of the system of equations (5.11) is zero. This is the form of the requiredequality (5.5).To investigate the asymptotic behavior of the functions Θ n and ˆΘ n when n → ∞ , we introduce the Toeplitzoperator (infinity dimensional) which corresponds to the matrix ( b i − j ) ∞ i,j =0 . Let P : L ( T ) → H ( T ) be theprojector defined by ( P f ) ( t ) = ∞ (cid:88) j =0 f j t j , where ∞ (cid:88) j = −∞ f j t j ∈ L ( T ) , and H (( T )) is the Hardy’s famous space. Then[ T ( b ) f ]( t ) = [ P bf ]( t ) (5.12)called the Toeplitz operator with symbol b (see [3]).Subsequent reasoning is essentially based on the general theory of projection methods (see [2], [32], [33]). Recallthe definition of Wiener-Hopf factorization. Let the function f belong to the Wiener class, i.e. f ∈ W α and f ( t ) (cid:54) = 0, t ∈ T . Then there is the representation of the function f as the following product: f = f + t κ f − , where κ = wind( f ), f ± ∈ W α ± , and wind ( f ± ) = 0. Here we assume W α ± = { f ∈ W α : f ( t ) = ∞ (cid:88) j =0 f ± j t ± j } . By virtue of Lemma 4.6 and equation (4.25), a function b ( t, λ ) is factorisable in the space W α − (see Lemma4.4 i)), while the factorization factors ˆ b ± ( t, λ ) can be written in the form:ˆ b ± ( t, λ ) = exp (cid:32)
12 log(ˆ b ( t, λ )) ± π i (cid:90) T log(ˆ b ( τ, λ )) τ − t dτ (cid:33) , (5.13)where the integral is understood in the sense of the principal value. The functions ˆ b ± ( t, λ ) can be analyticallycontinued inside and outside respectively of the unit circle T by the formula:ˆ b ± ( t, λ ) = exp (cid:32) ± π i (cid:90) T log(ˆ b ( τ, λ )) τ − t dτ (cid:33) , | t ± | < . Note that due to the symmetry b ( t − , λ ) = b ( t, λ ), the factorization factors satisfy the following relationˆ b − ( t, λ ) = ˆ b + ( t − , λ ) /χ b ( λ ) , (5.14)where the number χ b can be calculated by the formula: χ b ( λ ) = exp (cid:26) π (cid:90) π log ˆ b ( e i ϕ , λ ) dϕ (cid:27) . (5.15)19imilarly, according to Lemmas 4.6 and (4.26), the function ˆ b n ( t, λ ) also has the Wiener-Hopf factorization:ˆ b n ( t, λ ) = ˆ b n, + ( t, λ )ˆ b n, − ( t, λ ) . (5.16)Note that the functions b n, ± ( t, λ ) represent polynomials of degree ( n −
1) of the variables t and t − respectively. Inaddition, due to (4.11) we have (cid:107) ˆ b n, ± ( · , λ ) (cid:107) α − ≤ const (cid:107) a (cid:107) α . (5.17)Next we need to consider the functions ˆ b n ( t, λ ) and b n, ± ( t, λ ) in the ring area K n = (cid:26) z ∈ C | − Cn ≤ | z | ≤ Cn (cid:27) , C > . (see the definition of Π n ( a ) in (2.8)). Consider the numbers r ± n = 1 ± Cn , C > Lemma 5.2.
Let the function f n ( t ) ∈ L ( n )2 then (cid:107) f n ( r ± n t ) (cid:107) α ≤ const (cid:107) f n ( t ) (cid:107) α . Proof.
Let f n ( t ) = n − (cid:88) j = − ( n − f j t j . Then (cid:107) f n ( r ± n t ) (cid:107) α = n +1 (cid:88) j = − ( n − | f j | · | r ± n | j (1 + | j | ) α . Since | j | ≤ n − | r ± n | j = (cid:18) Cn (cid:19) j ≤ exp (cid:18) | j | Cn (cid:19) ≤ exp( C ) . Thus (cid:107) f n ( r ± n t ) (cid:107) α ≤ exp( C ) n +1 (cid:88) j = − ( n − | f j | · (1 + | j | ) α = const (cid:107) f n ( t ) (cid:107) α . The following statement follows from the above Lemma.
Lemma 5.3.
Let a ∈ CSL α , α ≥ , then the functions ˆ b ( r ± n t, λ ) ∈ W α − , with sup λ ∈R n ( a ) (cid:107) ˆ b n ( r ± n t, λ ) (cid:107) α − ≤ const (cid:107) a (cid:107) α . (5.18) Besides, ˆ b n, ± ( r ± n t, λ ) ∈ W α − and sup λ ∈R n ( a ) (cid:107) ˆ b n, ± ( r ± n t, λ ) (cid:107) α − ≤ const (cid:107) a (cid:107) α . (5.19) Proof.
According to Lemma 5.2 we get (5.18). Further, by Lemma 4.5 we obtain that for sufficiently large n ,ˆ b n ( r ± n t, λ ) (cid:54) = 0 and wind ˆ b n ( r ± n t, λ ) = 0 for all t ∈ T . Thus, the functions ˆ b n ( r ± n t, λ ) admit a Wiener-Hopf factoriza-tion in the space W α − and the following inequalities hold (cid:107) ˆ b n, ± ( r ± n t, λ ) (cid:107) α − ≤ const (cid:107) ˆ b n ( r ± n t, λ ) (cid:107) α − . Applying Lemma 5.2 and inequality (5.18) we get (5.19).Now we are ready to get an asymptotic representation of the functions Θ n +2 ( t, λ ), from the Theorem 4. To dothis, we note that the inverse of the Toeplitz operator (5.12), calculated by the formula: T − ( b ) = b − ( t ) P b − − ( t ) , where b ( t ) = b + ( t ) b − ( t ), is a Wiener-Hopf factorization in the space W α (see (5.13)). Thus T − (ˆ b n ) χ = ˆ b − n, + ( t ) P (cid:16) ˆ b − n, − ( · ) · (cid:17) ( t ) = ˆ b − n, + ( t ) . (5.20)20 emma 5.4. Let a ∈ CSL α , α ≥ . Then the following asymptotic representation holds Θ n +2 ( t, λ ) = ˆ b − n, + ( t, λ ) + ˜ R ( n )1 ( t, λ ) , ˆΘ n +2 ( t, λ ) = ˆ b − n, − ( t − , λ ) /χ b ( λ ) + ˜ R ( n )2 ( t, λ ) , where, for n → ∞ sup (cid:110)(cid:12)(cid:12)(cid:12) ˜ R ( n ) j ( z, λ ) (cid:12)(cid:12)(cid:12) : ( z, λ ) ∈ K n × R n ( a ) (cid:111) = o ( / n α − ) , j = 1 , , and χ b ( λ ) is given by (5.15).Proof. From the definition of Θ n ( t, λ ) (see (5.20)) it follows thatΘ n ( t, λ ) = ˆ b − n, + ( t, λ ) + R ( n )1 ( t, λ ) , where ˜ R ( n )1 ( t, λ ) = T − n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) χ − T − (cid:16) ˆ b n ( · , λ ) χ (cid:17) = T − n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) (cid:104) T (cid:16) ˆ b n ( · , λ ) (cid:17) − T n +2 (cid:16) ˆ b n ( · , λ ) (cid:17)(cid:105) T − (cid:16) ˆ b n ( · , λ ) (cid:17) χ − Q n +2 T − (cid:16) ˆ b n ( · , λ ) (cid:17) χ . By using the obvious equalities
P bP = P n bP n + P n bQ n + Q n bP n + Q n bQ n and (5.20) we get ˜ R ( n )1 ( t, λ ) = (cid:104) T − n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) P n +2 ˆ b n ( · , λ ) Q n +2 ˆ b − n ( · , λ ) (cid:105) ( t ) − (cid:104) Q n +2 (cid:16) ˆ b − n, + ( · , λ ) (cid:17)(cid:105) ( t ) , t ∈ T . (5.21)Thus (cid:107) ˜ R ( n )1 ( · , λ ) (cid:107) α − ≤ (cid:16) (cid:107) T − n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) (cid:107) α − · (cid:107) P n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) (cid:107) α − + 1 (cid:17) × (cid:107) Q n +2 (cid:16) ˆ b − n, + ( · , λ ) (cid:17) (cid:107) α − . From (5.2) (cid:107) T − n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) (cid:107) α − ≤ M. Further (cid:107) P n +2 (cid:16) ˆ b n ( · , λ ) (cid:17) (cid:107) α − ≤ const (cid:107) ˆ b n ( · , λ ) (cid:107) α − ≤ const (cid:107) ˆ b ( · , λ ) (cid:107) α − . Finally, Lemma 4.1 i) gives (cid:107) Q n +2 (cid:16) ˆ b − n, + ( · , λ ) (cid:17) (cid:107) α − = o ( / n α − ) . (5.22)Thus (cid:107) ˜ R ( n )1 (cid:107) α − = o ( / n α − ) . (5.23)In the above calculations t ∈ T . Consider case z ∈ K n , i.e. z = rt, r ∈ [ r − n , r + n ] . (5.24)Denote the first and second term in (5.21), respectively, R ( n )1 , ( t, λ ) and R ( n )1 , ( t, λ ). Note that R ( n )1 , ( t, λ ) ∈ L ( n +2)2 and with (5.22)–(5.23), we get (cid:107) ˜ R ( n )1 , ( t, λ ) (cid:107) α − = o ( / n α − ) , t ∈ T . Thus, Lemma 5.2 implies that sup r ∈ [ r − n ,r + n ] (cid:107) ˜ R ( n )1 , ( rt, λ ) (cid:107) α − = o ( / n α − ) . Consider now ˜ R ( n )1 , ( t, λ ) := (cid:104) Q n +2 (cid:16) ˆ b − n, + ( · , λ ) (cid:17)(cid:105) ( t ). The function b n, + ( t, λ ) ∈ L ( n +2)2 and by Lemma 5.3 we havesup r ∈ [ r − n ,r + n ] (cid:107) b n, + ( rt, λ ) (cid:107) α − ≤ const (cid:107) a (cid:107) α .
21n addition, Lemma 4.5 implies that sup z ∈ K n | b n, + ( z, λ ) | ≥ δ, where δ does not depend on n and λ . According to a standard theorem of the theory of Banach algebras, for all r ∈ [1 − C/n,
C/n ], b − n, + ( rt, λ ) ∈ W α − , and in addition (cid:107) b − n, + ( rt, λ ) (cid:107) α − ≤ const δ − (cid:107) b n, + ( rt, λ ) (cid:107) α − ≤ const δ − (cid:107) a ( t, λ ) (cid:107) α , ( | t | = 1) . Thus, according to Lemma 4.1 i) it is possible to show thatsup r ∈ [ r − n ,r + n ] (cid:107) ˜ R ( n )1 , ( rt, λ ) (cid:107) α − = o ( / n α − )and therefore sup r ∈ [ r − n ,r + n ] (cid:107) ˜ R ( n )1 ( rt, λ ) (cid:107) α − = o ( / n α − ) . Since (cid:107) f (cid:107) ≤ (cid:107) f (cid:107) α − , this Lemma is proved for the case j = 1. The case j = 2 treated similarly.Denote ˆ b n ( t, g n ( s )) := b n ( t, s ) , ˆ b n, ± ( t, g n ( s )) := b b, ± ( t, s ) , ˜ R j ( t, g n ( s )) := R j ( t, s ) , j = 1 , . Note that as in (5.14) we have b n, − ( t − , s ) = b n, + ( t, s ) /χ b . (5.25)We introduce a continuous function η n ( s ) satisfying the relation b n, + ( e i s , s ) b n, + ( e − i s , s ) = e − i η n ( s ) , s ∈ Π n ( a ) , (5.26)We take the continuous branch of the function η n ( s ) assuming η n (0) = 0. It is not difficult to see that η n ( π ) = η n (0) = 0 . (5.27) Lemma 5.5.
Let a ∈ CSL α , α ≥ . Then there is such a large enough natural N that for all n ≥ N the number λ is an eigenvalue of T n ( a ) if and only if j ∈ Z and s ∈ Π n ( a ) satisfy the equation ( n + 1) s + η n ( s ) + R ( n )6 ( s ) = πj, (5.28) λ = g n ( s ) where R ( n )6 ( s ) satisfies the following asymptotic relation with respect to n → ∞ : R ( n )6 ( s ) = o ( / n α − ) (5.29) – uniformly in parameter s ∈ Π n ( a ) .Proof. Considering the results of Lemma 5.4, we rewrite equality (5.5) in the form e − i( n +1) s (cid:16) b n, + ( e i s , s ) + R ( n )1 ( e i s , s ) (cid:17) (cid:16) b n, − ( e − i s , s ) + R ( n )2 ( e − i s , s ) (cid:17) = e i( n +1) s (cid:16) b n, + ( e − i s , s ) + R ( n )1 ( e − i s , s ) (cid:17) (cid:16) b n, − ( e i s , s ) + R ( n )2 ( e i s , s ) (cid:17) ,e n +1) s = b n, + ( e i s , s ) b n, − ( e − i s , s ) (cid:16) R ( n )3 ( s ) (cid:17) b n, + ( e − i s , s ) b n, − ( e i s , s ) (cid:16) R ( n )4 ( s ) (cid:17) R ( n )3 ( s ) = b − n, + ( e i s , s ) R ( n )1 ( e i s , s ) + b − n, − ( e − i s , s ) R ( n )2 ( e − i s , s )+ b − n, + ( e i s , s ) b − n, − ( e − i s , s ) R ( n )1 ( e i s , s ) R ( n )2 ( e − i s , s ) ,R ( n )4 ( s ) = R ( n )3 ( − s ) . Considering (5.25), we get e n +1) s = e − η n ( s ) R ( n )3 ( s )1 + R ( n )4 ( s ) . Let R ( n )5 ( s ) := log (cid:32) R ( n )3 ( s )1 + R ( n )4 ( s ) (cid:33) then e n +1) s = e − η n ( s )+ R ( n )5 ( s ) . The last equation is equivalent to the following set of equations:2i( n + 1) s = − η n ( s ) + R ( n )5 ( s ) + 2i πj, j ∈ Z . Assuming now R ( n )6 ( s ) := − R ( n )5 ( s )2iwe get ( n + 1) s + η n ( s ) + R ( n )6 ( s ) = πj, j ∈ Z . Considering now the relations connecting R ( n )6 ( s ) with R ( n )1 ( e ± i s , s ) and R ( n )2 ( e ± i s ), we get now the asymptoticexpansion (5.29).
6. Solvability Analysis of Equation (5.28)
Rewrite the equation (5.28) in the form F n ( s ) + R ( n )6 ( s ) n + 1 = d j,n , (6.1)where F n ( s ) := s + η n ( s ) n + 1 , (6.2)and d j,n := πjn + 1 . Along with (6.1), consider the approximating equation F n ( s ) = d j,n , j = 1 , , . . . , n. (6.3)We introduce the notion of the modulus of continuity in the complex domain. Let a function f ( z ) be continuousin some bounded domain G of the complex plane. Then the modulus of continuity f ( z ) is the function: w f ( δ ) := sup z , ∈ G, | z − z |≤ δ | f ( z ) − f ( z ) | , < δ ≤ δ . Let us introduce the domains: Π j,n ( a ) := (cid:26) s ∈ Π n ( a ) : | s − e j,n | ≤ c n n + 1 (cid:27) , (6.4)23here e j,n := d j,n − η n ( d j,n ) n + 1 and c n := 2 (cid:13)(cid:13)(cid:13) R ( n )6 ( s ) (cid:13)(cid:13)(cid:13) ∞ + w η n (cid:18) (cid:107) η n (cid:107) ∞ n + 1 (cid:19) , (6.5)and the norm (cid:107) · (cid:107) ∞ is defined in the standard way on the set G , where G = Π n ( a ). Recall, thatΠ n ( a ) = (cid:8) s = s + i δ | s ∈ [ cn − , π − cn − ] , δ ∈ [ − Cn − , Cn − ] (cid:9) , where c , C are some fixed positive numbers such that c is small enough and C is large enough.The following statement will apply the principle of contractive mappings to the analysis of the solvability of(6.1), (6.3).Let’s introduce the mappings Φ j,n ( s ) := d j,n − η n ( s ) n + 1 . Lemma 6.1.
Let the function a ∈ CSL α , α ≥ . Then, if s ∈ Π j,n ( a ) ,i) Φ j,n ( s ) ∈ Π j,n ( a ) .ii) (cid:32) Φ j,n ( s ) + R ( n )6 ( s ) n + 1 (cid:33) ∈ Π j,n ( a ) .Proof. We prove ii). Let s ∈ Π j,n ( a ), then for sufficiently large n , we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ j,n ( s ) + R ( n )6 ( s ) n + 1 − e j,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | η n ( s ) − η n ( d j,n ) | n + 1 + | R ( n )6 ( s ) | n + 1 ≤ w η n ( | s − d j,n | ) n + 1 + (cid:107) R ( n )6 (cid:107) ∞ n + 1 ≤ w η n (cid:16) | s − e j,n | + | η n ( d j,n ) | n +1 (cid:17) n + 1 + (cid:107) R ( n )6 (cid:107) ∞ n + 1 ≤ w η n (cid:16) c n n +1 + (cid:107) η n (cid:107) ∞ n +1 (cid:17) n + 1 + (cid:107) R ( n )6 (cid:107) ∞ n + 1 ≤ w η n (cid:16) (cid:107) η n (cid:107) ∞ n +1 (cid:17) n + 1 + (cid:107) R ( n )6 (cid:107) ∞ n + 1 ≤ c n n + 1 . Thus, item ii) of the Lemma is proved.The item i) is proved similarly if we put R ( n )6 ( s ) ≡ Theorem 5.
Let the function a ∈ CSL α . Theni) For α ≥ the equation (6.1) has a unique solution s j,n ∈ Π j,n ( a ) , j = 1 , , . . . , n , and all s j,n are different.ii) Let s ∗ j,n be a solution of the equation (6.3) belonging to Π j,n ( a ) . Then for α ≥ (cid:107) s j,n − s ∗ j,n (cid:107) = O ( / n α − ) , where the estimate is uniform in n and j .Proof. Let us prove statement i). For this purpose, consider the sequence s (0) j,n = e j,n , s ( k +1) j,n = Φ j,n ( s ( k ) j,n ) + R ( n )6 ( s j,n ) n + 1 , k = 1 , , . . . . According to Lemma 6.1 ii), the sequence { s ( k ) j,n } ∞ k =1 is contained in the domain Π j,n ( a ). Choose from it someconvergent subsequence and denote its limit by ˜ s j,n . Obviously, ˜ s j,n satisfies (6.1). Note that for any j (cid:54) = j Π j ,n ( a ) ∩ Π j ,n ( a ) = ∅ , | e j ,n − e j ,n | ≥ ∆ n +1 , (∆ > j,n = o (1 /n ). Thus, according to the Lemma 5.5, the numbers g (˜ s j,n ), j = 1 , , . . . , n are eigenvalues of the matrix T n ( a ). Since this matrix has at most n , then ˜ s j,n is a uniquesolution of equation (6.1) in the domain Π j,n ( a ). Denoting ˜ s j,n := s j,n , we completed the proof of i).Let us prove ii). Substituting into the equations (6.1) and (6.3) respectively, s j,n and s ∗ j,n , and subtracting thesecond expression from the first one, we get( s j,n − s ∗ j,n ) + η n ( s j,n ) − η n ( s ∗ j,n ) n + 1 = − R ( n )6 ( s j,n ) n + 1 . (6.6)Since α ≥
3, according to Lemma 4.9, η n ( s ) has a derivative that is bounded uniformly in n . In this way we obtain | η n ( s j,n ) − η n ( s ∗ j,n ) | ≤ η | s j,n − s ∗ j,n | , where η = sup n ∈ N sup s ∈ Π n ( a ) | η (cid:48) n ( s ) | < ∞ . From (6.6) we get | s j,n − s ∗ j,n | ≤ η | s j,n − s ∗ j,n | n + 1 + (cid:12)(cid:12)(cid:12) R ( n )6 ( s j,n ) (cid:12)(cid:12)(cid:12) n + 1 . From (5.29) it follows that | s j,n − s ∗ j,n | (cid:18) − η n + 1 (cid:19) ≤ (cid:12)(cid:12)(cid:12) R ( n )6 ( s j,n ) (cid:12)(cid:12)(cid:12) n + 1and finally follows | s j,n − s ∗ j,n | = o ( / n α − ) . The statement proved above shows that the roots s j,n of the equation (6.1) can be approximated by the roots s ∗ j,n of the equation (6.3) for large values of n . Besides, the values s ∗ j,n can be approximated using the method ofsuccessive approximations by the values s ∗ ( k ) j,n defined in the following way: s ∗ (0) j,n = e j,n , s ∗ ( k +1) j,n = Φ n ( e ∗ ( k ) j,n ) , k = 0 , , . . . . (6.7) Lemma 6.2.
Let the function a ∈ CSL α , α ≥ , then the equation (6.3) has a unique solution s ∗ j,n ∈ Π j,n ( a ) , j = 1 , , . . . , n and for sufficiently large n , the following estimate is valid: | s ∗ j,n − s ∗ ( k ) j,n | ≤ ηη (cid:18) η n + 1 (cid:19) k +2 , (6.8) where η = sup n ∈ N sup s ∈ Π n ( a ) | η n ( s ) | ,η = sup n ∈ N sup s ∈ Π n ( a ) | η (cid:48) n ( s ) | . Proof.
We show that sequence (cid:110) s ∗ ( k ) j,n (cid:111) ∞ k =1 is convergent. Indeed, according to Lemma 4.8, the functions η n ( s ) and η (cid:48) n ( s ) are bounded uniformly respect to n . That is, the valuesup n ∈ N sup s ∈ Π n ( a ) | η (cid:48) n ( s ) | = η < ∞ . Then we have: | s ∗ (1) j,n − s ∗ (0) j,n | = | η n ( e j,n ) − η n ( d j,n ) | n + 1 ≤ η | e j,n − d j,n | n + 1= η | η n ( d j,n ) | ( n + 1) ≤ η η ( n + 1) . | s ∗ (2) j,n − s ∗ (1) j,n | = | Φ j,n ( s ∗ (1) j,n ) − Φ j,n ( s ∗ (0) j,n ) | = | η n ( s ∗ (1) j,n ) − η n ( s ∗ (0) j,n ) | n + 1 ≤ η | s ∗ (1) j,n − s ∗ (0) j,n | n + 1 ≤ η η ( n + 1) . Similarly | s ∗ ( k +1) j,n − s ∗ ( k ) j,n | ≤ ηη k ( n + 1) k +2 . (6.9)Since η / n + 1 < n , then the sequence (cid:110) s ∗ ( k ) j,n (cid:111) ∞ k =1 converges to s ∗ j,n .From the estimate (6.9) it follows that | s ∗ ( k + m ) j,n − s ∗ ( k ) j,n | ≤ ηη (cid:18) η n + 1 (cid:19) k +2 − ( η / n + 1 ) m +1 − η / n + 1 . Assuming in this inequality that n + 1 > η and passing to the limit when m → ∞ , we get that | s ∗ j,n − s ∗ ( k ) j,n | ≤ ηη (cid:18) η n + 1 (cid:19) k +2 − η / n + 1 . The assertion of the Lemma obviously follows from this inequality.Now we are ready to prove the main results of the work.
7. Proof of the main results
Theorem 1 follows from Lemma 5.5 and Theorem 5.
Let 2 ≤ α <
3. We estimate the error term∆ ( n )2 ( j ) := s j,n − e j,n . We express s j,n from equation (6.1) and obtain | s j,n − e j,n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η n ( d j,n ) − η n ( s j,n ) n + 1 + R (6) n ( s ) n + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ w η n ( | s j,n − d j,n | ) n + 1 + | R (6) n ( s ) | n + 1 ≤ w η n (cid:16) | s j,n − e j,n | + η n ( d j,n ) n +1 (cid:17) n + 1 + | R (6) n ( s ) | n + 1 ≤ w η n (cid:16) c n + (cid:107) η (cid:107) ∞ n +1 (cid:17) n + 1 + (cid:107) R (6) n (cid:107) ∞ n + 1 , where the value c n is given in (6.5). Thus, for sufficiently large n we get | s j,n − e j,n | ≤ w η n (cid:16) ηn +1 (cid:17) + (cid:107) R (6) n (cid:107) ∞ n + 1 , (7.1)where η n is given by formula (5.26).Let now 2 < α <
3. Then, according to Lemma 4.8, η n ∈ H α − (Π n ( a )) with norm bounded uniformly in n .Thus | s j,n − e j,n | = O (cid:32)(cid:18) ηn + 1 (cid:19) α − · n + 1 (cid:33) + o ( / n α − ) n + 1 = O ( / n α − ) , (7.2)where the estimate is uniform in n . Let α = 2. Then η ( s ) ∈ C and η n ( s ) has a modulus of continuity with evaluationuniform in n . Thus, (7.1) implies ∆ ( n )2 ( j ) = o ( / n ) . s j,n = d j,n − η n ( d j,n ) n + 1 + ∆ ( n )2 ( j ) . (7.3)Thus, from the formulas (7.2), (7.3) we get the following equality | ∆ ( n )2 ( j ) | = (cid:40) o (1 /n ) , α = 2 ,O (cid:0) n − ( α − (cid:1) , < α < . Take into account that η n ( d j,n ) = o ( / n α − )(7.2) give Theorem 2 for case 0 ≤ α < ≤ α <
4. Then consider the difference˜∆ ( n )2 ( j ) := s j,n − s ∗ (1) j,n . From Theorem 5, ii) | s j,n − s ∗ j,n | = o ( / n α − ) . On the other hand, the estimate (6.8) gives | s ∗ j,n − s ∗ (1) j,n | = O ( / n ) . In this way we have | ˜∆ ( n )2 ( j ) | = o ( / n α − ) , (7.4)and we can write s j,n = s ∗ (1) j,n + ˜∆ ( n )2 ( j ) = d j,n − η n ( s ∗ (0) j,n ) n + 1 + ˜∆ ( n )2 ( j )= d j,n − η n (cid:16) d j,n − η n ( d j,n ) n +1 (cid:17) n + 1 + ˜∆ ( n )2 ( j ) . Since the function η n ( d j,n ) has a derivative, according to Lemma 4.8, with H α − (Π n ( a ))-norm bounded uni-formly on n , then we have that s j n = d j,n − η n ( d j,n ) n + 1 + η (cid:48) n ( d j,n ) η n ( d j,n )( n + 1) + O ( / n ) + ˜∆ (2) n ( j ) . Using the Lemma 4.8 again and the relation (7.4), we obtain the statement of the Theorem 2 for the case3 ≤ α < (cid:96) ≤ α < (cid:96) + 1, (cid:96) ≥ ϕ ∗ ( (cid:96) − j,n as an approximatingexpression. From the proved theorem 2 and the definition of the function g we obtain the assertions of the Theorem 3.Indeed, since λ ( n ) j = g n ( ϕ ( n ) j ), consider the Taylor series decomposition at the point d j,n for the function g n .We prove formula (2.22) for the first two terms of the expansion in the case [ α ] = 2. As an increment of theargument ∆ x consider the expression − η ( d j,n ) n + 1 + ∆ ( n )2 ( j ). According Taylor’s formula we have g n ( x + ∆ x ) = g n ( x ) + g (cid:48) n ( x )∆ x + O (∆ x ) . Hence, taking into account the definitions of the functions g n , we obtain the decomposition (2.22): λ ( n ) j = g n ( d j,n ) − g (cid:48) n ( d j,n ) (cid:18) η ( d j,n ) n + 1 + ∆ ( n )2 ( j ) (cid:19) + O (cid:18) η ( d j,n ) n + 1 + ∆ ( n )2 ( j ) (cid:19) = g n ( d j,n ) − g (cid:48) n ( d j,n ) η ( d j,n ) n + 1 − g (cid:48) n ( d j,n )∆ ( n )2 ( j ) + O (cid:18) η ( d j,n ) n + 1 + ∆ ( n )2 ( j ) (cid:19) . α = 2, g (cid:48) n ( d j,n ) = O ( d j,n ( π − d j,n )) , ∆ ( n )2 ( j ) = o (1 /n ) , η ( d j,n ) = O ( d j,n ( π − d j,n )) . The error term is o (cid:18) d j,n ( π − d j,n ) n (cid:19) + O (cid:18) η ( d j,n ) n + 1 + ∆ ( n )2 ( j ) (cid:19) = o (cid:18) d j,n ( π − d j,n ) n (cid:19) . It now remains to note that for points ϕ lying on the real line, by virtue of Lemma 4.2, we have the equality g ( k ) n ( ϕ ) = g ( k ) ( ϕ ) + o ( n − ( α − k ) ) , k = 0 , , . . . , [ α ] . Thus, we obtain that λ ( n ) j = g ( d j,n ) − g (cid:48) ( d j,n ) η ( d j,n ) n + 1 + o (cid:18) d j,n ( π − d j,n ) n (cid:19) + o ( / n ) + o ( / n ) . We now consider the case of 2 < α <
3. Repeating the above reasoning, we obtain the required estimate of theremainder: O (cid:18) d j,n ( π − d j,n ) n α − (cid:19) . Consider the case of [ α ] = 3: λ ( n ) j = g n ( d j,n ) + g (cid:48) n ( d j,n ) (cid:18) − η ( d j,n ) n + 1 + η ( d j,n ) η (cid:48) ( d j,n )( n + 1) + O ( / n α − ) (cid:19) + g (cid:48)(cid:48) n ( d j,n )2 (cid:18) − η ( d j,n ) n + 1 + η ( d j,n ) η (cid:48) ( d j,n )( n + 1) + O ( / n α − ) (cid:19) + O (cid:18) d j,n ( π − d j,n ) n (cid:19) = g n ( d j,n ) − g (cid:48) n ( d j,n ) η ( d j,n ) n + 1 + g (cid:48)(cid:48) n ( d j,n ) η ( d j,n ) + g (cid:48) n ( d j,n ) η ( d j,n ) η (cid:48) ( d j,n ) n + 1 + O (cid:18) d j,n ( π − d j,n ) n α − (cid:19) Also, as above, using Lemma 4.2, we obtain that λ ( n ) j = g ( d j,n ) + g (cid:48) ( d j,n ) (cid:18) − η ( d j,n ) n + 1 + η ( d j,n ) η (cid:48) ( d j,n )( n + 1) + O ( / n α − ) (cid:19) + g (cid:48)(cid:48) ( d j,n )2 (cid:18) − η ( d j,n ) n + 1 + η ( d j,n ) η (cid:48) ( d j,n )( n + 1) + O ( / n α − ) (cid:19) + O (cid:18) d j,n ( π − d j,n ) n (cid:19) + o ( / n α )= g ( d j,n ) − g (cid:48) ( d j,n ) η ( d j,n ) n + 1 + g (cid:48)(cid:48) n ( d j,n ) η ( d j,n ) + g (cid:48) n ( d j,n ) η ( d j,n ) η (cid:48) ( d j,n ) n + 1 + O (cid:18) d j,n ( π − d j,n ) n α − (cid:19) Thus, we obtain the formula (2.22) and the estimate∆ ( n )3 ( j ) = O (cid:18) d j,n ( π − d j,n ) n α − (cid:19) . 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Research of the authors S.M. Grudsky and E. Ram´ırez de Arellano was supported by CONACYT grant 238630.Research of the author I.S. Malisheva was supported by the Ministry of Education and Science of the RussianFederation, Southern Federal University (Project №№