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1. Preliminaries and introduction

Consider a linear functional L defined in the linear space of Laurent polynomials Λ = span{zⁿ}n∈Z such that L is Hermitian, i.e.,

Then, a bilinear functional can be defined in the linear space P = span{zⁿ}n≥0 of polynomials with complex coefficients by

The sequence of complex numbers {cn}n∈Z is called the sequence of moments associated with L. On the other hand, the Gram matrix associated with the canonical basis {zⁿ}n≥0 of P is

which is known in the literature as Toeplitz matrix [7]. A sequence of monic polynomials {φn}_n≥0, with deg (φn) = n, is said to be orthogonal with respect to L if the condition

where k_n ≠ 0, holds for every n, m ­ ≽ 0. Notice that the sequence {φn}n≥0 can be obtained by using the Gram-Schmidt orthogonalization process with respect to the basis {zⁿ}n­ ≽ 0. The necessary and sufficient conditions for the existence of such a sequence can be expressed in terms of the Toeplitz matrix T: {φn}n≥0 satisfies the orthogonality condition if and only if Tn, the (n + 1) × (n + 1) principal leading submatrix of T, is non-singular for every n ≥ 0. In such a case, L is said to be quasi-definite (or regular). On the other hand, if det T_n > 0 for every n ≥ 0, then L is said to be positive definite and it has the integral representation

where σ is a nontrivial positive Borel measure supported on the unit circle T = {z : |z| = 1}. In such a case, there exists a (unique) family of polynomials {ϕn}n≥0, with deg ϕn = n and positive leading coefficient, such that

{ϕn}n≥0 is said to be the sequence of orthonormal polynomials with respect to σ. If we denote by κn the leading coefficient of ϕn(z), then we have φn(z) = ϕn(z)/κn. These polynomials satisfy the following forward and backward recurrence relations (see [7], [10], [11]):

where φ^∗_n (z) = zⁿφn(z⁻¹) is the so-called reversed polynomial and the complex numbers {φn(0)}n≥1 are known as Verblunsky (Schur, reflection) parameters. It is important to notice that in the positive definite case we get |φn(0)| < 1, n ≥ 1, and

Moreover, we have

The n-th kernel polynomial K_n(z,y) associated with {φ}_n≥0 is defined by

and the right hand side is known in the literature as Christoffel-Darboux formula and it holds if yz ≠ 1. It satisfies the so called reproducing property

for every polynomial p of degree at most n. K_n^(i,j) (z,y) will denote the i-th and j-th partial derivative of K_n(z,y) with respect to z and y, respectively. Notice that we have φ∗n(z) = k_nK_n(z, 0), n ≥ 0.

Furthermore, in terms of the moments, an analytic function can be defined by

If L is a positive definite functional, then (9) is analytic in D and its real part is positive in D. In such a case, (9) is called a Carathéodory function, and can be represented by the Riesz-Herglotz transform

where σ is the positive measure associated with L. By extension, for a quasi-definite linear functional, (9) will denote its corresponding Carathéodory function.

On the other hand, given a positive, nontrivial Borel measure α supported in [−1, 1], we can define a positive, nontrivial Borel measure σ supported in [−π, π] in such a way that if dα(x) = ω(x)dx, then

There exists a relation between the corresponding families of orthogonal polynomials (see [6]). On the other hand, since the moments {c_n}n≥0 are real (see [6]), F(z), the Carathéodory function associated with σ, has real coefficients. Therefore, we have

and then dσ(θ) + dσ(2π − θ)=0. Thus, there exists a simple relation between the Stieltjes function (the real line analog of the Carathédory ­ functions, given by S(x) =∑^∞_n=0 µn^x−(n+1), where {µn}n≥0 are the moments associated with the measure on the real line) and the Carathéodory function associated with α and σ, respectively, given by (see [9])

where x = z+z⁻¹/2, z = x + √ x² − 1. In the literature, this relation is known as the Szegó transformation. Conversely, if σ is a positive, nontrivial Borel measure with support in the unit circle such that its moments are real, then there exists a positive, nontrivial Borel measure α, supported in [−1, 1], such that (10) holds. This is called the inverse Szegó transformation.

Given a measure σ supported on the unit circle, the perturbations

are called Christoffel, Uvarov, and Geronimus transformations, respectively. They are the unit circle analogue of the Christoffel, Uvarov and Geronimus transformations on the real line (see [12]). In general, a linear spectral transformation of a Stieltjes function is another Stieltjes function S(x) that has the form

where A, B and D are polynomials in x. The three transformations defined above are important due to the fact that any linear spectral transformation of a given Stieltjes function (i.e., for any polynomials A, B and D) can be obtained as a combination of Christoffel and Geronimus transformations (see [12]). A similar result holds for linear spectral transformations of Carathéodory functions, which are defined in a similar way (see [4]).

In [2], the authors studied the perturbation associated with the linear functional

where m ∈ R, p, q ∈ P, and L is (at least) a quasi-definite Hermitian linear functional defined in the linear space of Laurent polynomials. Notice that all moments associated with L are equal to the moments associated with L, except for the first moment, which is c₀ = c₀ + m. The corresponding Toeplitz matrix T is the result of adding m to the main diagonal of T. Later on, the linear functional

where j ∈ N is fixed and (·, ·)L_θ is the bilinear functional associated with the normalized Lebesgue measure on the unit circle was studied in [3]. It is easily seen that the moments associated with Lj are equal to those of L, except for the moments of order j and −j, which are perturbed by adding m and m, respectively. In other words, the corresponding Toeplitz matriz is perturbed on the j-th and −j-th subdiagonals. In both cases, the authors obtained the regularity conditions for such a linear functional and deduced connection formulas between the corresponding orthogonal sequences.

Assuming that both L and Lj are positive definite, the perturbation (13) can be expressed in terms of the corresponding measures as

On the other hand, the connection between the measure (14) and its corresponding measure supported in [−1, 1] via the inverse Szegó transformation was analyzed in [5], and it is deduced that the perturbed moments on the real line depend on the Chebyshev polynomials of the first kind.

In this contribution, we will extend those results to the case where a perturbation of a finite number of moments is introduced in (13). In Section 2, necessary and sufficient conditions for the regularity of the perturbed functional are obtained, as well as a connection formula that relates the corresponding families of monic orthogonal polynomials. For the positive definite case, the study of the perturbation through the inverse Szegó transformation will be analyzed in Section 3. An illustrative example will be presented in Section 4.

2. A perturbation on a finite number of moments associated with a linear functional L

Let L be a quasi-definite linear functional on the linear space of Laurent polynomials, and let {c_n}_n∈Z be its associated sequence of moments.

Definition 2.1. Let Ω be a finite set of non negative integers. The linear functional LΩ is defined such that the associated bilinear functional satisfies

where Mr ∈ C, p, q ∈ P, and (·, ·)L_θ is the bilinear functional associated with the normalized Lebesgue measure in the unit circle.

Notice that, from (15), one easily sees that

In other words, LΩ represents an additive perturbation of the moments c_r and c_−r of L, with r ∈ Ω. The rest of the moments remain unchanged. This is, the Toeplitz matrix associated with LΩ is

and therefore LΩ is also Hermitian. Moreover, if L is a positive definite functional, then the above perturbation can be expressed in terms of the corresponding measures as

On the other hand, if FΩ(z) is the Carathéodory function associated with LΩ, then

which is a linear spectral transformation of F(z). The following notation will be used hereinafter:

1. § A_{(s1,s2;l1,l2;r)} will denote a (s₂ − s₁ + 1) × (l₂ − l₁ + 1) matrix whose entries are a_(s,l)r, where s₁ ≤ s ≤ s₂ and l₁ ≤ l ≤ l₂,. For instance,

   A(2,3;4,5;6) = [a(2,4)₆ a(3,4)₆ a(2,5)₆ a(3,5)₆]

   § Ψ_n(0) = [ψ_n⁽⁰⁾ (0), ··· , ψ_n⁽ⁿ⁻¹⁾ (0)]^T and Φ_n(0) = [φ_n⁽⁰⁾ (0), ··· , φ_n⁽ⁿ⁻¹⁾ (0)]^T.

   § I_n will denote the n × n identity matrix.

   § Derivatives of negative order are defined as zero. For instance, K_n^(0,−2) (z,y) ≡ 0.

Necessary and sufficient conditions for the regularity of LΩ, as well as the relation between the corresponding sequences of orthogonal polynomials, are given in the next result.

Proposition 2.2. Let L be a quasi-definite linear functional and let {φ_n}_n≥0 be its associated monic orthogonal polynomials sequence (MOPS). The following statements are equivalent^[1]

1. 1. 1. LΩ is a quasi-definite linear functional.

   2. The matrix I_n + ∑_r∈Ω S^r_n is nonsingular, and

With

And

where the entries of the matrices A, B and C are given by

Furthermore, if {ψn}n≥0 denotes the MOPS associated with LΩ, then

for every n ≥ 1.

Remark 2.3. Notice that Q_n and S^r_n are n×n matrices, whereas Y^r_n, W_n and K^r_n−1(z, 0) are n-th dimensional column vectors.

Proof. Assume LΩ is a quasi-definite linear functional, and denote by {ψ_n}n≥0 its associated MOPS. Let us write

where, for 0 ≤ k ≤ n − 1,

and notice that (ψ_n(z), φ_k(z))L ≠ 0 (in general), and (ψ_n(z), φ_k(z))LΩ = 0 for n > k.

Substituting in (23) and using (7), we get

From the power series expansion of ψ_n(y) and K_n−1(z,y), we have

and for |y| = 1,

and since ʃ_T y^r−t dy/2πiy = 1 if r = t and zero otherwise, we arrive at

Similarly,

As a consequence, we get

which after a reorganization of the terms becomes

In order to find the constant values ψ^(l)_n (0), we take s derivatives, 0 ≤ s ≤ n, with respect to the variable z and evaluate at z = 0 to obtain the (n + 1) × (n + 1) linear system

If we denote

then the linear system becomes

Notice that the last equation (i.e., when s = n) gives no information, since a_(n,l)r = b_(n,l)r = c_(n,l)r = 0 and ψ_n⁽ⁿ⁾ (0) = φ_n⁽ⁿ⁾ (0) = n!. As a consequence, the (n + 1) × (n + 1) linear system can be reduced to an n × n linear system that can be expressed in matrix form as

Since LΩ is quasi-definite, it has a unique MOPS and therefore the linear system has a unique solution. As a consequence, the matrix I_n + ∑_r∈Ω S^r_n is nonsingular and

This is, we have

which is (22). On the other hand, for n ≥ 0,

Using (20) and (27), we get

Conversely, assume I_n + ∑_r∈Ω S^r_n is nonsingular for every n ≥ 1 and define {ψ_n}n≥0 as in (22). For 0 ≤ k ≤ n − 1, we have

and, similarly,

In addition,

and also

Thus, for 0 ≤ k ≤ n − 1, and taking into account (24) and the previous equations, we have

On the other hand,

which is different from 0 by assumption. Therefore, LΩ is quasi-definite.

Notice that if r_m = min{r : r ∈ Ω}, then from Proposition 2.2 we conclude that if n < r_m then K^rm_n−1 is the zero vector, K^(0,n−rm)_n−1 (z, 0) = 0 and according to (22) we have ψn(z) = φn(z). This means that the only affected polynomials are those with degree n ­ ⋟ rm.

3. Finite moments perturbation through the inverse Szegó transformation

Let σ be a positive measure supported on the unit circle such that its corresponding moments {c_n}_n∈Z are real. Assume also that the perturbed measure σΩ, defined by (17), is also positive and that Mr with r ∈ Ω is real, so that the moments associated with σΩ are also real. Our goal in this section is to determine the relation between the positive Borel measures α and αΩ, supported in [−1, 1], which are associated with σ and σ˜Ω, respectively, via the inverse Szegó transformation. This relation will be stated in terms of the corresponding measure and their sequences of moments.

Proposition 3.1. Let σ be a positive nontrivial Borel measure with real moments supported in the unit circle, and let α be its corresponding measure supported in [−1, 1], obtained through the inverse Szegó transformation. Let {c_n}_n∈Z and {µ_n}_n≥0 be their corresponding sequences of moments. Assume that σΩ, defined by (17) with r ∈ Ω and M_r ∈ R, is positive. Then, the measure αΩ, obtained by applying the inverse Szegó transformation to σΩ, is given by

where T_r(x) := cos(rθ) is the r-th degree Chebyshev polynomial of the first kind. Its corresponding sequence of moments is

With

Proof. Notice that, setting z = e^iθ, x = cos θ, and taking into account that the inverse Szegó transformation applied to the normalized Lebesgue measure dθ/2π yields the Chebyshev measure of the first kind dx/π √1−x², the measure αΩ obtained by applying

the inverse Szegó transformation to σΩ is given by

Notice that a measure that changes its sign in the interval [−1, 1] is added to dα. Then, the moments associated with αΩ are given by

As a consequence, by the orthogonality of Tr(x), we obtain for the n-th moments with n /∈ Ω

Furthermore (see [8]), we have

where [r/2] = r/2 if r is even and [r/2] = (r − 1)/2 if r is odd. Therefore,

and, since

(31) becomes (30).

From the previous proposition we can conclude that a perturbation of the moments cr and c_−r with r ∈ Ω, associated with a measure σ supported in the unit circle, results in a perturbation, defined by (30), of the moments µ_n, n ⋟ rm, associated with a measure α supported in [−1, 1], when both measures are related through the inverse Szegó transformation.

4. Example

Let L be the Christoffel transformation of the normalized Lebesgue measure with parameter ξ = 1, i.e., (p(z), q(z))L = ((z − 1)p(z),(z − 1)q(z))Lθ , and let Ω = {1, 2}. Then,

i.e., the moments of order 1 and 2 are perturbed. Since the sequence {zⁿ}n≥0 is orthogonal with respect to L_θ, the MOPS associated with ((z −1)p(z),(z −1)q(z))L_θ is given by (see [1])

or, equivalently,

and its corresponding reversed polynomial is

Furthermore, if 0 ≤ s ≤ n, we have

and if 0 ≤ t, s ≤ n − 1,

As a consequence, we have

We now proceed to obtain the MOPS associated with L_{1,2}, denoted by {ψ_n}n≥0. Notice that we have c_(s,n)1 = (s+1)!/n!(n+1) , c_(s,n)2 = 2(s+1)!/n!(n+1) if 0 ≤ s ≤ n − 2 and c_(n−1,n)2 = n−1/n(n+1) , and thus

On the other hand, for n ≥ 2 we have

and for n ≥ 3,

For illustrative purposes, we compute the first polynomials of the sequence:

§ Degree one:

since K¹₀ (z, 0) = 0, K²₀(z, 0) = 0 and φ₁(z) = z + 1/2φ₀(z).

§ Degree two:

since K²₁(z, 0) = 0, φ2(z) = z² + 2/3 φ1(z) and A = 1/det(I2+S1 2+S2 2) = 9/|M1+3|2−4|M1|2 .

§ In general, the n-th degree polynomial is

since φn(z) = zⁿ + n/n+1 φn−1(z).

On the other hand, assuming that the linear functional (32) is positive definite, the associated measure is

and the corresponding moments are given by

Thus, the perturbed Toeplitz matrix is

i.e., the first and second subdiagonals are perturbed. Furthermore, since dα = 2/π √1−x/1+x dx is the measure obtained applying the inverse Szegó transformation to dσ = |z − 1|² dθ/2π, then according to (29) the measure supported in [−1, 1] is

Then, according to (30), the perturbed moments associated with the measure (33) are

where {µn}n≥0 are the moments associated with the measure dα = 2 π √1−x/1+x, and

Acknowledgements

The work of the second author was supported by Consejo Nacional de Ciencia y Tecnología of México, grant 156668. Both authors thank the anonymous referees for her/his valuable comments and suggestions. They contributed greatly to improve the presentation and contents of the manuscript.

References

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Notes

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