Abstract
For Jacobi parameters belonging to one of three classes: asymptotically periodic, periodically modulated, and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.
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1 Introduction
Let \(\mu \) be a probability measure on the real line with infinite support such that for every \(n \in \mathbb {N}_0\),
Let \(L^2(\mathbb {R}, \mu )\) be the Hilbert space of square-integrable functions equipped with the scalar product
By performing the Gram–Schmidt orthogonalization process on the sequence of monomials \((x^n : n \in \mathbb {N}_0)\) one obtains the sequence of polynomials \((p_n : n \in \mathbb {N}_0)\) satisfying
where \(\delta _{nm}\) is the Kronecker delta. Moreover, \((p_n : n \in \mathbb {N}_0)\) satisfies the following recurrence relation
where
Notice that for every n, \(a_n > 0\) and \(b_n \in \mathbb {R}\). The pair \((a_n)\) and \((b_n)\) is called the Jacobi parameter.
Let \(\mathbb {P}_n\) be the orthogonal projection in \(L^2(\mathbb {R}, \mu )\) on the space of polynomials of degree at most n. Then \(\mathbb {P}_n\) is given by the Christoffel–Darboux kernel \(K_n\), that is
where from (1.1) one can verify that
To motivate the study of Christoffel–Darboux kernels see surveys [12] and [19].
The asymptotic behavior of \(K_n\) is well understood in the case when the measure \(\mu \) has compact support. In this setup one of the most general results has been proven in [29]. Namely, if I is an open interval contained in \({\text {supp}}(\mu )\) such that \(\mu \) is absolutely continuous on I with continuous positive density \(\mu '\), then
locally uniformly with respect to \(x \in I\) and \(u, v \in \mathbb {R}\), provided that \(\mu \) is regular (see [21, Definition 3.1.2]). In formula (1.4), \(\omega '\) denotes the density of the equilibrium measure corresponding to the support of \(\mu \), see (2.4) for details. In the case when \({\text {supp}}(\mu )\) is a finite union of compact intervals, \(\mu \) is regular provided that \(\mu ' > 0\) almost everywhere in the interior of \({\text {supp}}(\mu )\). To give some historical perspective, let us also mention three earlier results. The case \({\text {supp}}(\mu )=[-1,1]\) and \(u=v=0\) has been examined in [14, Theorem 8], and its extension to a general compact \({\text {supp}}(\mu )\) has been obtained in [28, Theorem 1]. The extension to all \(u,v \in \mathbb {R}\) has been proven in [11].
The best understood class of measures with unbounded support is the class of exponential weights (see the monograph [8]). In [10] and [9, Theorem 7.4] under a number of regularity conditions on the function \(Q(x) = -\log \mu '(x)\), the following analogue of (1.4) was shown
locally uniformly with respect to \(x,u,v \in \mathbb {R}\) where \(\tilde{K}_n(x,y) = \sqrt{\mu '(x) \mu '(y)} K_n(x,y)\). Unlike (1.4), the formula (1.5) does not give any information if \(u = v = 0\). It was recently proved in [4] that under some additional regularity on Q
locally uniformly with respect to \(x \in \mathbb {R}\) where
Let us comment that \(\rho _n\) is comparable to n if the sequences \((a_n)\) and \((a_n^{-1})\) are bounded. By combining (1.5) with (1.6) one obtains
where \(\omega '(x) = \big ( \pi \sqrt{4 - x^2} \big )^{-1}\) is the density of the equilibrium measure for the interval \([-2, 2]\).
Instead of taking the measure \(\mu \) as the starting point one can consider polynomials \((p_n : n \in \mathbb {N}_0)\) satisfying the three-term recurrence relation (1.2) with \(a_n > 0\) and \(b_n \in \mathbb {R}\) for any \(n \in \mathbb {N}_0\). Then the Favard’s theorem (see, e.g., [18, Theorem 5.10]) states that there is a probability measure \(\mu \) such that \((p_n)\) is orthonormal in \(L^2(\mathbb {R}, \mu )\). The measure \(\mu \) is unique, if and only if there is exactly one measure with the same moments as \(\mu \). It is always the case when the Carleman condition
is satisfied (see, e.g. [18, Corollary 6.19]). Moreover, the measure \(\mu \) has compact support, if and only if the Jacobi parameters are bounded.
In this article our starting point is the three-term recurrence relation. We study analogues of (1.6) and (1.7) for three different classes of Jacobi parameters: asymptotically periodic, periodically modulated and a blend of these two; for the definitions, see Sects. 3.1, 3.2 and 3.3, respectively. The first class only gives rise to measures with compact supports. The second class introduced in [5] has the Jacobi parameters uniformly unbounded in the sense that \(\liminf a_n = \infty \). The third class has been studied in [1] as an example of unbounded Jacobi parameters corresponding to measures with absolutely continuous parts having supports equal a finite union of closed intervals.
To simplify the exposition in the introduction, we shall focus on the periodic modulations only. Before we formulate our results, let us state some definitions. Let N be a positive integer. We say that sequences \((a_n), (b_n)\) are N-periodically modulated if there are two N-periodic sequences \((\alpha _n : n \in \mathbb {Z})\) and \((\beta _n : n \in \mathbb {Z})\) of positive and real numbers, respectively, such that
-
(a)
\(\lim _{n \rightarrow \infty } a_n = \infty ,\)
-
(b)
\(\lim _{n \rightarrow \infty }\Big | \frac{a_{n-1}}{a_n} -\frac{\alpha _{n-1}}{\alpha _n} \Big | = 0 ,\)
-
(c)
\(\lim _{n \rightarrow \infty }\Big | \frac{b_n}{a_n} - \frac{\beta _n}{\alpha _n} \Big | = 0.\)
The crucial rôle is played by the N-step transfer matrices defined by
and
The name is justified by the following property
We are interested in the class of Jacobi matrices associated with slowly oscillating sequences introduced in [22]. Let r be a positive integer. We say that the sequence \((x_n : n \in \mathbb {N})\) of vectors from a normed space V belongs to \(\mathcal {D}_r (V)\), if it is bounded and for each \(j \in \{1, \ldots , r\}\),
where
If X is the real line with a Euclidean norm we abbreviate \(\mathcal {D}_{r} = \mathcal {D}_{r}(X)\). Given a compact set \(K \subset \mathbb {C}\) and a normed space R, by \(\mathcal {D}_{r}(K, R)\) we denote the case when X is the space of all continuous mappings from K to R equipped with the supremum norm.
Our first result is the following theorem, see Theorem 4.4. By \({\text {GL}}(2, \mathbb {R})\) we denote \(2 \times 2\) real invertible matrices equipped with the spectral norm. For a matrix
we set \([X]_{i, j} = x_{i, j}\). Lastly, a discriminant of X is \({\text {discr}}X = ({\text {tr}}X)^2 - 4 \det X\).
Theorem 1.1
Let N and r be positive integers and \(i \in \{0, 1, \ldots , N-1\}\). Suppose that K is a compact interval with non-empty interior contained in
Assume that
and
Suppose that \(\mathcal {X}\) is the limit of \(\big (X_{jN+i} : j \in \mathbb {N}\big )\). If
then for \(x \in K\)
where
Let us remark that in Theorem 5.4 we have obtained the quantitative bounds on \(E_n\) in the \(\mathcal {D}_1\) setting.
Theorem 1.1 is an important step in proving the analogues of (1.6) and (1.7). The following theorem (see Theorem 4.6) provides the analogue of (1.6) for periodic modulations. Similar results are obtained also for the remaining classes, that is for asymptotically periodic in Theorem 4.7, and for the blend in Theorem 4.10.
Theorem 1.2
Let \((a_n)\) and \((b_n)\) be N-periodically modulated Jacobi parameters. Suppose that there is \(r \ge 1\) such that for every \(i \in \{0, 1, \ldots , N-1\}\),
and the Carleman condition (1.8) is satisfied. If \(|{{\text {tr}}\mathfrak {X}_0(0)} | < 2\), then
where
locally uniformly with respect to \(x \in \mathbb {R}\), where \(\omega \) is the equilibrium measure of
and
Again, in \(\mathcal {D}_1\) setup, we obtained the quantitative bound on \(E_n\), see Theorem 5.5 (periodic modulations) and Theorem 5.7 (asymptotically periodic).
We emphasize that Theorem 1.2 solves [3, Conjecture 1] for a larger class of Jacobi parameters than it was originally stated, see Sect. 4.3 for details.
Lastly, we provide the analogue of (1.7) for periodic modulations (see Theorem 5.15). In view of [27, Corollary 7], the asymptotically periodic case follows from [29]. For the blend, see Theorem 5.17.
Theorem 1.3
Suppose that the hypotheses of Theorem 1.2 are satisfied for \(r=1\). Then
locally uniformly with respect to \(x,u,v \in \mathbb {R}\).
Let us mention that the hypotheses of Theorems 1.2 and 1.3 for \(N=1\) are satisfied by Hermite polynomials, Meixner–Pollaczek polynomials and Freud weights, see [23, Section 5] for detailed proofs.
Let us present some ideas of the proofs. The basic strategy commonly used is to exploit the Christoffel–Darboux formula, that is
However, it requires the precise asymptotic of the polynomials as well as its derivatives in terms of both n and x. Unfortunately, for the classes of Jacobi parameters we are interested in they are not available. In the recent article [27], we managed to obtain the asymptotic of \((p_n(x) : n \in \mathbb {N}_0)\) locally uniformly with respect to x. Based on it we develop a method to study \(K_n(x, y)\). Namely, we use the formula (1.3), which leads to the need of estimation of the oscillatory sums of a form
where
To deal with the sums we prove two auxiliary results (see Lemma 4.1 and Theorem 5.11) that are valid for sequences not necessarily belonging to \(\mathcal {D}_r\).
The organization of the article is as follows. In Sect. 2 we present basic definitions used in the article. In Sect. 3 we collect the definitions and basic properties of the three classes of sequences. Section 4 is devoted to the general \(\mathcal {D}_r\) setting. In particular, we present there the proofs of Theorems 1.1 and 1.2, and we provide a solution of Ignjatović conjecture. In Sect. 5 we study the case \(\mathcal {D}_1\) where we derive quantitative bound on the error in the asymptotic of the polynomials. We also provide the quantitative versions of Theorems 1.1 and 1.2. Finally, we prove Theorem 1.3.
1.1 Notation
By \(\mathbb {N}\) we denote the set of positive integers and \(\mathbb {N}_0 = \mathbb {N}\cup \{0\}\). Throughout the whole article, we write \(A \lesssim B\) if there is an absolute constant \(c>0\) such that \(A\le cB\). Moreover, c stands for a positive constant whose value may vary from occurrence to occurrence.
2 Definitions
Given two sequences \(a = (a_n : n \in \mathbb {N}_0)\) and \(b = (b_n : n \in \mathbb {N}_0)\) of positive and real numbers, respectively, and \(k \in \mathbb {N}\), we define kth associated orthonormal polynomials as
For \(k=0\) we usually omit the superscript. A sequence \((u_n : n \in \mathbb {N}_0)\) is a generalized eigenvector associated with \(x \in \mathbb {C}\), if for all \(n \ge 1\),
where
Let A be the closure in \(\ell ^2\) of the operator acting on sequences having finite support by the matrix
The operator A is called Jacobi matrix. If the Carleman condition
is satisfied then the operator A has the unique self-adjoint extension (see e.g., [18, Corollary 6.19]). Let us denote by \(E_A\) its spectral resolution of the identity. Then for any Borel subset \(B \subset \mathbb {R}\), we set
where \(\delta _0\) is the sequence having 1 on the 0th position and 0 elsewhere. The polynomials \((p_n : n \in \mathbb {N}_0)\) form an orthonormal basis of \(L^2(\mathbb {R}, \mu )\).
In this article the central object is the Christoffel–Darboux kernel defined as
Given a compact set \(K \subset \mathbb {R}\) with non-empty interior, there is the unique probability measure \(\omega _K\), called the equilibrium measure corresponding to K, minimizing the energy
among all probability measures \(\nu \) supported on K. The measure \(\omega _K\) is absolutely continuous in the interior of K with continuous density, see [16, Theorem IV.2.5, pp. 216].
3 Classes of Sequences
In this article we are interested in Jacobi matrices having entries in one of the three classes defined in terms of periodic sequences. Let us start by fixing some notation.
By \((\alpha _n : n \in \mathbb {Z})\) and \((\beta _n : n \in \mathbb {Z})\) we denote N-periodic sequences of real and positive numbers, respectively. For each \(k \ge 0\), let us define polynomials \((\mathfrak {p}^{[k]}_n : n \in \mathbb {N}_0)\) by relations
Let
where for a sequence of square matrices \((C_n : n_0 \le n \le n_1)\) we set
By \(\mathfrak {A}\) we denote the Jacobi matrix corresponding to
Let \(\omega \) be the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\). Since \(\sigma _{\text {ess}}(\mathfrak {A})\) is a finite union of compact intervals with non-empty interiors (see, e.g., [20, Theorem 5.2.4 and Theorem 5.4.2]), \(\omega \) is absolutely continuous. Moreover, for x in the interior of \(\sigma _{\text {ess}}(\mathfrak {A})\) we obtain
Indeed, by [24, Proposition 3] (see also Lemma 3.2 below),
Since
we have
Therefore,
Let us recall that the first formula on page 214 of [30] reads
hence
Now, in view of [30, the formula (3.2)] we obtain (3.1).
3.1 Asymptotically Periodic
Definition 3.1
The Jacobi matrix A associated with \((a_n : n \in \mathbb {N}_0)\) and \((b_n : n \in \mathbb {N}_0)\) has asymptotically N-periodic entries, if there are two N-periodic sequences \((\alpha _n : n \in \mathbb {Z})\) and \((\beta _n : n \in \mathbb {Z})\) of positive and real numbers, respectively, such that
-
(a)
\(\lim _{n \rightarrow \infty } \big |a_n - \alpha _n\big | = 0\),
-
(b)
\( \lim _{n \rightarrow \infty } \big |b_n - \beta _n\big | = 0 \).
Let us recall the following lemma.
Lemma 3.2
([24, Proposition 3]) Let \((p_n : n \in \mathbb {N}_0)\) be a sequence of orthonormal polynomials associated with \((a_n : n \in \mathbb {N}_0)\) and \((b_n : n \in \mathbb {N}_0)\). Then for all \(n \ge 1\) and \(k \ge 1\),
Proposition 3.3
Suppose that A has asymptotically N-periodic entries. Then for each \(i \in \{ 0, 1, \ldots , N-1 \}\) and \(n \ge 0\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
Since for each \(i \in \{ 0, 1, \ldots , N-1 \}\), we have
uniformly on compact subsets of \(\mathbb {C}\), the conclusion follows by the continuity of \(B_n\). \(\square \)
For a Jacobi matrix having asymptotically N-periodic entries, we set
Let us denote by \(\mathcal {X}_i\) the limit of \((X_{jN+i} : j \in \mathbb {N})\). Then, by Proposition 3.3, we conclude that \(\mathcal {X}_i = \mathfrak {X}_i\) for all \(i \in \{0, 1, \ldots , N-1\}\).
Proposition 3.4
Suppose that a Jacobi matrix A has asymptotically N-periodic entries. Then for each \(i \in \{0, 1, \ldots , N-1 \}\), and \(n \ge 0\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
Lemma 3.2 together with Proposition 3.3 easily gives (3.3a). Since
the uniform convergence in (3.3a) entails (3.3b) and (3.3c). \(\square \)
Corollary 3.5
Suppose that a Jacobi matrix A has asymptotically N-periodic entries. Then for each \(i \in \{0, 1, \ldots , N-1\}\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
Since for \(x \in \mathbb {C}\),
we have
By Lemma 3.2,
which together with Proposition 3.4, implies that
uniformly on compact subsets of \(\mathbb {C}\). Hence, by (3.5) we easily obtain (3.4a).
For \(s \in \{1, 2 \}\) we write
hence, by Proposition 3.4, we obtain
Therefore, by Lemma 3.2,
which finishes the proof. \(\square \)
3.2 Periodic Modulations
Definition 3.6
We say that the Jacobi matrix A associated with \((a_n : n \in \mathbb {N}_0)\) and \((b_n : n \in \mathbb {N}_0)\) has N-periodically modulated entries, if there are two N-periodic sequences \((\alpha _n : n \in \mathbb {Z})\) and \((\beta _n : n \in \mathbb {Z})\) of positive and real numbers, respectively, such that
-
(a)
\(\lim _{n \rightarrow \infty } a_n = \infty ,\)
-
(b)
\(\lim _{n \rightarrow \infty } \Big | \frac{a_{n-1}}{a_n} - \frac{\alpha _{n-1}}{\alpha _n} \Big | = 0,\)
-
(c)
\(\lim _{n \rightarrow \infty } \Big | \frac{b_n}{a_n} - \frac{\beta _n}{\alpha _n} \Big | = 0.\)
Suppose that A is a Jacobi matrix with N-periodically modulated entries. Observe that, by setting
we obtain
Hence, A is an N-periodic modulation of the Jacobi matrix corresponding to the sequences \((\tilde{a}_n : n \in \mathbb {N}_0)\) \((\tilde{b}_n : n \in \mathbb {N}_0)\) in the usual sense.
Proposition 3.7
If a Jacobi matrix A has N-periodically modulated entries and \(i \in \mathbb {N}\), then
In particular,
Proof
Since
one has
Hence, for some \(c > 0\),
and the proposition follows. \(\square \)
Proposition 3.8
Suppose that A has N-periodically modulated entries. Then for each \(i \in \{ 0, 1, \ldots , N-1 \}\) and \(n \ge 0\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
Since for each \(i \in \{ 0, 1, \ldots , N-1 \}\), we have
uniformly on compact subsets of \(\mathbb {C}\), the conclusion follows by the continuity of \(B_n\). \(\square \)
For a Jacobi matrix with N-periodically modulated entries, we set
Let us denote by \(\mathcal {X}_i\) the limit of \((X_{jN+i} : j \in \mathbb {N})\). Then, by Proposition 3.8, we have \(\mathcal {X}_i(x) = \mathfrak {X}_i(0)\) for all \(i \in \{0, 1, \ldots , N-1\}\) and \(x \in \mathbb {C}\).
Proposition 3.9
Suppose that a Jacobi matrix A has N-periodically modulated entries. Then for each \(i \in \{0, 1, \ldots , N-1 \}\), and \(n \ge 0\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
By Lemma 3.2 and Proposition 3.8 we obtain that for every \(i \ge 0\) and \(n \ge 0\),
uniformly on compact subsets of \(\mathbb {C}\), which is (3.6a). Next, let us recall that (see e.g. [24, Proposition 2])
therefore for every \(n \in \mathbb {N}\),
Hence,
By (3.7) the right-hand side of (3.9) tends to
Therefore, by (3.8) we conclude (3.6b). For the proof of (3.6c), let us observe that from (3.9) we get
Therefore, by (3.6b) and (3.7), the first sum in (3.10) approaches to
In view of (3.6b), for each \(m \in \mathbb {N}_0\),
Hence, by (3.6b), (3.12), and (3.7), we obtain
Consequently, the second sum in (3.10) tends to
Finally, putting (3.11) and (3.13) into (3.10), by (3.8), we obtain (3.6c). This completes the proof. \(\square \)
By reasoning analogous to the proof of Corollary 3.5 we obtain the following corollary.
Corollary 3.10
Suppose that a Jacobi matrix A has N-periodically modulated entries. Then for each \(i \in \{0, 1, \ldots , N-1\}\),
locally uniformly with respect to \(x \in \mathbb {C}\).
3.3 A Blend of Bounded and Unbounded Parameters
Definition 3.11
The Jacobi matrix A associated with sequences \((a_n : n \in \mathbb {N}_0)\) and \((b_n : n \in \mathbb {N}_0)\) is an N-periodic blend if there are an asymptotically N-periodic Jacobi matrix \(\tilde{A}\) associated with sequences \((\tilde{a}_n : n \in \mathbb {N}_0)\) and \((\tilde{b}_n : n \in \mathbb {N}_0)\), and a sequence of positive numbers \((\tilde{c}_n : n \in \mathbb {N}_0)\), such that
-
(i)
\( \lim _{n \rightarrow \infty } \tilde{c}_n = \infty , \qquad \text {and}\qquad \lim _{m \rightarrow \infty } \frac{\tilde{c}_{2m+1}}{\tilde{c}_{2m}} = 1 \),
-
(ii)
\( a_{k(N+2)+i} = {\left\{ \begin{array}{ll} \tilde{a}_{kN+i} &{} \text {if } i \in \{0, 1, \ldots , N-1\}, \\ \tilde{c}_{2k} &{} \text {if } i = N, \\ \tilde{c}_{2k+1} &{} \text {if } i = N+1, \end{array}\right. } \)
-
(iii)
\( b_{k(N+2)+i} = {\left\{ \begin{array}{ll} \tilde{b}_{kN+i} &{} \text {if } i \in \{0, 1, \ldots , N-1\}, \\ 0 &{} \text {if } i \in \{N, N+1\}. \end{array}\right. } \)
Proposition 3.12
For \(i \in \{1, 2, \ldots ,N-1\}\),
locally uniformly with respect to \(x \in \mathbb {R}\). Moreover,
locally uniformly with respect to \(x \in \mathbb {R}\), where
Proof
The argument is contained in the proof of [27, Corollary 9]. \(\square \)
For a Jacobi matrix that is an N-periodic blend, we set
In view of Proposition 3.12, for \(i \in \{1, 2, \ldots , N\}\), the sequence \((X_{j(N+2)+i} : j \in \mathbb {N})\) converges to \(\mathcal {X}_i\) locally uniformly on \(\mathbb {C}\) where
We set
Theorem 3.13
There are non-empty open and disjoint intervals \((I_j : 1 \le j \le N)\) such that
Moreover, for \(x \in \Lambda \),
where \(\omega \) is the equilibrium measure corresponding to \(\overline{\Lambda }\).
Proof
Let us begin with the case \(N = 1\). Then
Therefore, by (3.1) one obtains
which ends the proof for \(N=1\).
Suppose that \(N \ge 2\). For \(k \in \mathbb {Z}\) and \(i \in \{0, 1, \ldots , N-1\}\), we set
Let
By (3.1),
where \(\tilde{\omega }\) is the equilibrium measure corresponding to the closure of
In particular, \(\tilde{\Lambda }\) is the union of N non-empty open and disjoint intervals, see [20, Theorem 5.4.2]. Notice that
We next show the following claim. \(\square \)
Claim 3.14
If \(N \ge 2\), then
For \(N = 2\), the identities (3.17) and (3.19) can be checked by direct computation. For \(N \ge 3\), we first observe that
Consequently,
Similarly, one can show (3.18) and (3.19).
Now, using (3.17), we easily get
which implies that \(\Lambda = \tilde{\Lambda }\). Hence, \(\tilde{\omega } = \omega \). Moreover, by Claim 3.14, we obtain
Therefore, for \(x \in \Lambda \), formula (3.16) gives
which finishes the proof. \(\square \)
Corollary 3.15
Proof
In view of [30, formula (3.2)],
By Claim 3.14,
which together with (3.23) concludes the proof. \(\square \)
Remark 3.16
We want to emphasize that Theorem 3.13 says that \(\Lambda \) is the disjoint union of exactly N non-empty open intervals. This should be compared with the discussion at the beginning of Section 5 in [1].
Proposition 3.17
Suppose that a Jacobi matrix A is an N-periodic blend. Then for each \(i \in \{1, 2, \ldots , N\}\) and \(n \ge 0\),
locally uniformly with respect to \(x \in \mathbb {C}\).
Proof
Lemma 3.2 together with Proposition 3.3 easily gives (3.24a). Since
the uniform convergence in (3.24a) entails (3.24b) and (3.24c). \(\square \)
Corollary 3.18
Suppose that a Jacobi matrix A is an N-periodic blend. Then for each \(i \in \{1, 2, \ldots , N\}\),
locally uniformly with respect to \(x \in \mathbb {C}\).
4 Christoffel Functions for \(\mathcal {D}_r\), \(r \ge 1\)
4.1 General Case
In this section we determine the asymptotic behavior of the Christoffel–Darboux kernel (2.3) on the diagonal. We start by showing the following lemma. In its proof we use the following well-known fact, called the Stolz–Cesáro theorem: if \((c_n : n \in \mathbb {N})\) is a sequence of real numbers strictly increasing and approaching infinity, and \((f_n : n \in \mathbb {N})\) is a sequence of real functions on a compact set K, such that
uniformly with respect to \(x \in K\), then
uniformly with respect to \(x \in K\). The classical scalar version (see, e.g. [15, Section 3.1.7]) quickly generalizes to the form above.
Lemma 4.1
Let \((\gamma _n : n \ge 0)\) be a sequence of positive numbers and \((\theta _n : n \ge 0)\) be a sequence of continuous functions on some compact set \(K \subset \mathbb {R}^d\) with values in \((0, 2 \pi )\). Suppose that there is a function \(\theta : K \rightarrow (0, 2\pi )\) such that
uniformly with respect to \(x \in K\). Then there is \(c > 0\) such that for all \(x \in K\) and \(n \in \mathbb {N}\),
In particular, if
then
uniformly with respect to \(x \in K\).
Proof
Let us observe that
where
and
Hence, by the summation by parts we get
Since
the first term in (4.3) is bounded by a constant multiple of \(\gamma _n\). Moreover,
because
We treat the second term in (4.3) similarly. Finally, the third term in (4.3) can be bounded by a constant multiple of
which together with the identity
entails that (4.3) is bounded by a constant multiple of
proving (4.1).
Lastly, we observe that by the Stolz–Cesáro theorem,
Similarly, we obtain
thus
uniformly with respect to \(x \in K\) proving (4.2), and the lemma follows. \(\square \)
The following theorem has been proved in [27].
Theorem 4.2
[27, Theorem C] Let N and r be positive integers and \(i \in \{0, 1, \ldots , N-1\}\). Suppose that K is a compact interval contained in
Assume that
and
Suppose that \(\mathcal {X}\) is the limit of \((X_{jN+i} : j \in \mathbb {N})\). Then there is a probability measure \(\nu \) such that \((p_n : n \in \mathbb {N}_0)\) are orthonormal in \(L^2(\mathbb {R}, \nu )\), which is purely absolutely continuous with continuous and positive density \(\nu '\) on K. Moreover, there are \(M > 0\) and a real continuous function \(\eta : K \rightarrow \mathbb {R}\) such that for all \(k \ge M\),
where
The proof of the following proposition is inspired by [2].
Proposition 4.3
Under the hypotheses of Theorem 4.2, we have
Proof
First, let us consider the case when the moment problem for \(\nu \) is indeterminate, that is
But (4.6) implies that
which easily gives (4.5).
Assume now that the moment problem for \(\nu \) is determinate. By Theorem 4.2, the measure \(\nu \) has non-trivial absolutely continuous part. To obtain a contradiction, suppose that there is a strictly increasing sequence \((L_j : j \in \mathbb {N}_0)\) such that
Without loss of generality we may assume that
The Jacobi matrix (2.1) can be written in the following block-form
where each \(J_i\) is a finite Jacobi matrix with dimension
and each \(P_i\) is a rectangular matrix
Let
By A and B we denote the restrictions of \(\mathcal {A}\) and \(\mathcal {B}\) to the maximal domains, respectively, that is
and
The determinacy of the moment problem for \(\nu \) is equivalent to A being self-adjoint. Moreover,
where \(E_A\) is the spectral resolution of A and \(\delta _0\) is the sequence having 1 on the zero position and zero elsewhere. In view of (4.7), the operator \(A - B\) is self-adjoint and belongs to the trace class. Hence, by the Kato–Rosenblum theorem (see, e.g., [17, Theorem 9.29])
Since B is unitary equivalent to an operator acting by the multiplication by a real-valued sequence, B has only discrete spectrum. Therefore,
and consequently, by (4.8), the measure \(\nu \) has no non-trivial absolutely continuous part, which leads to the contradiction. \(\square \)
For \(i \in \{0, 1, \ldots , N-1\}\) and \(n \in \mathbb {N}\) we set
and
Let us recall that the Carleman condition (2.2) implies that there is a unique probability measure \(\mu \) such that \((p_n : n \in \mathbb {N}_0)\) are orthonormal in \(L^2(\mathbb {R}, \mu )\).
Theorem 4.4
Let N and r be positive integers and \(i \in \{0, 1, \ldots , N-1\}\). Suppose that K is a compact interval with non-empty interior contained in
Assume that
and
Suppose that \(\mathcal {X}\) is the limit of \(\big (X_{jN+i} : j \in \mathbb {N}\big )\). If
then
where
Proof
Fix a compact interval K with non-empty interior contained in \(\Lambda \). In view of Theorem 4.2, there is \(M > 0\) such that for all \(k \ge M\),
where
Since \(2 \sin ^2(x) = 1 - \cos (2x)\), we have
Notice that the Stolz–Cesàro theorem gives
uniformly with respect to \(x \in K\). Moreover, by Lemma 4.1
Finally, since there is \(c > 0\) such that
we conclude that
which completes the proof. \(\square \)
Remark 4.5
In view of [27, Proposition 7], for each compact set \(K \subset \mathbb {R}\) with non-empty interior there is \(c > 0\) such that for all \(n \in \mathbb {N}\),
Moreover, the right-hand side is comparable to a constant multiple of
4.2 Application to the Classes
We are now ready to prove the main theorem of this section.
Theorem 4.6
Let A be a Jacobi matrix with N-periodically modulated entries. Suppose that there is \(r \ge 1\) such that for every \(i \in \{0, 1, \ldots , N-1\}\),
and
If \(|{{\text {tr}}\mathfrak {X}_0(0)} | < 2\), then
where
locally uniformly with respect to \(x \in \mathbb {R}\), where \(\omega \) is the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\), with \(\mathfrak {A}\) being the Jacobi matrix associated with \((\alpha _n : n \in \mathbb {N}_0)\) and \((\beta _n : n \in \mathbb {N}_0)\), and
Proof
Let K be a compact interval in \(\mathbb {R}\) with non-empty interior. Observe that, by Remark 4.5, for each \(i \in \{0, 1, \ldots , N-1\}\) the sequence \((X_{jN+i} : j \ge 0)\) belongs to \(\mathcal {D}_r \big ( K, {\text {GL}}(2, \mathbb {R}) \big )\). Moreover, by Proposition 3.8 we have
which together with \({\text {discr}}\mathfrak {X}_i(0) < 0\) implies that \(\Lambda = \mathbb {R}\). Since for each \(n, n' \in \mathbb {N}_0\), by Proposition 3.7, we have
by the Stolz–Cesàro theorem
Consequently, the Carleman condition (4.11) implies that
for each \(i \in \{0, 1, \ldots , N-1\}\). Thus, by Theorem 4.4, we obtain
Fix \(i \in \{0, 1, \ldots , N-1\}\) and consider \(n = kN + i\) for \(k \in \mathbb {N}_0\). We write
Observe that
hence, by (4.13),
Since
we immediately get
Finally, putting (4.12) and (4.15) into (4.14), we obtain
which together with (3.1) completes the proof. \(\square \)
The following theorem has essentially the same proof as Theorem 4.6.
Theorem 4.7
Let A be a Jacobi matrix with asymptotically N-periodic entries. Suppose that there is \(r \ge 1\) such that for every \(i \in \{0, 1, \ldots , N-1\}\),
Let K be a compact interval with non-empty interior contained in
Then
with
where \(\omega \) is the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\), with \(\mathfrak {A}\) being the Jacobi matrix associated with \((\alpha _n : n \in \mathbb {N}_0)\) and \((\beta _n : n \in \mathbb {N}_0)\), and
Remark 4.8
If the Jacobi matrix A has asymptotically N-periodic entries, then the condition (4.16) is equivalent to
To see this, let us observe that
Hence, by [27, Corollary 2]
thus by [27, Corollary 1]
Analogously, one can prove the opposite implication.
Remark 4.9
If the Jacobi matrix A has asymptotically N-periodic entries, then
Indeed, by the Stolz–Cesàro theorem, we have
Let us recall that in this case \({\text {supp}}(\mu )\) is compact and \(\mu '\) is continuous and positive in the interior of \({\text {supp}}(\mu )\). Thus, in view of (4.17), Theorem 4.7 follows from [28, Theorem 1]. We want to point out that our approach is completely different.
Theorem 4.10
Let A be a Jacobi matrix that is an N-periodic blend. Suppose that there is \(r \ge 1\) such that for every \(i \in \{0, 1, \ldots , N-1\}\),
and
Let K be a compact subset with non-empty interior contained in
where \(\mathcal {X}_1\) is the limit of \((X_{j(N+2)+1} : j \in \mathbb {N}_0)\). Then
where \(\omega \) is the equilibrium measure corresponding to \(\overline{\Lambda }\), and
with
Proof
Let K be a compact set with non-empty interior contained in \(\Lambda \). Fix \(i \in \{1,2, \ldots , N\}\). Let us recall that the sequence \((X_{j(N+2)+i} : j \in \mathbb {N})\) converges to \(\mathcal {X}_i\). Moreover, we claim the following holds.
Claim 4.11
The sequence \((X_{j(N+2)+i} : j \in \mathbb {N})\) belongs to \(\mathcal {D}_r\big (K, {\text {GL}}(2, \mathbb {R})\big )\).
For the proof, let us observe that if \(i \in \{0, 1, \ldots , N-1\}\),
thus, by [27, Corollary 2],
and consequently, by [27, Corollary 1(i)],
Moreover,
Since
by [27, Corollary 1], we get
Now, the conclusion follows from the proof of [27, Corollary 9].
Since
and, in view of (3.15),
therefore, by Theorem 4.4, we obtain
where
Next, we show
and
First, we prove the following claim.
Claim 4.12
There is \(c > 0\) such that for all \(k \in \mathbb {N}_0\),
and
For the proof, let us notice that [27, Theorem A] with \(i = 1\), implies
Since \((a_{k(N+2)} : k \in \mathbb {N}_0)\) is bounded, by [27, Corollary 9] with \(i = 1\),
consequently we get
Analogous reasoning for \(i = N\) shows that
Recall that we have the following recurrence relation
Therefore, by (4.23) and (4.24), we easily get (4.21). Moreover, we have
thus
which together with (4.21) and (4.25) entails (4.22).
Now using Claim 4.12 together with (4.24) and (4.25), we easily see that
Since \((a_{k(N+2)+N-1} : k \in \mathbb {N}_0)\) is bounded, by the Stolz–Cesàro theorem, we obtain
and
which gives (4.19). To prove (4.20), we reason analogously. Namely, by Claim 4.12, we have
thus, by the Stolz–Cesàro theorem, we get
Notice that, by [27, Corollary 9] and (4.21), we have
Next we show the following statement.
Claim 4.13
For each \(i \in \{0, 1, \ldots , N+1\}\) and \(i' \in \{0, 1, \ldots , N-1\}\),
For the proof, let us consider \(i \in \{0, 1, \ldots , N-2\}\). By the Stolz–Cesàro theorem, we have
For \(i = N-1\) we obtain
Finally, we observe that
which entails (4.27) for \(i \in \{N, N+1\}\).
Now, writing
by (4.19), (4.20) and (4.26), we obtain
uniformly with respect to \(x \in K\). Using (4.18) together with (4.27), we arrive at
uniformly with respect to \(x \in K\). To finish the proof, it is sufficient to invoke Theorem 3.13. \(\square \)
Remark 4.14
If A is a Jacobi matrix that is an N-periodic blend then
Indeed, for each \(i \in \{0, 1, \ldots , N+1\}\), by Claim 4.13, we have
Now, by the Stolz–Cesàro theorem, we obtain
and the formula (4.28) follows.
Let us recall that \({\text {supp}}(\mu )\) is not compact. In [1, Theorem 5] examples were provided of Jacobi parameters from this class such that the set of the accumulation points of \({\text {supp}}(\mu )\) is equal to the compact set \(\overline{\Lambda }\). From [27, Corollary 9] the density \(\mu '\) is continuous and positive in \(\Lambda \). Hence, in view of (4.28), the hypothesis on the compactness of \({\text {supp}}(\mu )\) from [28, Theorem 1] cannot be replaced by compactness of the set of its accumulation points.
4.3 Ignjatović Conjecture
In this section we show the relation between Theorem 4.6 and the conjecture due to Ignjatović [3, Conjecture 1].
Conjecture 4.15
(Ignjatović, 2016) Suppose that
- \((\mathcal {C}_1)\):
-
\( \lim _{n \rightarrow \infty } a_n = \infty \);
- \((\mathcal {C}_2)\):
-
\( \lim _{n \rightarrow \infty } \Delta a_n = 0 \);
- \((\mathcal {C}_3)\):
-
There exist \(n_0, m_0\) such that \(a_{n+m} > a_n\) holds for all \(n \ge n_0\) and all \(m \ge m_0\);
- \((\mathcal {C}_4)\):
-
\( \sum _{n=0}^\infty \frac{1}{a_n} = \infty \);
- \((\mathcal {C}_5)\):
-
There exists \(\kappa > 1\) such that \(\sum _{n=0}^\infty \frac{1}{a_n^\kappa } < \infty \);
- \((\mathcal {C}_6)\):
-
\( \sum _{n=0}^\infty \frac{|\Delta a_n|}{a_n^2} < \infty \);
- \((\mathcal {C}_7)\):
-
\( \sum _{n=0}^\infty \frac{\big |\Delta ^2 a_n \big |}{a_n} < \infty \).
If
then for any \(x \in \mathbb {R}\), the limit
exists and is positive.
In [3, Corollary 3] the conclusion was shown to hold in the case \(b_n \equiv 0\). Later, in [23, Corollary 3], the result was extended to a more general class of sequences with
and it was shown that
Our results imply the following corollary.
Corollary 4.16
Let N be a positive integer and let \(r \ge 1\). Suppose that for each \(i \in \{0, 1, \ldots , N-1\}\),
and
If \(|q|<2\) with
and the Carleman condition is satisfied, then
locally uniformly with respect to \(x \in \mathbb {R}\).
Proof
Let
By (3.1),
Hence, the conclusion follows from Theorem 4.6 provided that we can show
To do so, let us observe that
Hence, by Lemma 3.2
where \((U_n : n \in \mathbb {N}_0)\) is the sequence of the Chebyshev polynomials of the second kind (see [13, formula (1.6)]). By [13, formula (1.7)]
Hence, (4.29) implies (4.31). The proof is complete. \(\square \)
Remark 4.17
The case when (4.29) is violated is more complicated and demands stronger hypotheses. We refer to [26] for more details.
As it was shown in [23, Section 4] conditions \((\mathcal {C}_1)\)–\((\mathcal {C}_7)\) imply the hypotheses of Corollary 4.16 with \(b_n \equiv 0\), \(N=1\) and \(r=1\). On the other hand, in Conjecture 4.15 no regularity assumptions on \((b_n)\) were imposed, whereas in Corollary 4.16 we asked for
for each \(i \in \{0, 1, \ldots , N-1\}\). The following example illustrates that some regularity assumption on \((b_n)\) is necessary.
Example 4.18
Let
Then the measure \(\mu \) is absolutely continuous on \(\mathbb {R}\setminus [0, 1]\) with continuous and positive density, 0 is not a mass point of \(\mu \), and
For the proof, we set
that is
Since
we obtain
Consequently,
Therefore, by [24, Theorem D] and [24, Proposition 11], the measure \(\mu \) is purely absolutely continuous on \(\Lambda \) with positive continuous density proving the first assertion.
Next, let us observe that, by (4.33),
Thus
and
By (4.34) and (4.35), \((p_{2n+1}(0) : n \in \mathbb {N}_0)\) satisfies
It can be verified that
is the only solution of (4.36). Hence,
Using Stirling’s formula, we can find that
Now, by the Stolz–Cesàro theorem
proving (4.32). Moreover, by (4.37), the sequence \(\big ( p_n(0) : n \in \mathbb {N}_0 \big )\) is not square summable. Hence, 0 is not a mass point of \(\mu \).
Remark 4.19
It turns out that in the previous example the measure \(\mu \) on (0, 1) can have only mass points without any accumulation points on this set (see [25]). Moreover, we have recently shown [26] that in this case
locally uniformly with respect to \(x \in \mathbb {R}\setminus [0,1]\). So the value of the limit is different than in (4.30).
5 Christoffel–Darboux Kernel for \(\mathcal {D}_1\)
For \(\mathcal {D}_1\) class, we can describe the speed of convergence in Theorem 4.2 and Theorem 4.4. First, let us show a refined asymptotic of polynomials.
5.1 Asymptotics of Polynomials
Theorem 5.1
Let N be a positive integer and \(i \in \{0, 1, \ldots , N-1\}\). Suppose that K is a compact interval with non-empty interior contained in
Assume that
and
Let \(\mathcal {X}\) denote the limit of \((X_{jN+i} : j \in \mathbb {N})\). Then there is a probability measure \(\nu \) such that \((p_n : n \in \mathbb {N}_0)\) are orthonormal in \(L^2(\mathbb {R}, \nu )\) which is purely absolutely continuous with continuous and positive density \(\nu '\) on K. Moreover, there are \(M > 0\) and a real continuous function \(\eta : K \rightarrow \mathbb {R}\), such that for all \(k \ge M\),
where
and
Proof
Let us fix a compact interval K with non-empty interior. Since \(\mathcal {X}\) is the uniform limit of \((X_{jN+i} : j \in \mathbb {N})\), there are \(\delta > 0\) and \(M > 0\) such that for all \(x \in K\) and \(k \ge M\),
Therefore, the matrix \(X_{kN+i}(x)\) has two eigenvalues \(\lambda _{kN+i}\) and \(\overline{\lambda _{kN+i}}\) where
Let us next observe that for \(k \ge M\),
Moreover,
where
which is well-defined since \([X_{kN+i}(x)]_{1,2} \ne 0\) for any \(x \in K\). Let
We claim the following holds true.
Claim 5.2
There is \(c > 0\) such that for all \(m \ge n \ge M\), and \(x \in K\),
We start by writing
Let us now introduce two auxiliary functions
and
Notice that
where
In view of [27, Propositon 1], we have
thus
Next, by [27, Claim 2], there is \(c > 0\) such that for all \(n \ge M\) and \(x \in K\),
and consequently, for all \(m \ge n \ge M\),
In particular, we obtain
Next, we notice that
Since
we obtain
which together with (5.3) implies that for all \(m \ge n > M\) and \(x \in K\),
In particular, the sequence \((\phi _{mN+i} : m \in \mathbb {N})\) converges. Let us denote by \(\varphi \) its limit. Since polynomials \(p_n\) have real coefficients, by taking the imaginary part we arrive at
Because
we obtain
Therefore, by [27, Theorem 6],
Observe that, by (5.2),
thus
Since
we finish the proof. \(\square \)
Remark 5.3
Under the assumption of Theorem 5.1, we have
where
Indeed, since for \(m \ge n \ge M\),
we get
which proves our statement.
5.2 Christoffel Functions
We are now in the position to prove the main theorem of this section.
Theorem 5.4
Let N be a positive integer and \(i \in \{0, 1, \ldots , N-1\}\). Suppose that K is a compact interval with non-empty interior and contained in
Assume that
and
If
then
where \(\mathcal {X}\) is the limit of \(\left( X_{jN+i} : j \in \mathbb {N}\right) \), and
Proof
Let \(K \subset \Lambda \) be a compact interval with non-empty interior. By Theorem 5.1, there are \(c > 0\) and \(M \in \mathbb {N}\) such that for all \(k \ge M\),
where
In view of the identity \(2 \sin ^2(x) = 1 - \cos (2x)\), we get
Since there is \(c > 0\) such that
by Lemma 4.1 and Remark 5.3, we obtain
which completes the proof. \(\square \)
Theorem 5.5
Let A be a Jacobi matrix with N-periodically modulated entries. Suppose that for each \(i \in \{0, 1, \ldots , N-1\}\),
and
If \(|{{\text {tr}}\mathfrak {X}_0(0)} | < 2\) then
where \(\omega \) is the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\) with \(\mathfrak {A}\) being the Jacobi matrix associated with \((\alpha _n : n \in \mathbb {N}_0)\) and \((\beta _n : n \in \mathbb {N}_0)\),
and for each compact interval \(K \subset \mathbb {R}\) with non-empty interior
Proof
Let K be a compact interval with non-empty interior and contained in \(\mathbb {R}\). By Remark 4.5, for each \(i \in \{0, 1, \ldots , N-1\}\), the sequence \((X_{jN+i} : j \in \mathbb {N})\) belongs to \(\mathcal {D}_1\big (K, {\text {GL}}(2, \mathbb {R}) \big )\), thus, by Proposition 3.8, we have
uniformly with respect to \(x \in K\). Since
we have
By Theorem 5.4,
where
For \(k \in \mathbb {N}_0\) and \(i \in \{0, 1, \ldots , N-1\}\) we write
Since
we obtain
where
We next claim the following holds true. \(\square \)
Claim 5.6
For each \(i', i'' \in \{0, 1, \ldots , N-1\}\),
For the proof let us observe that, by Proposition 3.7, we have
Therefore,
which implies (5.5).
Now, using Claim 5.6, we can write
Hence, by (5.4), we obtain
which together with (3.1), concludes the proof. \(\square \)
The following theorem has a proof analogous to Theorem 5.5.
Theorem 5.7
Let A be a Jacobi matrix with asymptotically N-periodic entries. Suppose that for each \(i \in \{0, 1, \ldots , N-1\}\),
Let K be a compact interval with non-empty interior contained in
Then
where \(\omega \) is the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\) with \(\mathfrak {A}\) being the Jacobi matrix associated to \((\alpha _n : n \in \mathbb {N}_0)\) and \((\beta _n : n \in \mathbb {N}_0)\),
and
In the following two examples we want to compare the estimate (5.6) with some known results.
Example 5.8
(Generalized Jacobi) Let h be a real-analytic positive function on the neighborhood of \([-1, 1]\). Let \(\mu \) be a probability measure supported on \([-1,1]\) with the density
where \(\gamma _1, \gamma _2 > -1\), and c is the normalizing constant. Then (see [6, Theorem 1.10])
Therefore, by Theorem 5.5 we obtain
Hence, we obtain the same rate as in [7, Theorem 1.1(a)].
Example 5.9
(Pollaczek-type) Let \(\mu \) be a probability measure supported on \([-1,1]\) with the density
where \(\gamma \in (0, \tfrac{1}{2})\), and c is the normalizing constant. Then (see, [32, Corollary 4])
Hence, Theorem 5.5 implies
It should be compared with [31, Theorem 1(i)].
5.3 Auxiliary Results
Lemma 5.10
Let \((\gamma _k : k \ge 0)\) be a sequence of positive numbers such that
Assume that \((\theta _n : n \ge 0)\) is a sequence of continuous functions on some open set \(U \subset \mathbb {R}^d\) with values in \((0, 2\pi )\). Suppose that there is \(\theta : U \rightarrow (0, 2\pi )\) such that
locally uniformly with respect to \(x \in U\). Let \((r_n : n \in \mathbb {N})\) be a sequence of positive numbers such that
For \(x \in U\), and \(a, b \in \mathbb {R}\), we set
Then for each compact subset \(K \subset U\), \(L > 0\), and any function \(\sigma : U \rightarrow \mathbb {R}\),
uniformly with respect to \(x \in K\), and \(a, b \in [-L, L]\).
Proof
Let us fix K a compact subset of U and \(L > 0\). Select \(R > 0\) so that
for all \(x \in K\), and let \(N \in \mathbb {N}\) be such that \(r_n \ge R\) for all \(n \ge N\). For \((x, a, b) \in U \times (-2L, 2L)^2\), we set
Then
uniformly with respect to \((x, a, b) \in K \times [-L, L]^2\). By Lemma 4.1, there is \(c > 0\) such that
for all \(x \in K\), \(a, b \in [-L, L]\), and \(n \in \mathbb {N}\). Since
by (5.8),
for all \(x \in K\), \(a, b \in [-L, L]\), and \(n \ge N\). Finally,
which together with (5.9) implies that there are \(c > 0\) and \(N \in \mathbb {N}\), such that for any function \(\sigma : U \rightarrow \mathbb {R}\) and all \(x \in K\), \(a, b \in [-L, L]\) and \(n \ge N\),
Finally, (5.7) follows from (5.10) by the Stolz–Cesàro theorem, since
and
\(\square \)
Theorem 5.11
Let U be an open subset of \(\mathbb {R}\). Let \((\gamma _k : k \ge 0)\) be a sequence of positive numbers such that
Assume that \((\theta _k : k \ge 0)\) is a sequence of \(\mathcal {C}^2(U)\) functions with values in \((0, 2 \pi )\) such that for each compact set \(K \subset U\) there are functions \(\theta : K \rightarrow (0, 2\pi )\) and \(\psi : K \rightarrow (0, \infty )\), and \(c > 0\) such that
-
(a)
\( \lim _{n \rightarrow \infty } \sup _{x \in K}{\big |\theta _n(x) - \theta (x)\big |} = 0, \)
-
(b)
\( \lim _{n \rightarrow \infty } \sup _{x \in K}{\big | \gamma _n^{-1} \cdot \theta _n'(x) - \psi (x) \big |} =0, \)
-
(c)
\( \sup _{n \in \mathbb {N}} \sup _{x \in K}{\big |\gamma _n^{-2} \cdot \theta _n''(x)\big |} \le c. \)
For \(x \in U\) and \(a, b \in \mathbb {R}\) we set
Then for any continuous function \(\sigma : U \rightarrow \mathbb {R}\),
locally uniformly with respect to \(x \in U\), and \(a, b \in \mathbb {R}\).
Proof
Let us fix a compact set \(K \subset U\) and \(L > 0\). We write
In view of Corollary 5.10, the second term has no contribution to the limit (5.12). To deal with the first term in (5.13), we write
Now, by the continuity of \(\sigma \), the second term in (5.14) has no contribution to the limit (5.12). Hence, it is enough to show that
uniformly with respect to \(x \in K\), and \(a, b \in [-L, L]\). We first prove the following claim. \(\square \)
Claim 5.12
There is \(c > 0\) such that for all \(j \in \mathbb {N}\) and \(n \in \mathbb {N}\),
For the proof, let us write Taylor’s polynomial for \(\theta _j\) centered at x, that is,
where
Therefore,
which leads to (5.15).
Let us now observe that, by the mean value theorem and Claim 5.12, we have
Hence,
Now, by the Stolz–Cesàro theorem, we have
thus, by repeated application of the Stolz–Cesàro theorem we obtain
In view of (c), it is enough to show that
For the proof, we write
We now claim the following.
Claim 5.13
For each \(m \in \mathbb {N}\),
By (b) and the Stolz–Cesàro theorem, we get
Since
we get
Therefore, another application of the Stolz–Cesàro theorem leads to (5.16).
Let us notice that for some \(c > 0\),
thus, we have the estimate
Hence, by the dominated convergence theorem and Claim 5.13, we can compute
which finishes the proof of the theorem. \(\square \)
5.4 Christoffel–Darboux Kernel
Let us recall the definition
Proposition 5.14
Let A be a Jacobi matrix with N-periodically modulated entries. Then for every compact subset \(K \subset \mathbb {R}\), we have
and
for some \(c > 0\).
Proof
Let us fix \(i \in \{ 0, 1, \ldots , N-1 \}\). Since
by Proposition 3.7, we conclude that
The chain rule applied to (5.1) leads to
thus, by (5.20) and Corollary 3.10, we obtain (5.17). Consequently, in view of (5.19),
Therefore, the estimate (5.18) is a consequence of Corollary 3.10. \(\square \)
Theorem 5.15
Let A be a Jacobi matrix with N-periodically modulated entries. Suppose that for each \(i \in \{0, 1, \ldots , N-1\}\),
and
If \(|{{\text {tr}}\mathfrak {X}_0(0)} | < 2\), then
locally uniformly with respect to \(x, u, v \in \mathbb {R}\), where
and \(\omega \) is the equilibrium measure corresponding to \(\sigma _{\text {ess}}(\mathfrak {A})\) with \(\mathfrak {A}\) being the Jacobi matrix associated with \((\alpha _n : n \in \mathbb {N}_0)\) and \((\beta _n : n \in \mathbb {N}_0)\).
Proof
Let us fix a compact set \(K \subset \Lambda \) with non-empty interior and \(L > 0\). Let \(\tilde{K} \subset \Lambda \) be a compact set containing K in its interior. There is \(n_0 > 0\) such that for all \(x \in K\), \(n \ge n_0\), \(u \in [-L, L]\), and \(i \in \{0, 1, \ldots , N-1\}\),
Given \(u, v \in [-L, L]\), we set
Remark 4.5 together with (5.21) entails that \((X_{jN+i} : j \in \mathbb {N})\) belongs to \(\mathcal {D}_1\). Moreover,
uniformly with respect to \(x \in K\). Let us recall that
In view of (4.12), the Carleman condition (5.22) implies that
Hence, by Theorem 5.1, there are \(c > 0\) and \(M \in \mathbb {N}\) such that for all \(x, y \in \tilde{K}\) and \(k \ge M\),
where
Hence, we obtain
Let
By Proposition 5.14
Therefore, by Theorem 5.11 we get
Now, by uniformness and (4.12), for any \(i' \in \{0, 1, \ldots , N-1\}\), we obtain
uniformly with respect to \(x \in K\) and \(u, v \in [-L, L]\). Here, we have also used that
Since
by (5.25) and (5.24), we obtain
which together with (3.1) concludes the proof. \(\square \)
Proposition 5.16
Let A be a Jacobi matrix that is N-periodic blend. Let K be a non-empty compact interval contained in
where \(\mathcal {X}_1\) is the limit of \((X_{j(N+2)+1} : j \in \mathbb {N}_0)\). Then for each \(i \in \{1, 2, \ldots , N\}\),
and
for some \(c > 0\).
Proof
Let us fix \(i \in \{1, 2, \ldots , N\}\). Since
we conclude that
The chain rule applied to (5.1) leads to
thus, by (5.29) and Corollary 3.18, we obtain (5.26). Consequently, in view of (5.28),
Therefore, the estimate (5.27) is a consequence of Corollary 3.18. \(\square \)
Theorem 5.17
Let A be a Jacobi matrix that is N-periodic blend. Suppose that for each \(i \in \{0, 1, \ldots , N-1\}\),
and
Let K be a compact interval with non-empty interior contained in
where \(\mathcal {X}_1\) is the limit of \((X_{j(N+2)+1} : j \in \mathbb {N}_0)\). Then
locally uniformly with respect to \(x \in K\), and \(u, v \in \mathbb {R}\), where \(\omega \) is the equilibrium measure corresponding to \(\overline{\Lambda }\), and
Proof
Let \(K \subset \Lambda \) be a compact interval with non-empty interior and let \(L > 0\). Let \(\tilde{K} \subset \Lambda \) be a compact set containing K in its interior. There is \(n_0 > 0\) such that for all \(x \in K\), \(n \ge n_0\), \(i \in \{0, 1, \ldots , N+1\}\), and \(u \in [-L, L]\),
Given \(x \in K\) and \(u, v \in [-L, L]\), we set
For each \(i \in \{1, 2, \ldots , N\}\) and \(i' \in \{0, 1, \ldots , N+1\}\), by the reasoning analogous to the proof of Theorem 5.15 one can show that
uniformly with respect to \(x \in K\) and \(u, v \in [-L, L]\). By Claim 4.11, for each \(i \in \{1, 2, \ldots , N\}\) the sequence \((X_{j(N+2)+i} : j \in \mathbb {N}_0)\) belongs to \(\mathcal {D}_1\big (K, {\text {GL}}(2, \mathbb {R}) \big )\) and converges to \(\mathcal {X}_i\) satisfying
For \(i = 0\), by (4.26) and Claim 4.12,
Therefore,
Similarly, for \(i = N+1\), one can show
Now, let \(i \in \{0, 1, \ldots , N+1\}\). We write
By (4.26), (5.31) and (5.32), we obtain
uniformly with respect to \(x, y \in \tilde{K}\). Therefore, by (5.30) and (4.12),
which together with Theorem 3.13 finishes the proof. \(\square \)
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The first author was partially supported by the Foundation for Polish Science (FNP) and by long term structural funding, Methusalem grant of the Flemish Government.
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Świderski, G., Trojan, B. Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I. Constr Approx 54, 49–116 (2021). https://doi.org/10.1007/s00365-020-09519-w
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DOI: https://doi.org/10.1007/s00365-020-09519-w