Spectral Distribution Results Beyond the Algebra Generated by Variable-Coefficient Toeplitz Sequences: The GLT Approach

Abstract

We show that, under suitable assumptions on the function $a:[0,1]^2\times [-\pi ,\pi ]\rightarrow \mathbb C$, the sequence of variable-coefficient Toeplitz matrices generated by a is a generalized locally Toeplitz (GLT) sequence with symbol $a(x,x,\theta )$. This property, in combination with the theory of GLT sequences, allows us to derive a lot of spectral distribution (Szegő-type) results for various sequences of matrices, including those obtained from algebraic and non-algebraic operations on variable-coefficient Toeplitz sequences.

1 Introduction and Main Result

Let $\mathcal L_1$ be the space of functions $a:[0,1]^2\times [-\pi ,\pi ]\rightarrow \mathbb C$ such that $a(x,y,\cdot )\in L^1([-\pi ,\pi ])$ for all $(x,y)\in [0,1]^2$. Every $a\in \mathcal L_1$ can be formally represented by its Fourier series in the last variable,

$$\begin{aligned} a(x,y,\theta )=\sum _{k\in \mathbb Z}\hat{a}_k(x,y)\mathrm{e}^{\mathrm{i}k\theta }, \end{aligned}$$

where

$$\begin{aligned} \hat{a}_k(x,y):=\frac{1}{2\pi }\int _{-\pi }^\pi a(x,y,\theta )\mathrm{e}^{-\mathrm{i}k\theta }d\theta ,\quad k\in \mathbb Z, \end{aligned}$$

are the Fourier coefficients of $a(x,y,\cdot )$. For $n\ge 2$, the $n\times n$ variable-coefficient Toeplitz matrix associated with a is defined as

$$\begin{aligned} A_n(a):=\left[ \hat{a}_{i-j}\Bigl (\frac{i-1}{n-1},\frac{j-1}{n-1}\Bigr )\right] _{i,j=1}^n. \end{aligned}$$

Throughout this paper, by a matrix-sequence (or sequence of matrices) we mean a sequence of the form $\{A_n\}_n$, in which the n-th term $A_n$ is a $n\times n$ matrix. The specific matrix-sequence $\{A_n(a)\}_n$ is referred to as the variable-coefficient Toeplitz sequence associated with a, which in turn is called the generating function of $\{A_n(a)\}_n$.

Variable-coefficient Toeplitz matrices are also known as generalized convolutions and appear in many different contexts. As testified by the literature, this kind of matrices has received a certain attention in the last years; see, e.g., [3,4,5, 10, 17,18,19, 21, 22]. We also refer the reader to [6, 8, 9] for a numerical-oriented literature about orthogonal polynomials with varying recurrence coefficients: the associated Jacobi matrices can be interpreted as tridiagonal symmetric Toeplitz matrices with variable coefficients.

In this paper we show that, under suitable assumptions on a, $\{A_n(a)\}_n$ is a generalized locally Toeplitz (GLT) sequence with symbol $a(x,x,\theta )$. This property, in combination with the theory of GLT sequences [7], allows one to derive a lot of spectral distribution (Szegő-type) results for various matrix-sequences, including those obtained from algebraic and non-algebraic operations on variable-coefficient Toeplitz sequences. Let us formulate our main result in a precise way. For any $\varepsilon >0$, define the strip

$$\begin{aligned} S_\varepsilon :=\{(x,y)\in [0,1]^2:\ |x-y|\le \varepsilon \}. \end{aligned}$$

Set

$$\begin{aligned} \mathcal W&:=\biggl \{a\in \mathcal L_1:\quad \sum _{k\in \mathbb Z}\sup _{(x,y)\in [0,1]^2}|\hat{a}_k(x,y)|<\infty ,\\&\qquad \text {for all }k\in \mathbb Z \,\text {there is}\,\varepsilon (k)>0\,\text {such that}\,\hat{a}_k(\cdot ,\cdot )\,\text{ is } \text{ continuous } \text{ on }\, S_{\varepsilon (k)}\biggr \}, \end{aligned}$$

$$\begin{aligned} \mathcal X:= & {} \biggl \{\,\sum _{r=1}^q\alpha _r(x,y)\beta _r(\theta ):\quad \alpha _r\in C([0,1]^2) \text{ and } \beta _r\in L^2([-\pi ,\pi ])\\&\text{ for } \text{ all } \, r=1,\ldots ,q,\quad q\in \mathbb N\biggr \}, \end{aligned}$$

$$\begin{aligned} \mathcal Y:= & {} \biggl \{\,\sum _{r=1}^q\alpha _r(x)\beta _r(y)\gamma _r(\theta ): \quad \alpha _r,\beta _r\in C([0,1]) \text{ and } \gamma _r\in L^1([-\pi ,\pi ])\\&\text{ for } \text{ all } r=1,\ldots ,q,\quad q\in \mathbb N\biggr \}. \end{aligned}$$

Note that $\mathcal W$ contains every continuous function $a\in C([0,1]^2\times [-\pi ,\pi ])$ satisfying the Wiener-type condition

$$\begin{aligned} \sum _{k\in \mathbb Z}\sup _{(x,y)\in [0,1]^2}|\hat{a}_k(x,y)|<\infty . \end{aligned}$$

(1)

Our main result is the following.

Theorem 1

If $a\in \mathcal W\cup \mathcal X\cup \mathcal Y$ then $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$, i.e., $\{A_n(a)\}_n$ is a GLT sequence with symbol $a(x,x,\theta )$.

The meaning of this statement will become clear in Sect. 2, where we present a full overview of the theory of GLT sequences. This is done because, in addition to the proof of the main result and its corollaries, a further purpose of this paper is to provide the reader with a complete and concise picture of the theory of GLT sequences. In Sect. 3 we prove the main result, and in Sect. 4 we investigate some of its consequences. Finally, in Sect. 5 we make some concluding remarks and illustrate possible future lines of research.

2 GLT Sequences

In this section we present the theory of GLT sequences, which goes back to the pioneering works by Tilli [20] and the second author [14, 15], and is developed in detail in [7]. Despite its conciseness, our presentation contains all the necessary ingredients for a full understanding of this theory. All the content of this section can be found in [7].

Singular value and eigenvalue distribution of a matrix-sequence Let $\mu _k$ be the Lebesgue measure in $\mathbb R^k$ and let $C_c(\mathbb C)$ be the space of continuous complex-valued functions with bounded support defined on $\mathbb C$. If A is a $n\times n$ matrix, the singular values and the eigenvalues of A are denoted by $\sigma _1(A),\ldots ,\sigma _n(A)$ and $\lambda _1(A),\ldots ,\lambda _n(A)$, respectively.

Definition 1

Let $\{A_n\}_n$ be a matrix-sequence and let $f:D\subset \mathbb R^k\rightarrow \mathbb C$ be a measurable function defined on a set D with $0<\mu _k(D)<\infty$.

• We say that $\{A_n\}_n$ has a singular value distribution described by f, and we write $\{A_n\}_n\sim _\sigma f$, if

  $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=1}^nF(\sigma _i(A_n))=\frac{1}{\mu _k(D)}\int _DF(|f(y_1,\ldots ,y_k)|)dy_1\ldots dy_k \end{aligned}$$

  for all $F\in C_c(\mathbb C)$.

• We say that $\{A_n\}_n$ has an eigenvalue distribution described by f, and we write $\{A_n\}_n\sim _\lambda f$, if

  $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=1}^nF(\lambda _i(A_n))=\frac{1}{\mu _k(D)}\int _DF(f(y_1,\ldots ,y_k))dy_1\ldots dy_k \end{aligned}$$

  for all $F\in C_c(\mathbb C)$.

Toeplitz matrices If $n\in \mathbb N$ and $f:[-\pi ,\pi ]\rightarrow \mathbb C$ is a function in $L^1([-\pi ,\pi ])$, the Toeplitz matrix $T_n(f)$ associated with f is the $n\times n$ matrix defined as

$$\begin{aligned} T_n(f):=\bigl [\hat{f}_{i-j}\bigr ]_{i,j=1}^n, \end{aligned}$$

where the numbers $\hat{f}_k$ are the Fourier coefficients of f,

$$\begin{aligned} \hat{f}_k:=\frac{1}{2\pi }\int _{-\pi }^\pi f(\theta )\mathrm{e}^{-\mathrm{i} k\theta }d\theta ,\qquad k\in \mathbb Z. \end{aligned}$$

It is not difficult to see that all the matrices $T_n(f)$ are Hermitian when f is real almost everywhere (a.e.). Moreover, the variable-coefficient Toeplitz matrix $A_n(a)$ coincides with the Toeplitz matrix $T_n(a)$ whenever a is independent of x and y. The next proposition is proved in [13, 16] and contains important inequalities involving Toeplitz matrices and the so-called Schatten p-norms. If A is a $n\times n$ matrix and $1\le p\le \infty$, the Schatten p-norm of A is denoted by $\Vert A\Vert _p$ and is defined as the p-norm of the vector $(\sigma _1(A),\ldots ,\sigma _n(A))$ formed by the singular values of A; see [1]. The Schatten $\infty$-norm $\Vert A\Vert _\infty$ is the largest singular value of A and coincides with the spectral norm $\Vert A\Vert$. The Schatten 1-norm $\Vert A\Vert _1$ is the sum of all the singular values of A and is often referred to as the trace-norm of A. The Schatten 2-norm $\Vert A\Vert _2$ can be proved to be equal to $(\sum _{i,j=1}^n|a_{ij}|^2)^{1/2}$, i.e., it is just the Frobenius norm of A. The Schatten p-norms satisfy the Hölder-type inequality

$$\begin{aligned} \Vert AB\Vert _1\le \Vert A\Vert _p\Vert B\Vert _q, \quad A,B\in \mathbb C^{n\times n}, \end{aligned}$$

(2)

where $1\le p,q\le \infty$ are conjugate exponents, i.e., $1/p+1/q=1$; see [1, Problem III.6.2 and Corollary IV.2.6]. For convenience, throughout this paper we use the natural convention $1/\infty =0$.

Proposition 1

Let $f\in L^p([-\pi ,\pi ])$, $n\in \mathbb N$ and $1\le p\le \infty$. Then

$$\begin{aligned} \Vert T_n(f)\Vert _p\le \frac{n^{1/p}}{(2\pi )^{1/p}}\Vert f\Vert _{L^p}. \end{aligned}$$

Diagonal sampling matrices If $n\in \mathbb N$ and a is a function from [0, 1] to $\mathbb C$, the diagonal sampling matrix $D_n(a)$ associated with a is the $n\times n$ diagonal matrix defined as

$$\begin{aligned} D_n(a):=\mathop {\mathrm{diag}}\limits _{i=1,\ldots ,n}a\Bigl (\frac{i}{n}\Bigr ). \end{aligned}$$

Zero-distributed sequences Any matrix-sequence $\{Z_n\}_n$ such that $\{Z_n\}_n\sim _\sigma 0$ is referred to as a zero-distributed sequence. In other words, $\{Z_n\}_n$ is zero-distributed if and only if $\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=1}^nF(\sigma _i(Z_n))=F(0)$ for all $F\in C_c(\mathbb C)$. An important characterization of zero-distributed sequences is stated in Proposition 2 together with a useful criterion to understand whether a given matrix-sequence is zero-distributed.

Proposition 2

The following properties hold.

• $\{Z_n\}_n$ is zero-distributed if and only if $Z_n=R_n+N_n$ with $\displaystyle \lim _{n\rightarrow \infty }\frac{\mathrm{rank}(R_n)}{n}=\lim _{n\rightarrow \infty }\Vert N_n\Vert =0$.

• $\{Z_n\}_n$ is zero-distributed if $\displaystyle \lim _{n\rightarrow \infty }\frac{\Vert Z_n\Vert _p}{n^{1/p}}=0$ for some $p\in [1,\infty ]$.

Approximating classes of sequences The notion of approximating classes of sequences (a.c.s.es) is due to the second author [12] and is deeply studied in [7]. This is indeed the fundamental concept on which the theory of GLT sequences is based.

Definition 2

Let $\{A_n\}_n$ be a matrix-sequence. A sequence of matrix-sequences $\{\{B_{n,m}\}_n\}_m$ is said to be an approximating class of sequences (a.c.s.) for $\{A_n\}_n$ if the following condition is met: for every m there exists $n_m$ such that, for $n\ge n_m$,

$$\begin{aligned} A_n=B_{n,m}+R_{n,m}+N_{n,m},\qquad \mathrm{rank}(R_{n,m})\le c(m)n,\qquad \Vert N_{n,m}\Vert \le \omega (m), \end{aligned}$$

where the quantities $n_m,\,c(m),\,\omega (m)$ depend only on m, and

$$\begin{aligned} \lim _{m\rightarrow \infty }c(m)=\lim _{m\rightarrow \infty }\omega (m)=0. \end{aligned}$$

Roughly speaking, $\{\{B_{n,m}\}_n\}_m$ is an a.c.s. for $\{A_n\}_n$ if, for large m, the sequence $\{B_{n,m}\}_n$ approximates $\{A_n\}_n$ in the sense that $A_n$ is eventually equal to $B_{n,m}$ plus a small-rank matrix (with respect to the matrix size n) plus a small-norm matrix. It turns out that the notion of a.c.s.es is a notion of convergence in the space of matrix-sequences

$$\begin{aligned} \mathscr {E}:=\bigl \{\{A_n\}_n:\ \{A_n\}_n \text{ is } \text{ a } \text{ matrix-sequence }\bigr \}. \end{aligned}$$

(3)

More precisely, let $p(\cdot )$ be the function defined on $\mathscr {E}$ by

$$\begin{aligned} p(\{A_n\}_n):= & {} \inf \left\{ \limsup _{n\rightarrow \infty }\biggl (\frac{\mathrm{rank}(R_n)}{n}+\Vert N_n\Vert \biggr ):\quad \{A_n\}_n=\{R_n+N_n\}_n,\right. \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left. \quad \{R_n\}_n,\{N_n\}_n\in \mathscr {E}\right\} , \end{aligned}$$

(4)

and set

$$\begin{aligned} d_\mathrm{a.c.s.}(\{A_n\}_n,\{B_n\}_n):=p(\{A_n-B_n\}_n),\qquad \{A_n\}_n,\{B_n\}_n\in \mathscr {E}. \end{aligned}$$

(5)

Then $d_\mathrm{a.c.s.}$ is a distance on $\mathscr {E}$ which turns $\mathscr {E}$ into a topological (pseudometric) space $(\mathscr {E},\tau _\mathrm{a.c.s.})$ where the statement “$\{\{B_{n,m}\}_n\}_m$ converges to $\{A_n\}_n$” is equivalent to “$\{\{B_{n,m}\}_n\}_m$ is an a.c.s. for $\{A_n\}_n$”. In particular, we can reformulate Definition 2 in the following way: a sequence of matrix-sequences $\{\{B_{n,m}\}_n\}_m$ is said to be an a.c.s. for $\{A_n\}_n$ if $\{B_{n,m}\}_n$ converges to $\{A_n\}_n$ in $(\mathscr {E},\tau _\mathrm{a.c.s.})$ as $m\rightarrow \infty$, i.e., if $d_\mathrm{a.c.s.}(\{B_{n,m}\}_n,\{A_n\}_n)\rightarrow 0$ as $m\rightarrow \infty$. The theory of a.c.s.es may then be interpreted as an approximation theory for matrix-sequences, and for this reason we will use the convergence notation

$$\begin{aligned} \{B_{n,m}\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n\}_n \end{aligned}$$

to indicate that $\{\{B_{n,m}\}_n\}_m$ is an a.c.s. for $\{A_n\}_n$. A useful criterion to identify a.c.s.es is provided in the next proposition.

Proposition 3

Let $\{A_n\}_n$ be a matrix-sequence, let $\{\{B_{n,m}\}_n\}_m$ be a sequence of matrix-sequences, and let $p\in [1,\infty ]$. Suppose that for every m there exists $n_m$ such that, for $n\ge n_m$,

$$\begin{aligned} \Vert A_n-B_{n,m}\Vert _p\le \varepsilon (m,n)n^{1/p}, \end{aligned}$$

where $\displaystyle \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\varepsilon (m,n)=0$. Then $\{B_{n,m}\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n\}_n$.

Remark 1

It sometimes happens to hear about a.c.s.es parameterized by a positive $\varepsilon \rightarrow 0$ rather than an integer $m\rightarrow \infty$. A.c.s.es parameterized by a positive $\varepsilon \rightarrow 0$ appear, for example, in the definition of GLT sequences [7, Definition 8.1]. Thanks to the above topological interpretation, it is easy to guess what an a.c.s. of this kind should be. Here is the definition: a class of matrix-sequences $\{\{B_{n,\varepsilon }\}_n\}_{\varepsilon >0}$ is said to be an a.c.s. for $\{A_n\}_n$ as $\varepsilon \rightarrow 0$ if $\{B_{n,\varepsilon }\}_n$ converges to $\{A_n\}_n$ in $(\mathscr {E},\tau _\mathrm{a.c.s.})$ as $\varepsilon \rightarrow 0$, i.e., if $d_\mathrm{a.c.s.}(\{B_{n,\varepsilon }\}_n,\{A_n\}_n)\rightarrow 0$ as $\varepsilon \rightarrow 0$.

GLT sequences A GLT sequence $\{A_n\}_n$ is a special matrix-sequence equipped with a measurable function $\kappa :[0,1]\times [-\pi ,\pi ]\rightarrow \mathbb C$, the so-called symbol (or kernel). We use the notation $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ to indicate that $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$. The symbol of a GLT sequence is unique, in the sense that if $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $\{A_n\}_n\sim _\mathrm{GLT}\xi$ then $\kappa =\xi$ a.e. in $[0,1]\times [-\pi ,\pi ]$.

To formulate the precise definition of GLT sequences, we introduce the set of measurable functions

$$\begin{aligned} \mathcal M:=\{\kappa :[0,1]\times [-\pi ,\pi ]\rightarrow \mathbb C:\ \kappa \text{ is } \text{ measurable }\}. \end{aligned}$$

This is a *-algebra with respect to the natural operations, and it is also a topological (pseudometric) space with respect to the distance

$$\begin{aligned} d_\mathrm{measure}(\kappa ,\xi )=\int _{[0,1]\times [-\pi ,\pi ]}\frac{|\kappa -\xi |}{1+|\kappa -\xi |}, \end{aligned}$$

which induces the topology $\tau _\mathrm{measure}$ of convergence in measure [2, 11]. Consider the product space $\mathscr {E}\times \mathcal M$ equipped with the product topology $\tau _\mathrm{a.c.s.}\times \tau _\mathrm{measure}$ and with the structure of product *-algebra:

$$\begin{aligned} (\{A_n\}_n,\kappa )^*&:=(\{A_n^*\}_n,\overline{\kappa }),\\ \alpha (\{A_n\}_n,\kappa )+\beta (\{B_n\}_n,\xi )&:=(\{\alpha A_n+\beta B_n\}_n,\alpha \kappa +\beta \xi ),\\ (\{A_n\}_n,\kappa )(\{B_n\}_n,\xi )&:=(\{A_nB_n\}_n,\kappa \xi ). \end{aligned}$$

The set of “GLT pairs” is defined as the subset of $\mathscr {E}\times \mathcal M$ given by

$$\begin{aligned} \bigl \{(\{A_n\}_n,\kappa )\in \mathscr {E}\times \mathcal M:\ \{A_n\}_n\sim _\mathrm{GLT}\kappa \bigr \}. \end{aligned}$$

(6)

Let $C_\mathrm{a.e.}([0,1])$ be the space of functions $a:[0,1]\rightarrow \mathbb C$ that are continuous a.e. on [0, 1], i.e., the space of functions a from [0, 1] to $\mathbb C$ such that a is continuous at x for almost every $x\in [0,1]$, and consider the specific subset of $\mathscr {E}\times \mathcal M$ given by

$$\begin{aligned}&\bigl \{(\{T_n(f)\}_n,f):\ f\in L^1([-\pi ,\pi ])\bigr \}\cup \bigl \{(\{D_n(a)\}_n,a):\ a\in C_\mathrm{a.e.}([0,1])\bigr \}\nonumber \\&\cup \,\bigl \{(\{Z_n\}_n,0):\ \{Z_n\}_n\sim _\sigma 0\bigr \}. \end{aligned}$$

(7)

Definition 3

The set of GLT pairs (6) is the closed *-subalgebra of $\mathscr {E}\times \mathcal M$ generated by the pairs (7), i.e., the smallest closed *-subalgebra of $\mathscr {E}\times \mathcal M$ containing (7).

Definition 3 looks quite different from the traditional one [7, Definition 8.1]. However, as proved in [7, Sect. 8.5], the two definitions are equivalent. Moreover, Definition 3 is much easier to formulate and this is why we preferred it to [7, Definition 8.1].

The theory of GLT sequences is summarized in the following list of properties. Throughout this paper, the Moore–Penrose pseudoinverse of a matrix A is denoted by $A^\dag$; recall that $A^\dag =A^{-1}$ whenever A is invertible.

• GLT 1. If $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ then $\{A_n\}_n\sim _\sigma \kappa$. If, moreover, the matrices $A_n$ are Hermitian, then $\{A_n\}_n\sim _\lambda \kappa$.

• GLT 2. If $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $A_n=X_n+Y_n$, where

  • ${\bullet }$ $\Vert X_n\Vert ,\,\Vert Y_n\Vert \le C$ for some constant C independent of n,

  • ${\bullet }$ every $X_n$ is Hermitian,

  • ${\bullet }$ $\displaystyle \lim _{n\rightarrow \infty }\frac{\Vert Y_n\Vert _1}{n}=0$,

  then $\{A_n\}_n\sim _\lambda \kappa$.

• GLT 3. We have:

  • ${\bullet }$ $\{T_n(f)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=f(\theta )$ if $f\in L^1([-\pi ,\pi ])$;

  • ${\bullet }$ $\{D_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=a(x)$ if $a\in C_\mathrm{a.e.}([0,1])$;

  • ${\bullet }$ $\{Z_n\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=0$ if and only if $\{Z_n\}_n\sim _\sigma 0$.

• GLT 4. If $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ then $\{A_n^*\}_n\sim _\mathrm{GLT}\overline{\kappa }$.

• GLT 5. If $A_n=\sum _{i=1}^r\alpha _i\prod _{j=1}^{q_i}A_n^{(i,j)}$, with $r,q_1,\ldots ,q_r\in \mathbb N$, $\alpha _1,\ldots ,\alpha _r\in \mathbb C$, and $\{A_n^{(i,j)}\}_n\sim _\mathrm{GLT}\kappa _{ij}$, then $\{A_n\}_n\sim _\mathrm{GLT}\kappa :=\sum _{i=1}^r\alpha _i\prod _{j=1}^{q_i}\kappa _{ij}$.

• GLT 6. If $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $\kappa \ne 0$ a.e., then $\{A_n^\dag \}_n\sim _\mathrm{GLT}\kappa ^{-1}$.

• GLT 7. If $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and each $A_n$ is Hermitian, then $\{f(A_n)\}_n\sim _\mathrm{GLT}f(\kappa )$ for all continuous functions $f:\mathbb C\rightarrow \mathbb C$.

• GLT 8. $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ if and only if there exist GLT sequences $\{B_{n,m}\}_n\sim _\mathrm{GLT}\kappa _m$ such that $\{B_{n,m}\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n\}_n$ and $\kappa _m\rightarrow \kappa$ in measure over $[0,1]\times [-\pi ,\pi ]$.

The first properties GLT 1 – GLT 2 contain the main spectral distribution results for GLT sequences. GLT 3 lists the fundamental examples of GLT sequences, from which one can construct, via GLT 5, many other GLT sequences. Note that GLT 1 and GLT 3 imply the famous Szegő first limit theorem and the Avram–Parter theorem for Toeplitz sequences $\{T_n(f)\}_n$ with $f\in L^1([-\pi ,\pi ])$. From GLT 4 – GLT 5 we see that GLT sequences form a *-algebra. More specifically, let $\mathscr {E}$ be the space of matrix-sequences defined in (3), and note that $\mathscr {E}$ is a *-algebra over $\mathbb C$ with respect to the natural (pointwise) operations of Hermitian transposition, linear combination and product of matrix-sequences:

$$\begin{aligned} \{A_n\}_n^*&:=\{A_n^*\}_n,\\ \alpha \{A_n\}_n+\beta \{B_n\}_n&:=\{\alpha A_n+\beta B_n\}_n,\\ \{A_n\}_n\{B_n\}_n&:=\{A_nB_n\}_n. \end{aligned}$$

Then, the set

$$\begin{aligned} \mathscr {A}:=\bigl \{\{A_n\}_n:\ \{A_n\}_n\sim _\mathrm{GLT}\kappa \text{ for } \text{ some } \text{ measurable } \kappa :[0,1]\times [-\pi ,\pi ]\rightarrow {\mathbb C}\bigr \} \end{aligned}$$

is a *-subalgebra of $\mathscr {E}$, which is referred to as the GLT algebra. In view of GLT 3, the GLT algebra contains the algebra generated by zero-distributed sequences, Toeplitz sequences, and sequences of diagonal sampling matrices associated with a.e. continuous functions. To be precise, let

$$\begin{aligned} \mathscr {B}:= & {} \bigl \{\{T_n(f)\}_n:\ f\in L^1([-\pi ,\pi ])\bigr \}\cup \bigl \{\{D_n(a)\}_n:\ a\in C_\mathrm{a.e.}([0,1])\bigr \}\\&\cup \,\bigl \{\{Z_n\}_n:\ \{Z_n\}_n\sim _\sigma 0\bigr \} \end{aligned}$$

and let $\mathrm{algebra}(\mathscr {B})$ be the subalgebra of $\mathscr {E}$ generated by $\mathscr {B}$, i.e., the smallest subalgebra of $\mathscr {E}$ containing $\mathscr {B}$. Then,

$$\begin{aligned} \mathrm{algebra}(\mathscr {B})= & {} \left\{ \left\{ \sum _{i=1}^r\prod _{j=1}^{q_i}A_n^{(i,j)}\right\} :\ r,q_1,\ldots ,q_r\in \mathbb N,\ \right. \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left. \{A_n^{(i,j)}\}_n\in \mathscr {B} \text { for all } i=1,\ldots ,r \text { and } j=1,\ldots ,q_i\right\} \subseteq \mathscr {A}. \end{aligned}$$

GLT 6 – GLT 7 show that the GLT algebra is closed under more complex operations than sums and products, such as pseudoinversions and matrix functions. GLT 8 can be informally rephrased as follows: if a sequence of GLT sequences $\{B_{n,m}\}_n$ converges to a matrix-sequence $\{A_n\}_n$ (in the a.c.s. sense), and if the corresponding sequence of symbols $\kappa _m$ converges to a function $\kappa$ (in measure), then $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$. This is then the closure property of GLT pairs expressed in Definition 3.

3 Proof of the Main Result

This section is devoted to the proof of Theorem 1. If $\alpha :[0,1]^2\rightarrow \mathbb C$, $n\ge 2$ and $k\in \mathbb Z$, we define the $n\times n$ diagonal matrix

$$\begin{aligned} D_{n,k}(\alpha ):=\mathop {\mathrm{diag}}\limits _{h=1,\ldots ,n}\alpha \Bigl (\frac{(h-1+k)\,\mathrm{mod}\,n}{n-1},\frac{h-1}{n-1}\Bigr ). \end{aligned}$$

Lemma 1

For every $a\in \mathcal L_1$ and $n\ge 2$, we have

$$\begin{aligned} A_n(a)=\sum _{k=-(n-1)}^{n-1}T_n(\mathrm{e}^{\mathrm{i}k\theta })D_{n,k}(\hat{a}_k). \end{aligned}$$

Proof

The result of this lemma is already known in the literature; see, e.g., [10, p. 6]. Nevertheless, we include the details of the proof for the reader’s convenience. For every $n\ge 2$ and $k\in \mathbb Z$, we have

$$\begin{aligned} (T_n(\mathrm{e}^{\mathrm{i}k\theta }))_{ij}=\delta _{i-j,k}:=\left\{ \begin{array}{ll}1, &{} \mathrm{if}\ i-j=k,\\ 0, &{} \mathrm{otherwise}, \end{array}\right. \qquad i,j=1,\ldots ,n. \end{aligned}$$

Hence, for all $i,j=1,\ldots ,n$,

$$\begin{aligned}&\left( \sum _{k=-(n-1)}^{n-1}T_n(\mathrm{e}^{\mathrm{i}k\theta })D_{n,k}(\hat{a}_k)\right) _{ij} =\sum _{k=-(n-1)}^{n-1}\delta _{i-j,k}\,\hat{a}_k\Bigl (\frac{(j-1+k)\,\mathrm{mod}\,n}{n-1},\frac{j-1}{n-1}\Bigr )\nonumber \\&=\hat{a}_{i-j}\Bigl (\frac{i-1}{n-1},\frac{j-1}{n-1}\Bigr )=(A_n(a))_{ij}, \end{aligned}$$

and the lemma is proved. $\square$

Lemma 2

If $\alpha :[0,1]^2\rightarrow \mathbb C$ is continuous on the strip $S_\varepsilon$ for some $\varepsilon >0$ and $k\in \mathbb Z$, then

$$\begin{aligned} \{D_{n,k}(\alpha )\}_n\sim _\mathrm{GLT}\alpha (x,x). \end{aligned}$$

Proof

We show that, for all $n\ge 2$ and $k\in \mathbb Z$,

$$\begin{aligned} D_{n,k}(\alpha ) = D_n(\alpha (x,x))+R_{n,k}+N_{n,k}, \end{aligned}$$

(8)

where

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\mathrm{rank}(R_{n,k})}{n}=\lim _{n\rightarrow \infty }\Vert N_{n,k}\Vert =0. \end{aligned}$$

(9)

This implies that the sequence $\{Z_{n,k}:=R_{n,k}+N_{n,k}\}_n$ is zero-distributed (see Proposition 2), and the thesis follows from (8) in combination with GLT 3 and GLT 5.

Let $\omega _{\alpha ,\varepsilon }(\cdot )$ be the modulus of continuity of $\alpha$ over the strip $S_\varepsilon$,

$$\begin{aligned} \omega _{\alpha ,\varepsilon }(\delta ):=\max _{\begin{array}{c} (x,y),(x',y')\in S_\varepsilon \\ |x-x'|,|y-y'|\le \delta \end{array}}|\alpha (x,y)-\alpha (x',y')|,\quad \delta >0. \end{aligned}$$

If $k\ge 0$ and $n>\frac{k}{\varepsilon }+1$, for $h=1,\ldots ,n-k$ we have

$$\begin{aligned} \left| (D_{n,k}(\alpha ))_{hh}-(D_n(\alpha (x,x)))_{hh}\right|&= \left| \alpha \Bigg (\frac{h-1+k}{n-1},\frac{h-1}{n-1}\Bigg )- \alpha \Bigg (\frac{h}{n},\frac{h}{n}\Bigg )\right| \nonumber \\&\le \omega _{\alpha ,\varepsilon } \Bigg (\frac{k+1}{n-1}\Bigg ), \end{aligned}$$

(10)

which tends to 0 as $n\rightarrow \infty$. Write

$$\begin{aligned} D_{n,k}(\alpha )-D_n(\alpha (x,x)) = N_{n,k}+R_{n,k}, \end{aligned}$$

where $N_{n,k}$ (resp., $R_{n,k}$) is the matrix obtained from $D_{n,k}(\alpha )-D_n(\alpha (x,x))$ by setting to 0 all the diagonal elements corresponding to indices $h>n-k$ (resp., $h\le n\,{-}\,k$). The decomposition (8)–(9) follows from (10) and from the (obvious) inequality $\mathrm{rank}(R_{n,k})\le k$.

If $k<0$ and $n>\frac{|k|}{\varepsilon }+1$, for $h=|k|+1,\ldots ,n$ we have

$$\begin{aligned} \left| (D_{n,k}(\alpha ))_{hh}-(D_n(\alpha (x,x)))_{hh}\right|&= \left| \alpha \Bigg (\frac{h-1+k}{n-1},\frac{h-1}{n-1}\Bigg )-\alpha \Bigg (\frac{h}{n},\frac{h}{n}\Bigg )\right| \nonumber \\&\le \omega _{\alpha ,\varepsilon } \Bigg (\frac{|k|+1}{n-1}\Bigg ), \end{aligned}$$

(11)

which tends to 0 as $n\rightarrow \infty$. Write

$$\begin{aligned} D_{n,k}(\alpha )-D_n(\alpha (x,x)) = N_{n,k}+R_{n,k}, \end{aligned}$$

where $N_{n,k}$ (resp., $R_{n,k}$) is the matrix obtained from $D_{n,k}(\alpha )-D_n(\alpha (x,x))$ by setting to 0 all the diagonal elements corresponding to indices $h<|k|+1$ (resp., $h\ge |k|+1$). The decomposition (8)–(9) follows from (11) and from the (obvious) inequality $\mathrm{rank}(R_{n,k})\le |k|$. $\square$

We are now ready to prove the main result.

Proof of Theorem 1

The proof consists of three steps. In the first step we show that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ whenever $a\in \mathcal W$. In the second step we show that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ whenever $a\in \mathcal X$. Finally, in the third step we show that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ whenever $a\in \mathcal Y$.

Step 1. Suppose $a\in \mathcal W$. For $n\ge 2$ and $m\in \mathbb N$, consider the matrix

$$\begin{aligned} A_{n,m}(a) := \sum _{k=-m}^mT_n(\mathrm{e}^{\mathrm{i}k\theta })D_{n,k}(\hat{a}_k). \end{aligned}$$

Note that $A_{n,m}(a)$ “resembles” $A_n(a)$ because, by Lemma 1, the only difference between these two matrices is the range of indices where k varies. By Lemma 1 we also have $A_{n,m}(a)=A_n(a_m)$ for $n>m$, where

$$\begin{aligned} a_m(x,y,\theta ):=\sum _{k=-m}^m\hat{a}_k(x,y)\mathrm{e}^{\mathrm{i}k\theta } \end{aligned}$$

is the m-th Fuorier sum of $a(x,y,\theta )$. We are going to show that:

• i. $\{A_{n,m}(a)\}_n\sim _\mathrm{GLT}a_m(x,x,\theta )$ for every $m\in \mathbb N$;

• ii. $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure over $[0,1]\times [-\pi ,\pi ]$;

• iii. $\{A_{n,m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

Once this is done, the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ follows from GLT 8. By the hypothesis on a, each $\hat{a}_k$ is continuous on $S_{\varepsilon (k)}$ for some $\varepsilon (k)>0$, and so $\{D_{n,k}(\hat{a}_k)\}_n\sim _\mathrm{GLT}\hat{a}_k(x,x)$ by Lemma 2. Thus, by GLT 3 and GLT 5,

$$\begin{aligned} \{A_{n,m}(a)\}_n\sim _\mathrm{GLT}\sum _{k=-m}^m\mathrm{e}^{\mathrm{i}k\theta }\hat{a}_k(x,x)=a_m(x,x,\theta ), \end{aligned}$$

and item i is proved. Since a satisfies the Wiener-type condition (1), $a_m(x,y,\theta )\rightarrow a(x,y,\theta )$ uniformly on $[-\pi ,\pi ]$ for each fixed $(x,y)\in [0,1]^2$. In particular, the sequence of continuous functions $a_m(x,x,\theta )$ converges pointwise to $a(x,x,\theta )$ over $[0,1]\times [-\pi ,\pi ]$. Since the pointwise convergence on a set of finite measure implies the convergence in measure [2, 11], item ii is proved. Finally, by Lemma 1 and by the equality

$$\begin{aligned} \Vert T_n(\mathrm{e}^{\mathrm{i}k\theta })\Vert =1,\qquad |k|<n, \end{aligned}$$

for every $n>m$ we have

$$\begin{aligned} \Vert A_n(a)-A_{n,m}(a)\Vert&=\left\| \sum _{n>|k|>m}T_n(\mathrm{e}^{\mathrm{i}k\theta })D_{n,k}(\hat{a}_k)\right\| \\&\le \sum _{n>|k|>m}\Vert T_n(\mathrm{e}^{\mathrm{i}k\theta })\Vert \,\Vert D_{n,k}(\hat{a}_k)\Vert \\&\le \sum _{n>|k|>m}\sup _{x,y\in [0,1]}|\hat{a}_k(x,y)|\\&=:\varepsilon (m,n). \end{aligned}$$

Recalling that a satisfies the Wiener-type condition (1), we have

$$\begin{aligned} \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\varepsilon (m,n)=0, \end{aligned}$$

and Proposition 3 implies that $\{A_{n,m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$. We then conclude that item iii holds.

Step 2. Suppose $a\in \mathcal X$. In view of GLT 5 and the linearity of the map $A_n(\cdot )$, it suffices to prove that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ for any a of the form $a(x,y,\theta )=\alpha (x,y)\beta (\theta )$, with $\alpha \in C([0,1]^2)$ and $\beta \in L^2([-\pi ,\pi ])$. Fix then $a(x,y,\theta )=\alpha (x,y)\beta (\theta )$ with $\alpha \in C([0,1]^2)$ and $\beta \in L^2([-\pi ,\pi ])$. Let

$$\begin{aligned} a_m(x,y,\theta )=\alpha (x,y)p_m(\theta ), \end{aligned}$$

where $p_m$ is a sequence of trigonometric polynomials such that $p_m\rightarrow \beta$ in $L^2([-\pi ,\pi ])$. As in Step 1, we show that:

• i. $\{A_n(a_m)\}_n\sim _\mathrm{GLT}a_m(x,x,\theta )$ for every $m\in \mathbb N$;

• ii. $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure over $[0,1]\times [-\pi ,\pi ]$;

• iii. $\{A_n(a_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

Once this is done, the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ follows from GLT 8. Since $a_m$ belongs to $\mathcal W$ (note that only a finite number of Fourier coefficients of $a_m(x,y,\cdot )$ is nonzero), item i follows immediately from the result of Step 1. Since $p_m\rightarrow \beta$ in $L^2([-\pi ,\pi ])$, $a_m(x,x,\theta )=\alpha (x,x)p_m(\theta )\rightarrow \alpha (x,x)\beta (\theta )=a(x,x,\theta )$ in $L^2([0,1]\times [-\pi ,\pi ])$, and so $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure. This proves item ii. To prove item iii, we note that

$$\begin{aligned} A_n(a)-A_n(a_m)=A_n(a-a_m)=\left[ \alpha \Bigg (\frac{i-1}{n-1}, \frac{j-1}{n-1}\Bigg )\right] _{i,j=1}^n\circ \ T_n(\beta -p_m), \end{aligned}$$

where $\circ$ denotes the componentwise (or Hadamard) product of matrices. The Frobenius norm of $A_n(a)-A_n(a_m)$ can then be estimated by means of Proposition 1:

$$\begin{aligned} \Vert A_n(a)-A_n(a_m)\Vert _2^2 \le \Vert \alpha \Vert _{\infty }^2\Vert T_n(\beta -p_m)\Vert _2^2\le \Vert \alpha \Vert _{\infty }^2 \Vert \beta -p_m\Vert _{L^2}^2n. \end{aligned}$$

(12)

Item iii now follows from Proposition 3, taking into account that $\Vert \beta -p_m\Vert _{L^2}\rightarrow 0$.

Step 3. Suppose $a\in \mathcal Y$. In view of GLT 5 and the linearity of the map $A_n(\cdot )$, it suffices to prove that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ for any a of the form $a(x,y,\theta )=\alpha (x)\beta (y)\gamma (\theta )$, with $\alpha ,\beta \in C([0,1])$ and $\gamma \in L^1([-\pi ,\pi ])$. Fix then $a(x,y,\theta )=\alpha (x)\beta (y)\gamma (\theta )$ with $\alpha ,\beta \in C([0,1])$ and $\gamma \in L^1([-\pi ,\pi ])$. Let

$$\begin{aligned} a_m(x,y,\theta )=\alpha (x)\beta (y)p_m(\theta ), \end{aligned}$$

where $p_m$ is a sequence of trigonometric polynomials such that $p_m\rightarrow \gamma$ in $L^1([-\pi ,\pi ])$. As in Steps 1–2, we show that:

• i. $\{A_n(a_m)\}_n\sim _\mathrm{GLT}a_m(x,x,\theta )$ for every $m\in \mathbb N$;

• ii. $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure over $[0,1]\times [-\pi ,\pi ]$;

• iii. $\{A_n(a_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

Once this is done, the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ follows from GLT 8. Since $a_m$ belongs to $\mathcal W$ (note that only a finite number of Fourier coefficients of $a_m(x,y,\cdot )$ is nonzero), item i follows immediately from the result of Step 1. Since $p_m\rightarrow \gamma$ in $L^1([-\pi ,\pi ])$, $a_m(x,x,\theta )=\alpha (x)\beta (x)p_m(\theta )\rightarrow \alpha (x)\beta (x)\gamma (\theta )=a(x,x,\theta )$ in $L^1([0,1]\times [-\pi ,\pi ])$, and so $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure. This proves item ii. To prove item iii, let

$$\begin{aligned} \Delta _n(g)=\mathop {\mathrm{diag}}\limits _{i=1,\ldots ,n}g\Bigl (\frac{i-1}{n-1}\Bigr ),\qquad g:[0,1]\rightarrow \mathbb C,\qquad n\ge 2. \end{aligned}$$

Note that

$$\begin{aligned} A_n(a)-A_n(a_m)=A_n(a-a_m)=\Delta _n(\alpha )T_n(\gamma -p_m)\Delta _n(\beta ). \end{aligned}$$

The trace-norm of $A_n(a)-A_n(a_m)$ can then be estimated by means of the Hölder-type inequality (2) and Proposition 1:

$$\begin{aligned} \Vert A_n(a)-A_n(a_m)\Vert _1\le & {} \Vert \Delta _n(\alpha )\Vert \,\Vert \Delta _n(\beta )\Vert \,\Vert T_n(\gamma -p_m)\Vert _1\nonumber \\\le & {} \Vert \alpha \Vert _\infty \Vert \beta \Vert _\infty \Vert \gamma -p_m\Vert _{L^1}n. \end{aligned}$$

(13)

Item iii now follows from Proposition 3, taking into account that $\Vert \gamma -p_m\Vert _{L^1}\rightarrow 0$. $\square$

4 Consequences of the Main Result

Consequences of Theorem 1 (and of the theory of GLT sequences) are all the results that can be deduced from the list of properties GLT 1 – GLT 8 in which GLT 3 is extended to include the result of Theorem 1. For the sake of clarity, we use a different notation for the extended version of GLT 3 and we give it the label $\overline{\mathbf{GLT\,3}}$. Property $\overline{\mathbf{GLT\,3}}$ is then the following.

$\overline{\mathbf{GLT\,3 }}.$ We have:

• $\{T_n(f)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=f(\theta )$ if $f\in L^1([-\pi ,\pi ])$;

• $\{D_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=a(x)$ if $a\in C_\mathrm{a.e.}([0,1])$;

• $\{Z_n\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=0$ if and only if $\{Z_n\}_n\sim _\sigma 0$;

• $\{A_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=a(x,x,\theta )$ if $a\in \mathcal W\cup \mathcal X\cup \mathcal Y$.

In this section we discuss some consequences of GLT 1 – GLT 2, $\overline{\mathbf{GLT\,3}}$, GLT 4 – GLT 8.

Spectral distribution results on the algebra generated by variable-coefficient Toeplitz sequences Let $\mathscr {C}$ denote the *-algebra generated by the variable-coefficient Toeplitz sequences $\{A_n(a)\}_n$ with $a\in \mathcal W\cup \mathcal X\cup \mathcal Y$. It is not difficult to see that

$$\begin{aligned} \mathscr {C}= & {} \left\{ \left\{ \sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\right\} _n:\ r,q_1,\ldots ,q_r\in \mathbb N,\right. \\&\left. a_{ij}\in \mathcal W\cup \mathcal X\cup \mathcal Y \text{ for } \text{ all }\, i=1,\ldots ,r \mathrm{and}j=1,\ldots ,q_i\right\} . \end{aligned}$$

By $\overline{\mathbf{GLT\,3}}$ and GLT 5, $\mathscr {C}$ is a subalgebra of the GLT algebra $\mathscr {A}$, and for the generic element of $\mathscr {C}$ we have

$$\begin{aligned} \left\{ \sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\right\} _n\sim _\mathrm{GLT}\sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x,x,\theta ). \end{aligned}$$

Hence, by GLT 1,

$$\begin{aligned} \left\{ \sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\right\} _n\sim _\sigma \sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x,x,\theta ) \end{aligned}$$

(14)

and, if the matrices $\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})$ are Hermitian,

$$\begin{aligned} \left\{ \sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\right\} _n\sim _\lambda \sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x,x,\theta ). \end{aligned}$$

(15)

A result analogous to (14)–(15) was obtained by Silbermann and Zabroda in [18, Theorem 7.2]. In particular, in view of [18, pp. 185–186], a consequence of [18, Theorem 7.2] is that (15) holds whenever the functions $a_{ij}$ belong to

$$\begin{aligned}&C^\infty _\theta ([0,1]^2\times [-\pi ,\pi ]):=\Bigl \{a\in C([0,1]^2\times [-\pi ,\pi ]): \\&\mathrm{the}\,2\pi \hbox {-}\mathrm{periodic\,extension\,of}\,a(x,y,\cdot )\,\mathrm{from}\, [-\pi ,\pi )\, \mathrm{to} \, \mathbb R\\&\text{ belongs } \text{ to }\, C^\infty (\mathbb R)\, \text{ for } \text{ all }\,(x,y)\in [0,1]^2,\\&\text{ all } \text{ the } \text{ derivatives } \dfrac{\partial ^ra(x,y,\theta )}{\partial \theta ^r} \text{ belong } \text{ to }\, C([0,1]^2\times [-\pi ,\pi ])\Bigr \}. \end{aligned}$$

Since $C^\infty _\theta ([0,1]^2\times [-\pi ,\pi ])$ is a subset of $\mathcal W$ (as already observed in [18, pp. 171–172]), the relation (15) seems to add something to [18, Theorem 7.2], though it should be noted that the latter theorem implies more than the eigenvalue distribution (15) for functions $a_{ij}\in C^\infty _\theta ([0,1]^2\times [-\pi ,\pi ])$. Concerning the singular value distribution (14), one of its consequences is that, if $\sigma _1(A_n(a))\le \ldots \le \sigma _n(A_n(a))$ are the singular values of $A_n(a)$ arranged in non-decreasing order, and if a is a function in $\mathcal W\cup \mathcal X\cup \mathcal Y$ such that $\mathrm{ess\,inf}_{(x,\theta )\in [0,1]\times [-\pi ,\pi ]}a(x,x,\theta )=0$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\sigma _i(A_n(a))=0\ \text{ for } \text{ each }\,fixed \,i\ge 1; \end{aligned}$$

(16)

cf. with [10, Theorem 1]. To give a formal proof of (16) based on (14), suppose by contradiction that there exists an index $i\ge 1$ such that $\sigma _i(A_n(a))$ does not converge to 0 as $n\rightarrow \infty$. Then, there exists $\varepsilon >0$ and a subsequence $\{\sigma _i(A_{n_j}(a))\}_j$ such that $\sigma _i(A_{n_j}(a))\ge \varepsilon$ for all j. This implies that $\sigma _k(A_{n_j}(a))\ge \varepsilon$ for all $k\ge i$ and all j (recall that we are assuming the singular values are arranged in non-decreasing order). Take a function $F\in C_c(\mathbb C)$ such that $F=1$ over $[0,\varepsilon /2]$, $F=0$ over $[\varepsilon ,\infty )$, and $0\le F\le 1$ over $[0,\infty )$. For this function we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\frac{1}{n_j}\sum _{k=1}^nF(\sigma _k(A_{n_j}(a)))= \lim _{j\rightarrow \infty }\frac{1}{n_j}\sum _{k=1}^{i-1}F(\sigma _k(A_{n_j}(a)))\le \lim _{j\rightarrow \infty }\frac{i-1}{n_j}=0. \end{aligned}$$

On the other hand, since we are assuming that $\mathrm{ess\,inf}_{(x,\theta )\in [0,1]\times [-\pi ,\pi ]}a(x,x,\theta )=0$, the measure of the set $S=\{(x,\theta )\in [0,1]\times [-\pi ,\pi ]:\ 0\le a(x,x,\theta )\le \varepsilon /2\}$ is positive and the singular value distribution (14) yields

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^nF(\sigma _k(A_n(a)))=\frac{1}{2\pi } \int _{-\pi }^\pi \int _0^1F(a(x,x,\theta ))dxd\theta \\&\ge \frac{1}{2\pi } \int _SF(a(x,x,\theta ))dxd\theta =\frac{\mu _2(S)}{2\pi }>0, \end{aligned}$$

a contradiction.

Spectral distribution results beyond the algebra generated by variable-coefficient Toeplitz sequences It is clear that the relations (14)–(15) are far from exhausting the spectral distribution results that can be derived from GLT 1 – GLT 2, $\overline{\mathbf{GLT\,3}}$, GLT 4 – GLT 8. In particular, GLT 1, $\overline{\mathbf{GLT\,3}}$ and GLT 5 – GLT 7 allow one to compute the singular value and eigenvalue distribution of matrix-sequences that are obtained not only through sums and products of variable-coefficient Toeplitz sequences, but also through more complex operations involving all the GLT sequences listed in $\overline{\mathbf{GLT\,3}}$. For example, let $a\in \mathcal W\cup \mathcal X\cup \mathcal Y$. If $a(x,x,\theta )\ne 0$ a.e., then GLT 1, $\overline{\mathbf{GLT\,3}}$, GLT 6 yield

$$\begin{aligned}&\{A_n(a)^\dag \}_n\sim _\mathrm{GLT}\frac{1}{a(x,x,\theta )},\\&\{A_n(a)^\dag \}_n\sim _\sigma \frac{1}{a(x,x,\theta )}. \end{aligned}$$

If in addition the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{\mathbf{GLT\,3}}$, GLT 6 also yield

$$\begin{aligned}&\{A_n(a)^\dag \}_n\sim _\lambda \frac{1}{a(x,x,\theta )}. \end{aligned}$$

If the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{\mathbf{GLT\,3}}$, GLT 7 give

$$\begin{aligned}&\{\sin A_n(a)\}_n\sim _\mathrm{GLT}\sin (a(x,x,\theta )),\\&\{\sin A_n(a)\}_n\sim _\sigma \sin (a(x,x,\theta )),\\&\{\sin A_n(a)\}_n\sim _\lambda \sin (a(x,x,\theta )). \end{aligned}$$

If $a(x,x,\theta )\ne 0$ a.e. and the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{\mathbf{GLT\,3}}$, GLT 5 – GLT 7 yield

$$\begin{aligned}&\{T_n(|\theta |^{-1/2})A_n(a)^\dag \mathrm{e}^{A_n(a)}A_n(a)^\dag T_n(|\theta |^{-1/2})+8D_n(\log x)A_n(a)^\dag D_n(\log x)\}_n\\&\sim _\mathrm{GLT}\frac{\mathrm{e}^{a(x,x,\theta )}}{|\theta |a(x,x,\theta )^2}+\frac{8\log ^2x}{a(x,x,\theta )},\\&\{T_n(|\theta |^{-1/2})A_n(a)^\dag \mathrm{e}^{A_n(a)}A_n(a)^\dag T_n(|\theta |^{-1/2})+8D_n(\log x)A_n(a)^\dag D_n(\log x)\}_n\\&\sim _\sigma \frac{\mathrm{e}^{a(x,x,\theta )}}{|\theta |a(x,x,\theta )^2}+\frac{8\log ^2x}{a(x,x,\theta )},\\&\{T_n(|\theta |^{-1/2})A_n(a)^\dag \mathrm{e}^{A_n(a)}A_n(a)^\dag T_n(|\theta |^{-1/2})+8D_n(\log x)A_n(a)^\dag D_n(\log x)\}_n\\&\sim _\lambda \frac{\mathrm{e}^{a(x,x,\theta )}}{|\theta |a(x,x,\theta )^2}+\frac{8\log ^2x}{a(x,x,\theta )}. \end{aligned}$$

We could continue with this game indefinitely...

5 Final Remarks and Future Perspectives

In [18, Theorem 9.1] the anonymous reviewer of [18] formulated a result which extends Theorem 1. If this result were true, the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ would be satisfied for any function $a\in C([0,1]^2\times [-\pi ,\pi ])$ (and not only). As a consequence, we could replace the fourth item in $\overline{\mathbf{GLT\,3}}$ by the following,

• $\{A_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta ):=a(x,x,\theta )$ if $a\in \mathcal W\cup \mathcal X\cup \mathcal Y\cup C([0,1]^2\times [-\pi ,\pi ])$,

thus obtaining a property $\overline{\mathbf{GLT\,3}}$ stronger than the one considered in Sect. 4, and we could then generalize the results of Sect. 4 in this perspective. It should be noted, however, that the proof of [18, Theorem 9.1] provided in [18, Sect. 10] is unfortunately wrong. The problem in the proof is contained in [18, p. 195], where it is asserted that a bound on the Frobenius norm such as $\Vert A_n-A_{n,\varepsilon }\Vert _2\le \varepsilon n$ implies that $\{\{A_{n,\varepsilon }\}_n\}_{\varepsilon >0}$ is an a.c.s. for $\{A_n\}_n$ as $\varepsilon \rightarrow 0$. This is not true, as shown by the following example.

Example

Let $\{A_n\}_n$ be a matrix-sequence and let $A_{n,\varepsilon }:=A_n+c_{n,\varepsilon }I_n$, where $I_n$ is the $n\times n$ identity matrix and $c_{n,\varepsilon }$ is a coefficient depending on both n and $\varepsilon$. It is clear that

$$\begin{aligned} \Vert A_n-A_{n,\varepsilon }\Vert _2 = |c_{n,\varepsilon }|\sqrt{n}. \end{aligned}$$

If we choose $c_{n,\varepsilon }$ so that

$$\begin{aligned} 0\le c_{n,\varepsilon }\le \varepsilon \sqrt{n}, \end{aligned}$$

(17)

then the Frobenius norm $\Vert A_n-A_{n,\varepsilon }\Vert _2$ is bounded by $\varepsilon n$ for all n and $\varepsilon$. However, there are many choices for the coefficient $c_{n,\varepsilon }$ such that (17) is satisfied but $\{\{A_{n,\varepsilon }\}_n\}_{\varepsilon >0}$ is not an a.c.s. for $\{A_n\}_n$ as $\varepsilon \rightarrow 0$. For example, take any $c_{n,\varepsilon }$ satisfying (17) and such that, for every $\varepsilon >0$,

$$\begin{aligned} \lim _{n\rightarrow \infty }c_{n,\varepsilon }=c_\varepsilon \ge c>0, \end{aligned}$$

(18)

with c independent of $\varepsilon$. A possible choice is $c_{n,\varepsilon }=\varepsilon \sqrt{n}$. Then $\{\{A_{n,\varepsilon }\}_n\}_{\varepsilon >0}$ is not an a.c.s. for $\{A_n\}_n$ as $\varepsilon \rightarrow 0$. Let us prove it formally. If $R_{n,\varepsilon }$ and $N_{n,\varepsilon }$ are any two matrices such that

$$\begin{aligned} c_{n,\varepsilon }I_n=A_n-A_{n,\varepsilon }=R_{n,\varepsilon }+N_{n,\varepsilon }, \end{aligned}$$

then, by the minimax principle for singular values [1, Problem III.6.1],

$$\begin{aligned} c_{n,\varepsilon }&=\sigma _i(R_{n,\varepsilon }+N_{n,\varepsilon })\nonumber \\&=\max _{\begin{array}{c} V subspace of \mathbb C^n\\ \dim V=i \end{array}}\min _{\begin{array}{c} \mathbf x\in V\\ \Vert \mathbf x\Vert =1 \end{array}}\Vert R_{n,\varepsilon }\mathbf x+N_{n,\varepsilon }\mathbf x\Vert \nonumber \\&\le \max _{\begin{array}{c} V subspace of \mathbb C^n\\ \dim V=i \end{array}}\min _{\begin{array}{c} \mathbf x\in V\\ \Vert \mathbf x\Vert =1 \end{array}}\Vert R_{n,\varepsilon }\mathbf x\Vert +\Vert N_{n,\varepsilon }\Vert \nonumber \\&=\sigma _i(R_{n,\varepsilon })+\Vert N_{n,\varepsilon }\Vert , \end{aligned}$$

(19)

for all $i=1,\ldots ,n$. If $rank (R_{n,\varepsilon })<n$, then at least one singular value $\sigma _i(R_{n,\varepsilon })$ is zero, and by (18)–(19) we have $\Vert N_{n,\varepsilon }\Vert \ge c_{n,\varepsilon }\rightarrow c_\varepsilon \ge c>0$. In view of (4)–(5), this implies that $d_\mathrm{a.c.s.}(\{A_n\}_n,\{A_{n,\varepsilon }\}_n)\ge \min (c,1)>0$ for all $\varepsilon >0$ and, consequently, $\{\{A_{n,\varepsilon }\}_n\}_{\varepsilon >0}$ is not an a.c.s. for $\{A_n\}_n$ as $\varepsilon \rightarrow 0$.

We conclude this work by mentioning two possible ways to extend Theorem 1, i.e., to prove the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ for a space of functions a larger than $\mathcal W\cup \mathcal X\cup \mathcal Y$.

• 1. The first way originates from the observation that, when proving items i–iii in Step 1 of Sect. 3, we actually proved more than needed. We emphasize in particular the following aspect. When proving item iii, we showed that

  $$\begin{aligned} \Vert A_n(a)-A_{n,m}(a)\Vert \le \varepsilon (m,n) \end{aligned}$$

  (20)

  for some $\varepsilon (m,n)$ satisfying

  $$\begin{aligned} \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\varepsilon (m,n)=0. \end{aligned}$$

  (21)

  On the other hand, it would have been enough to show that $\{A_{n,m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$, by proving for example that

  $$\begin{aligned} \Vert A_n(a)-A_{n,m}(a)\Vert _p\le \varepsilon (m,n)n^{1/p} \end{aligned}$$

  (22)

  for some $p\in [1,\infty )$ and some $\varepsilon (m,n)$ satisfying (21); see Proposition 3. Note that (22) is a condition weaker than (20), because for all $p\in [1,\infty )$ we have

  $$\begin{aligned} \frac{\Vert A\Vert _p}{n^{1/p}}\le \Vert A\Vert , \end{aligned}$$

  with the equality holding if and only if all the singular values of A are equal. In view of these considerations, there is room to refine the argument used for the proof of Theorem 1 (Step 1), so as to weaken the hypotheses on a and, consequently, to enlarge the space of functions a such that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$.

• 2. Suppose that, for all functions a belonging to a certain class $\mathcal C\subset \mathcal L_1$ we can construct a sequence $\{a_m\}_m\subset \mathcal L_1$ such that

  • i. the relation $\{A_n(a_m)\}_n\sim _\mathrm{GLT}a_m(x,x,\theta )$ holds for all m,

  • ii. $a_m$ converges to a in some way which ensures that $a_m(x,x,\theta )\rightarrow a(x,x,\theta )$ in measure,

  • iii. $\{A_n(a_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

  Then $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x,x,\theta )$ for all $a\in \mathcal C$ (by GLT 8). This technique is the second possible way to extend Theorem 1, and it was already used in the proof of Theorem 1 with $\mathcal C=\mathcal W$ (Step 1), $\mathcal C=\mathcal X$ (Step 2) and $\mathcal C=\mathcal Y$ (Step 3). What is important to point out is that, now that we have proved Theorem 1, we are allowed to take any sequence in $\mathcal W\cup \mathcal X\cup \mathcal Y$ as the sequence $\{a_m\}_m$, because item i is automatically satisfied and item ii is satisfied as well, provided we choose an $a_m$ converging to a in a suitable way. In Steps 2–3 we precisely used the strategy outlined here by taking $\{a_m\}_m\subset \mathcal W$ such that $a_m\rightarrow a$ in $L^p$. Moreover, the strategy worked successfully because the $L^p$ convergence implied the a.c.s. convergence $\{A_n(a_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$, thanks to the estimates (12)–(13), that is, to the specific “separable” structure of the functions $a_m,a$ and to the crucial inequality $\Vert T_n(f)\Vert _p\le n^{1/p}\Vert f\Vert _{L^p}$. It would be interesting to see whether this same strategy also works for classes $\mathcal C$ other than $\mathcal X$ and $\mathcal Y$. In particular, does it work if $\mathcal C$ is one of the Hölder classes considered in [3, 4, 17]?

The extension of Theorem 1 using the suggestions in the above items 1–2 may form the content of a future research.