We consider the Jacobi operator $(T,D(T))$ associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmmetric with deficiency indices $(1,1)$. For a complex number $z$ let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences $(p_n(z))$ and $(q_n(z))$
corresponding to the orthonormal polynomials $p_n$ and  polynomials $q_n$ of the second kind. We determine whether linear combinations of $\mathfrak{p}_u, \mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for $u,v\in\C$ belong to $D(T)$ or to the domain of the self-adjoint extensions of $T$ in $\ell^2$.
The results depend on the four Nevanlinna functions of two variables associated with the moment problem. We also show that $D(T)$ is the common range of an explicitly constructed family of bounded operators on $\ell^2$.

Mathematics Subject Classification: Primary 47B25, 47B36, 44A60

Keywords. Jacobi matrices and operators, indeterminate moment problems.

§ INTRODUCTION
We shall consider the Jacobi matrix $J$ associated with a moment sequence $s=(s_n)_{n\ge 0}$ of the form

\begin{equation}\label{eq:mom}
s_n=\int x^n\,d\mu(x),\quad n=0,1,\ldots,
\end{equation}

where $\mu$ is a positive measure on $\R$  with infinite support and moments of every order. It is a tridiagonal matrix of the  form

\begin{equation}\label{eq:Jac}
b_0 & a_0 & 0 & \hdots\\
a_0 & b_1 & a_1 & \hdots\\
0 & a_1 & b_2 & \hdots\\
\vdots &\vdots & \vdots & \ddots
\end{pmatrix},
\end{equation}

where $a_n>0, b_n\in\R, n\ge 0$  are given by the three term recurrence relation

\begin{equation*}\label{eq:3term}
xp_n(x)=a_np_{n+1}(x)+b_np_n(x)+a_{n-1}p_{n-1}(x), n\ge 0, \quad a_{-1}:=0.
\end{equation*}

Here $(p_n)_{n\ge 0}$ is the sequence of orthonormal polynomials associated with $\mu$, hence satisfying

\begin{equation*}\label{eq:op}
\int p_n(x)p_m(x)\,d\mu(x)=\delta_{n,m},
\end{equation*}

and $p_n$ is a real polynomial of degree $n$ with positive leading coefficient. In this paper   we follow the terminology of [17]. Basic results about the classical moment problem can also be found in [1] and [16]. Recent  results about indeterminate moment problems can be found in [5], [6] [7], [8].

It is  clear that the proportional measures $\la\mu, \la>0$ lead to the same Jacobi matrix $J$, and the well-known Theorem of Favard (see <cit.>) states that any matrix of the form (<ref>) with $a_n>0, b_n\in\R$ comes from a unique  moment sequence  $(s_n)$ as above, normalized such that $s_0=1$. In the following we shall always assume that this normalization holds, and consequently the solutions $\mu$ of (<ref>) are probability measures  and $p_0=1$.

The Jacobi matrix acts as a symmetric operator  in the Hilbert space $\ell^2$ of square summable complex sequences. Its domain $\mathcal F$ consists of the complex sequences $(c_n)_{n\ge 0}$ with only finitely many non-zero terms, and the action is multiplication of the matrix $J$ by $c\in\mathcal F$ considered as a column, i.e.,

\begin{equation}\label{eq:J}
(Jc)_n:=a_{n-1}c_{n-1}+b_n c_n+a_n c_{n+1},\quad n\ge 0.
\end{equation}

Denoting $(e_n)_{n\ge 0}$ the standard orthonormal basis of $\ell^2$, we have

\begin{equation*}\label{eq:F}
\mathcal F=\span\{e_n|n\ge 0\}.
\end{equation*}

The Jacobi operator associated with $J$ is by definition the closure $(T,D(T))$ of the symmetric operator $(J,\mathcal F)$.

It is a classical fact that $(T,D(T))$ is a closed symmetric operator, and its deficiency indices are either $(0,0)$  or $(1,1)$. These cases occur precisely if the  moment sequence (<ref>) is determinate or indeterminate, i.e., there is exactly one or several solutions $\mu$ satisfying (<ref>).

By definition $D(T)$ consists of those $c\in\ell^2$ for which there exists a sequence
$(c^{(k)})\in\mathcal F$ such that $\lim_{k\to\infty}c^{(k)}=c$ and $(Jc^{(k)})$ is a convergent sequence in $\ell^2$. For  such $c$ we have $Tc=\lim_{k\to\infty} Jc^{(k)}$, and this limit is independent of the choice of approximating sequence $(c^{(k)})$.

Clearly, $D(T)$ is closed under complex conjugation and

T\overline{c}=\overline{Tc},\quad c\in D(T).

The purpose of the present paper is to study the Jacobi operator $(T,D(T))$ as well as its self-adjoint extensions $(T_t, D(T_t)), t\in\R^*:=\R\cup\{\infty\}$ in the indeterminate case. We shall in particular give some families of sequences $c\in\ell^2$ which belong to $D(T)$, see Theorem <ref>–Theorem <ref>.

Section 2 is devoted to the proof of Theorem <ref> after  a presentation of the deficiency spaces of $(T, D(T))$. The self-adjoint extensions of $(T, D(T))$ as well as their corresponding N-extremal solutions to (<ref>), cf. (<ref>), are introduced in Section 3.

In Theorem <ref>, Theorem <ref> and Theorem <ref> we describe vectors belonging to $D(T_t)\setminus D(T)$. Like the results in Theorem <ref>–Theorem <ref>, they depend on the Nevanlinna functions of two variables defined in (<ref>),  (<ref>), (<ref>), (<ref>).

In Section 4 we construct for each $z_0\in\C$ a bounded operator $\Xi_{z_0}$ in $\ell^2$ with range $D(T)$. The restriction of $\Xi_{z_0}$ to $(T-z_0I)(D(T))$ is a bijection onto $D(T)$ equal to $(T-z_0I)^{-1}$, see Theorem <ref>. It is based
on a study of the function space $\mathcal E$ defined in (<ref>), and known to be a de Branges space of entire functions by <cit.>. We prove in particular Theorem <ref>, showing that $\mathcal E$ is stable under the formation of difference quotients.

Various technical  results about the Nevanlinna functions are given in Section 5.

After this summary of the content of the present paper, we recall that the adjoint operator $(T^*,D(T^*))$ is the maximal operator associated with $J$, cf. <cit.>. In fact, the matrix product of $J$  and any column vector $c$ makes sense, cf. (<ref>), and $D(T^*)$ consists of those $c\in\ell^2$ for which the product $Jc$ belongs to $\ell^2$. For $c\in D(T^*)$ we have $T^*c=Jc$.

In the determinate case with a unique solution $\mu$ of (<ref>), the Jacobi operator is self-adjoint and $(p_n)$ is an orthonormal basis of $L^2(\mu)$. The self-adjoint operator of multiplication $M_\mu$ in $L^2(\mu)$ given by

D(M_\mu)=\{f\in L^2(\mu)\mid xf(x)\in L^2(\mu)\},\quad M_\mu f(x)=xf(x)

is unitarily equivalent with $(T, D(T))$ via the unitary operator $U:\ell^2\to L^2(\mu)$ given by $U(e_n)=p_n, n\ge 0$. We shall not study the determinate case in this paper, but concentrate on the  indeterminate case, where it is known that  the set of solutions $\mu$ to (<ref>) is an infinite convex set $V$. The polynomials of the second kind $(q_n)$ are given as

\begin{equation*}
q_n(z)=\int \frac{p_n(z)-p_n(x)}{z-x}\,d\mu(x), \quad z\in \C,
\end{equation*}

where $\mu\in V$ is arbitrary.

We define  and recall

\begin{equation}\label{eq:frak}
\mathfrak{p}_z:=(p_n(z)), \mathfrak{q}_z:=(q_n(z))\in\ell^2,\quad z\in\C,
\end{equation}

where we have followed the terminology of [17]. It is known that
$||\mathfrak{p}_z||$ and $||\mathfrak{q}_z||$ are positive continuous functions on $\C$.  It is therefore possible
for $c\in\ell^2$ to define entire functions $F_c, G_c$ as

\begin{equation}\label{eq:FGc}
F_c(z)=\sum_{n=0}^\infty c_np_n(z),\quad G_c(z)=\sum_{n=0}^\infty c_nq_n(z),\quad z\in\C.
\end{equation}

We also have the following four entire functions of two complex variables, called the Nevanlinna functions of the indeterminate moment problem:

\begin{eqnarray}
A(u,v)&=&(u-v)\sum_{k=0}^\infty q_k(u)q_k(v)\label{eq:A}\\
B(u,v)&=&-1+(u-v)\sum_{k=0}^\infty p_k(u)q_k(v) \label{eq:B}\\
C(u,v)&=&1+(u-v)\sum_{k=0}^\infty q_k(u)p_k(v)\label{eq:C}\\
D(u,v)&=&(u-v)\sum_{k=0}^\infty p_k(u)p_k(v)\label{eq:D},
\end{eqnarray}

see Section 7.1 in [17]. The two-variable  functions were introduced in [11] in a slightly different form, which was subsequently  used in [3],[14]. An approximation to the two-variable functions was already considered in <cit.>.
If the functions of [11] are marked with a $*$, we have

\begin{eqnarray*}\label{eq:A*-D*}
A^*(u,v)&=&-A(u,v),\; B^*(u,v)=-C(u,v),\\
C^*(u,v)&=&-B(u,v),\; D^*(u,v)=-D(u,v).
\end{eqnarray*}

In the following we need several formulas about these functions, see Theorem <ref> and Corollary <ref> in the Appendix, but at this point we just recall that

\begin{equation}\label{eq:det=1}
A(u,v)D(u,v)-B(u,v)C(u,v)=1,\quad u,v\in\C.
\end{equation}

We define entire functions of one variable by setting the second variable to 0, i.e.,

\begin{equation}\label{eq:A-D}
\end{equation}

and by specialization of (<ref>) we get

\begin{equation}\label{eq:det=1*}
A(u)D(u)-B(u)C(u)=1,\quad u\in\C.
\end{equation}

By Section 6.5 in [17] we have

\begin{equation}\label{eq:p,q}
\mathfrak{p}_z, \mathfrak{q}_z \in D(T^*),\quad T^*\mathfrak{p}_z=z\mathfrak{p}_z,
T^*\mathfrak{q}_z=e_0+z\mathfrak{q}_z,\quad z\in\C.
\end{equation}

Our first main result is the following:

For all $z\in\C$ we have $\mathfrak{p}_z, \mathfrak{q}_z\notin D(T)$.

Let $u,v\in\C$ be given.

(i) There exists $\alpha\in\C$ such that $\mathfrak{p}_u+\alpha \mathfrak{p}_v\in D(T)$ if and only if $D(u,v)=0$. In the affirmative case $\alpha$  is uniquely determined as

(ii) There exists $\beta\in\C$ such that $\mathfrak{q}_u+\beta \mathfrak{q}_v\in D(T)$ if and only if $A(u,v)=0$. In the affirmative case $\beta$  is uniquely determined as $\beta=-C(u,v)$.

(iii) There exists $\gamma\in\C$ such that $\mathfrak{p}_u+\gamma \mathfrak{q}_v\in D(T)$ if and only if $B(u,v)=0$. In the affirmative case $\gamma$  is uniquely determined as $\gamma=-D(u,v)$. In particular $\mathfrak{p}_u+\gamma\mathfrak{q}_u\notin D(T)$ for all $u, \gamma\in\C$.

The proof will be given in Section 2.

We shall next give results about  the  zero-sets of the entire functions $A,\ldots,D$ of two variables.

Let $F(u,v)$ denote any of the four functions $A,B,C,D$ on $\C^2$. For  $v\in\C$ define

\begin{equation}\label{eq:zset}
Z(F)_v:=\{u\in\C \mid F(u,v)=0\}.
\end{equation}

Then $Z(F)_v$ is countably infinite. If $v\in\R$ then $Z(F)_v\subset \R$, and if $v$ is in either the upper or lower half-plane, then $Z(F)_v$ belongs to the same half-plane.

As a follow up on the two previous theorems we have the following:
Let  $v\in\R$ be given and consider the set of real zeros $Z(F)_v$ from Theorem <ref>.

(i) Let $u\in Z(D)_v$ be such that $u<v$ and such that $]u,v[\cap Z(D)_v=\emptyset$. Then $B(u,v)>0$, where  $\mathfrak{p}_u+B(u,v)\mathfrak{p}_v\in D(T)$ according to Theorem <ref>.

(ii) Let $u\in Z(A)_v$ be such that $u<v$ and such that $]u,v[\cap Z(A)_v=\emptyset$. Then $C(u,v)<0$, where  $\mathfrak{q}_u-C(u,v)\mathfrak{q}_v\in D(T)$ according to Theorem <ref>.

The proofs of Theorem <ref> and Theorem <ref> will be given in Section 5.

§ PRELIMINAIRES AND PROOF OF THEOREM <REF>

Fix $z_0\in\C$ in the open upper half-plane and consider the deficiency spaces

\begin{eqnarray*}
\Delta^+(z_0) &=&\ker(T^*-z_0I)=\C \mathfrak{p}_{z_0}\\
\Delta^-(z_0) &=&\ker(T^*-\overline{z_0}I)=\C\mathfrak{p}_{\overline{z_0}},
\end{eqnarray*}

cf. (<ref>).

We know from <cit.> that

\begin{equation}\label{eq:direct}
D(T^*)=D(T)\oplus \Delta^+(z_0) \oplus \Delta^-(z_0),
\end{equation}

and the sum is direct as indicated by the $\oplus$ signs.

For any $\la\in \C$ we have the decomposition from (<ref>)

\begin{equation}\label{eq:p}
\mathfrak{p}_{\la}=s_\la +s_\la^+\mathfrak{p}_{z_0}+s_\la^-\mathfrak{p}_{\overline{z_0}},
\end{equation}

where $s_\la\in D(T)$ and

\begin{equation}\label{eq:s+-}
s_\la^-=-\frac{D(\la, z_0)}{2i\Im(z_0)||\mathfrak{p}_{z_0}||^2}.
\end{equation}

Similarly, we have the decomposition

\begin{equation}\label{eq:q}
\mathfrak{q}_{\la}=r_\la +r_\la^+\mathfrak{p}_{z_0}+r_\la^-\mathfrak{p}_{\overline{z_0}},
\end{equation}

where $r_\la\in D(T)$ and

\begin{equation}\label{eq:r+-}
r_\la^-=-\frac{C(\la, z_0)}{2i\Im(z_0)||\mathfrak{p}_{z_0}||^2}.
\end{equation}

Applying the operator $T^*-\overline{z_0}I$ to the Equation (<ref>) gives

(T^*-\overline{z_0}I)\mathfrak{p}_{\la}=(T-\overline{z_0}I)s_\la +s_\la^+(z_0-\overline{z_0})\mathfrak{p}_{z_0},

which is the splitting of the left-hand side   according to the orthogonal decomposition

\begin{equation}\label{eq:os}
\ell^2=(T-\overline{z_0}I)(D(T)) \oplus \Delta^+(z_0).
\end{equation}

Therefore, $s_\la^+(z_0-\overline{z_0})\mathfrak{p}_{z_0}$ is the orthogonal projection of
$(T^*-\overline{z_0}I)\mathfrak{p}_{\la}$ onto  $\Delta^+(z_0)$, and hence

\frac{\mathfrak{p}_{z_0}}{||\mathfrak{p}_{z_0}||^2},

which gives the first formula in (<ref>). The second formula is obtained similarly by applying the operator $(T^*-z_0I)$ to the Equation (<ref>). Notice that $||\mathfrak{p}_{z_0}||=||\mathfrak{p}_{\overline{z_0}}||$.

Applying the operator $T^*-\overline{z_0}I$ to the Equation (<ref>) gives

(T^*-\overline{z_0}I)\mathfrak{q}_{\la}=(T-\overline{z_0}I)r_\la +r_\la^+(z_0-\overline{z_0})\mathfrak{p}_{z_0},

which is the splitting of the left-hand side   according to the orthogonal decomposition
Therefore, $r_\la^+(z_0-\overline{z_0})\mathfrak{p}_{z_0}$ is the orthogonal projection of
$(T^*-\overline{z_0}I)\mathfrak{q}_{\la}$ onto  $\Delta^+(z_0)$, and hence

\frac{\mathfrak{p}_{z_0}}{||\mathfrak{p}_{z_0}||^2},

which gives the first formula in (<ref>) because $T^*(\mathfrak{q}_{\la})=\la\mathfrak{q}_{\la}+e_0$ by (<ref>). The second formula is obtained similarly by applying the operator $(T^*-z_0I)$ to the Equation (<ref>).

For all  $\la\in\C$  we have $\mathfrak{p}_{\la},\mathfrak{q}_{\la}\notin D(T)$.

For $\la =z_0$ we get from (<ref>) and (<ref>)

s_{z_0}=0,\quad s_{z_0}^+=1,\quad s_{z_0}^-=0,

showing that $\mathfrak{p}_{z_0}\notin D(T)$. The case $\la=\overline{z_0}$ follows because $D(T)$ is closed under complex conjugation. Since $z_0$ in the upper half-plane is arbitrary, the  assertion about $\mathfrak{p}_{\la}$ follows for $\la\in\C\setminus\R$.

For $\la\in\R$ we note that $s_\la^-=\overline{s_\la^+}\neq 0$ because $z\mapsto D(\la,z)$ has only real zeros, cf. <cit.>
or Theorem <ref>.

For $\la =z_0$ we get from (<ref>) that

r_{z_0}^-=\frac{-1}{2i\Im(z_0)||\mathfrak{p}_{z_0}||^2}\neq 0,

showing that $\mathfrak{q}_{z_0}\notin D(T)$ and hence also $\mathfrak{q}_{\overline{z_0}}\notin D(T)$. Since $z_0$ in the upper half-plane is arbitrary, the assertion about $\mathfrak{q}_{\la}$ follows for $\la\in\C\setminus\R$.

For $\la\in\R$ we note that $r_\la^{-}=\overline{r_\la^+}$, and by (<ref>) we have

C(\la,z_0)=D(z_0)[A(\la)-\rho C(\la)]\neq 0,\quad \rho:=B(z_0)/D(z_0).

because $D$ has only real zeros. Furthermore, $\Im\rho>0$ because $B/D$ is a Pick function, cf. Proposition <ref>, so also the second factor is non-zero.

Concerning Corollary <ref>, it is clear that $\mathfrak{p}_\la\notin D(T)$ for $\la\notin\R$ because otherwise  $\mathfrak{p}_\la$ would be an eigenvector for $T$ with eigenvalue $\la$, and as $T$ is symmetric, the eigenvalues are real.

A small modification yields also that $\mathfrak{q}_\la\notin D(T)$ for $\la\notin\R$.
In fact, otherwise by symmetry of $T$

\begin{equation}\label{eq:sym}
\langle T\mathfrak{q}_\la, \mathfrak{q}_\la\rangle=\langle \mathfrak{q}_\la,T \mathfrak{q}_\la\rangle.
\end{equation}

The left-hand side of (<ref>) equals $\langle \la\mathfrak{q}_\la+e_0, \mathfrak{q}_\la\rangle=
\la||\mathfrak{q}_\la||^2$
because $\langle e_0,\mathfrak{q}_\la\rangle=0$.

Similarly, the right-hand side of (<ref>) equals $\overline{\la}||\mathfrak{q}_\la||^2$,
and finally $\la$ must be real. We show later that $(T, D(T))$ has no eigenvalues at all, cf. (<ref>).

Proof of Theorem <ref>.

Corollary <ref> proves the first assertion,  and from this assertion it is clear that there exists at most one number $\alpha$ satisfying $\mathfrak{p}_u+\alpha\mathfrak{p}_v\in D(T)$, and similarly with $\beta, \gamma$.

Let us now prove assertion (i) of the theorem.

By (<ref>) we get

\mathfrak{p}_u+\alpha \mathfrak{p}_v=(s_u+\alpha s_v) + (s_u^+ +\alpha s_v^+)\mathfrak{p}_{z_0}+
(s_u^- +\alpha s_v^-)\mathfrak{p}_{\overline{z_0}},

so $\mathfrak{p}_u+\alpha\mathfrak{p}_v\in D(T)$ if and only if

s_u^+ +\alpha s_v^+=s_u^- +\alpha s_v^-=0,

which by (<ref>) is equivalent to

\begin{equation}\label{eq:Di}
D(u,\overline{z_0})+\alpha D(v,\overline{z_0})=D(u,z_0)+\alpha D(v,z_0)=0.
\end{equation}

The determinant of this linear system is

\begin{equation*}\label{eq:det}
\mathcal D:=D(u,\overline{z_0})D(v,z_0)-D(u,z_0)D(v,\overline{z_0})
\end{equation*}

and using Lemma <ref> with

x=\begin{pmatrix} B(u)\\D(u) \end{pmatrix},\; y=\begin{pmatrix}B(\overline{z_0})\\
D(\overline{z_0})\end{pmatrix},\; z=\overline{y},\;w=\begin{pmatrix} B(v)\\D(v)\end{pmatrix},

we get from Corollary <ref> and (<ref>)

\mathcal D=\left|\begin{array}{cc}

However, $D(\overline{z_0},z_0)=-2\Im(z_0)||\mathfrak{p}_{z_0}||^2\neq 0$ so $\mathcal D=0$ iff $D(u,v)=0$.
Therefore, if $\alpha$ is a solution to (<ref>) we have $D(u,v)=0$. Suppose next that
$D(u,v)=0$. To see that (<ref>) has a solution $\alpha$, we notice that $D(v,\overline{z_0})$ and $D(v,z_0)$ cannot both be zero.
In fact, defining $\rho:=B(z_0)/D(z_0)$ we have $\Im(\rho)>0$ because $B/D$ is a Pick function, cf. Proposition <ref>, and
by (<ref>) $D(v,\overline{z_0})=0$ iff $B(v)=\overline{\rho}D(v)$ while $D(v,z_0)=0$ iff $B(v)=\rho D(v)$, so both equations cannot hold. Here we use that $B(v)=D(v)=0$ is impossible because of (<ref>).

If $D(v,z_0)\neq 0$, then $\alpha:=-D(u,z_0)/D(v,z_0)$ satisfies (<ref>) because $\mathcal D=0$. Similarly $\alpha:=-D(u,\overline{z_0})/D(v,\overline{z_0})$ satisfies (<ref>) if $D(v,\overline{z_0})\neq 0$.

Furthermore, in the case $D(v,z_0)\neq 0$ we get using (<ref>) and $D(u,v)=0$ that

so finally $\alpha=B(u,v)$. The case $D(v,\overline{z_0})\neq 0$ is similar.

Proof of (ii):

By (<ref>) we get

\mathfrak{q}_u+\beta \mathfrak{q}_v=(r_u+\beta r_v) + (r_u^+ +\beta r_v^+)\mathfrak{p}_{z_0}+
(r_u^- +\beta r_v^-)\mathfrak{p}_{\overline{z_0}},

so $\mathfrak{q}_u+\beta\mathfrak{q}_v\in D(T)$ if and only if

r_u^+ +\beta r_v^+=r_u^- +\beta r_v^-=0,

which by (<ref>) is equivalent to

\begin{equation}\label{eq:Ci}
C(u,\overline{z_0})+\beta C(v,\overline{z_0})=C(u,z_0)+\beta C(v,z_0)=0.
\end{equation}

The determinant of this linear system is

\begin{equation*}\label{eq:det1}
{\mathcal D}_1:=C(u,\overline{z_0})C(v,z_0)-C(u,z_0)C(v,\overline{z_0})
\end{equation*}

and using Lemma <ref> with

x=\begin{pmatrix} A(u)\\C(u) \end{pmatrix},\; y=\begin{pmatrix}B(\overline{z_0})\\
D(\overline{z_0})\end{pmatrix},\; z=\overline{y},\;w=\begin{pmatrix} A(v)\\C(v)\end{pmatrix}

we get from Corollary <ref> combined with (<ref>), (<ref>), (<ref>)

{\mathcal D}_1=\left|\begin{array}{cc}

As in case (i) we see that
${\mathcal D}_1=0$ iff $A(u,v)=0$.
Therefore, if $\beta$ is a solution to (<ref>), we have  $A(u,v)=0$. Suppose next that
$A(u,v)=0$. To see that (<ref>) has a solution $\beta$, we notice as in (i) that $C(v,\overline{z_0})$ and $C(v,z_0)$ cannot both be zero. For this we use that $A/C$ is a Pick function by Proposition <ref>.

If $C(v,z_0)\neq 0$, then $\beta:=-C(u,z_0)/C(v,z_0)$ satisfies (<ref>). Similarly $\beta:=-C(u,\overline{z_0})/C(v,\overline{z_0})$ satisfies (<ref>) if $C(v,\overline{z_0})\neq 0$.

Furthermore, in the case $C(v,z_0)\neq 0$ we get using (<ref>) and $A(u,v)=0$ that

so finally $\beta=-C(u,v)$. The case $C(v,\overline{z_0})\neq 0$ is similar.

Proof of (iii): By (<ref>) and (<ref>) we get

\mathfrak{p}_u+\gamma \mathfrak{q}_v=(s_u+\gamma r_v) + (s_u^+ +\gamma r_v^+)\mathfrak{p}_{z_0}+
(s_u^- +\gamma r_v^-)\mathfrak{p}_{\overline{z_0}},

so $\mathfrak{p}_u+\gamma\mathfrak{q}_v\in D(T)$ if and only if

s_u^+ +\gamma r_v^+=s_u^- +\gamma r_v^-=0,

which by (<ref>) and (<ref>) is equivalent to

\begin{equation}\label{eq:DCi}
D(u,\overline{z_0})+\gamma C(v,\overline{z_0})=D(u,z_0)+\gamma C(v,z_0)=0.
\end{equation}

The determinant of this linear system is

\begin{equation*}\label{eq:det2}
{\mathcal D}_2:=D(u,\overline{z_0})C(v,z_0)-D(u,z_0)C(v,\overline{z_0})
\end{equation*}

and using Lemma <ref> with

x=\begin{pmatrix} B(u)\\D(u) \end{pmatrix},\; y=\begin{pmatrix}B(\overline{z_0})\\
D(\overline{z_0})\end{pmatrix},\; z=\overline{y},\;w=\begin{pmatrix} A(v)\\C(v)\end{pmatrix}

we get from Corollary <ref> combined with (<ref>), (<ref>), (<ref>)

{\mathcal D}_2=\left|\begin{array}{cc}

As in case (i) we see that
${\mathcal D}_2=0$ iff $B(u,v)=0$.
Therefore, if $\gamma$ is a solution to (<ref>), we have $B(u,v)=0$. Suppose next that $B(u,v)=0$. We see like in (ii) that if $C(v,z_0)\neq 0$, then $\gamma:=-D(u,z_0)/C(v,z_0)$ satisfies (<ref>), and if $C(v,\overline{z_0})\neq 0$, then $\gamma:=-D(u,\overline{z_0})/C(v,\overline{z_0})$ satisfies (<ref>).

We finally see that in both cases $\gamma=-D(u,v)$ because of (<ref>).

The case (ii) can be deduced from case (i) by using the observation that
the polynomials $(q_{n+1}(x)/q_1(x))_{n\ge 0}$ are the orthonormal polynomials associated with the truncated Jacobi matrix $J^{(1)}$ obtained from $J$ by removing the first row and column. See <cit.>. We have

J^{(1)}(Sc)=S(Jc)-a_0\langle c, e_0\rangle,\quad c\in\mathcal F,

where $S$ is the bounded shift operator in $\ell^2$ given by $(Sc)_n:=c_{n+1}, n\ge 0$.

If we let $(T^{(1)}, D(T^{(1)}))$ denote the Jacobi operator associated with $J^{(1)}$, one can prove that

v\in D(T^{(1)})\iff v=Su, u\in D(T)

T^{(1)}(Su)=S(Tu)-a_0\langle u, e_0\rangle,\quad u\in D(T).

§ SELF-ADJOINT EXTENSIONS OF THE JACOBI OPERATOR

As before $(s_n)$ is an indeterminate moment sequence with $s_0=1$. The corresponding Jacobi operator $(T,D(T))$ has deficiency indices $(1,1)$ and the self-adjoint extensions in
$\ell^2$ can be parametrized as the operators $T_t, t\in\R^*=\R\cup \{\infty\}$ with domain

\begin{equation}\label{eq:domTt}
D(T_t)=D(T)\oplus \C  (\mathfrak{q}_0 + t\mathfrak{p}_0) \;\mbox{for}\;t\in\R,\quad
D(T_\infty)=D(T)\oplus \C \mathfrak{p}_0
\end{equation}

and defined by the restriction of $T^*$ to the domain, cf. <cit.>.
We recall that $\mathfrak{p}_0, \mathfrak{q}_0$ are defined in (<ref>).

The purpose of this section is to give some results about the domains $D(T_t)$ of the self-adjoint operators $T_t, t\in\R^*$.

For $t\in\R^*$ we define the solutions to the moment problem

\begin{equation}\label{eq:Next}
\mu_t(\cdot):=\langle E_t(\cdot)e_0,e_0\rangle,
\end{equation}

where $E_t(\cdot)$ is the spectral measure of the self-adjoint operator $T_t$.

The measures $\mu_t, t\in\R^*$ are precisely those measures $\mu\in V$  for which the polynomials $\C[x]$ are dense in $L^2(\mu)$ according to a famous theorem of M. Riesz, cf. [15], and they are called N-extremal  in [1] and von Neumann solutions in [16].  They form a closed subset of $\ext(V)$, the set of extreme points of the convex set $V$. However, $\ext(V)$ is known to be a dense subset of $V$.  They are characterized by the formula

\begin{equation}\label{eq:Npar}
\int\frac{d\mu_t(x)}{x-z}=-\frac{A(z)+tC(z)}{B(z)+tD(z)},\quad z\in \C\setminus\R, t\in\R^*,
\end{equation}

where $A,\ldots,D$ are the entire functions given in (<ref>), cf. <cit.>. Recall that (<ref>) holds.

We summarize some of the properties of $\mu_t$, which can be found in [1] and [17].

(i) The  solution $\mu_t$ is a discrete measure with support equal to the countable zero set $\Lambda_t$ of the entire function $B(z)+tD(z)$, with the convention that $\Lambda_\infty$ is the zero set of $D$. In particular $\Lambda_t\subset\R$ for $t\in\R^*$.

(ii) The support of two different N-extremal solutions are disjoint, and each point $x_0\in\R$ belongs to the support of a unique N-extremal measure $\mu_t$, where
$t\in\R^*$ is given  as $t=-B(x_0)/D(x_0)$ if $D(x_0)\neq 0$ and $t=\infty$ if $D(x_0)=0$.

Let us consider the vector space

\begin{equation}\label{eq:E}
\mathcal E:=\{F_c(z)=\sum_{n=0}^\infty c_np_n(z) \mid c\in\ell^2\}
\end{equation}

of entire functions, cf. (<ref>).  It is a Hilbert space under the norm

||F_c||^2=\sum_{n=0}^\infty |c_n|^2=\int |F_c(x)|^2\,d\mu(x),

where $\mu\in V$ can be arbitrary. It is a reproducing kernel Hilbert space of functions with the reproducing kernel

K(u,v):=\sum_{n=0}^\infty p_n(u)p_n(v),\quad u,v\in\C,

in the sense that

\int K(u,x)F_c(x)\,d\mu(x)=F_c(u),\quad \mu\in V, u\in\C.

Note that $(p_n)$ is an orthonormal basis of $\mathcal E$, and the mapping $c\mapsto F_c$
is a unitary operator of the Hilbert space $\ell^2$ onto $\mathcal E$.

For each N-extremal measure $\mu_t$ the mapping $c\mapsto F_c|_{\supp(\mu_t)}$ is a unitary operator of $\ell^2$ onto $L^2(\mu_t)$. The inverse mapping is given by

f\mapsto \left(\langle f, p_n\rangle_{L^2(\mu_t)}\right)_{n\ge 0},\quad f\in L^2(\mu_t),

\begin{equation}\label{eq:pointwise}
f(x)=\sum_{n=0}^\infty \langle f, p_n\rangle_{L^2(\mu_t)}p_n(x),\quad x\in\supp(\mu_t).
\end{equation}

The series in (<ref>) converges to $f$ in $L^2(\mu_t)$ and converges also locally uniformly for $x\in\C$, but $f$ is apriori only defined on $\supp(\mu_t)$, so the equality holds pointwise for $x\in\supp(\mu_t)$. The series represents a holomorphic extension of $f$ to all of $\C$.

The self-adjoint operator $T_t$ from
(<ref>) is unitarily equivalent with the multiplication operator $M_{\mu_t}$ on
$L^2(\mu_t)$ given by

\begin{equation*}\label{eq:Mt}
M_{\mu_t}f(x)=xf(x),\quad f\in L^2(\mu_t), x\in \supp(\mu_t).
\end{equation*}

Let $\mu_t$ be an N-extremal measure and let $\lambda\in \C\setminus \supp\mu_t$.

\begin{equation}\label{eq:imp}w_{\mu_t}(\lambda)\mathfrak{p}_\lambda+\mathfrak{q}_\lambda \in D(T_t),
\end{equation}

\begin{equation}\label{eq:w}
w_{\mu_t}(\lambda):=\int {1\over x-\lambda}\,d\mu_t(x) =-{C(\lambda,x)\over D(\lambda,x)},\quad x\in \supp\mu_t.
\end{equation}

In particular, the ratio $C(\lambda,x)/D(\lambda,x)$
does not depend on $x\in \supp\mu_t.$

Since $\lambda \notin \supp\mu_t$
the functions $(x-\lambda)^{-1}$ and $x(x-\lambda)^{-1}$ are bounded on $\supp\mu_t$ and in particular they belong to $L^2(\mu_t)$. Thus $(x-\lambda)^{-1}\in D(M_{\mu_t})$ and we find

\int \frac{p_n(x)}{x-\lambda}\,d\mu_t(x)=w_{\mu_t}(\lambda)p_n(\lambda)+q_n(\lambda),

where $w_{\mu_t}(\lambda)$ is given by the first equality of  (<ref>).
Moreover, by (<ref>)

\begin{equation}\label{eq:x}(x-\lambda)^{-1}=\sum_{n=0}^\infty [w_{\mu_t}(\lambda)p_n(\lambda)+q_n(\lambda)]p_n(x), \quad x\in \supp\mu_t.
\end{equation}

In view of the unitary equivalence of $T_t$ and $M_{\mu_t}$ we get (<ref>).
Multiplying (<ref>) sidewise by $x-\lambda$ gives

1=w_{\mu_t}(\lambda)(x-\lambda)\sum_{n=0}^\infty p_n(\lambda)p_n(x)+(x-\lambda) \sum_{n=0}^\infty q_n(\lambda)p_n(x),\quad x\in\supp(\mu_t).

By (<ref>) and (<ref>) we therefore get

w_{\mu_t}(\lambda)D(\lambda,x)+C(\lambda,x)=0,\quad x\in \supp\mu_t.

Assume $D(\lambda,x)=0.$ Then $C(\lambda,x)=0,$ but this gives a contradiction to (<ref>).
Hence $D(\lambda,x)\neq 0$ and

w_{\mu_t}(\lambda)=-{C(\lambda, x)\over D(\lambda,x)},\quad x\in \supp\mu_t,

which gives the second part of (<ref>).

Using the formulas (<ref>) and (<ref>) in the last expression in (<ref>), we get formula (<ref>) with
$t=-B(x)/D(x)$ independent of $x\in\supp(\mu_t)$ if $D(x)\neq 0$, and $t=\infty$ if $D(x)=0$.

Note also that $w_{\mu_t}(\lambda)\mathfrak{p}_\lambda+\mathfrak{q}_\lambda \notin D(T)$ by Theorem <ref> (iii) because $B(\la,\la)=-1$.

We know from Theorem <ref> that $\mathfrak{p}_\la, \mathfrak{q}_\la\notin D(T)$ for every $\la\in\C$. We shall now clarify when  $\mathfrak{p}_\la, \mathfrak{q}_\la$  belong to the domain of the self-adjoint extension $T_t$ associated with the N-extremal measure $\mu_t$.

For $\la\in\C$ and $t\in\R^*$ we have

\begin{equation}\label{eq:pD(T_t)}
\mathfrak{p}_\la\in D(T_t) \iff D(\la,x)=0 \;\forall x\in\supp(\mu_t)\iff \la\in\supp(\mu_t).
\end{equation}

\begin{equation}\label{eq:qD(T_t)}
\mathfrak{q}_\la\in D(T_t) \iff C(\la,x)=0 \;\forall x\in\supp(\mu_t).
\end{equation}

We define the entire functions $g_\la, h_\la\in L^2(\mu_t)$ by

g_\la(x):=\sum_{k=0}^\infty p_k(\la)p_k(x),\quad h_\la(x):=\sum_{k=0}^\infty q_k(\la)p_k(x),\quad x\in\C.

If $\mathfrak{p}_\la\in D(T_t)$ then $(T_t-\la I)\mathfrak{p}_\la=0$ by (<ref>), because $T_t$ is a restriction of $T^*$. Furthermore, by the unitary equivalence between
$T_t$ and the multiplication operator $M_{\mu_t}$ on $L^2(\mu_t)$ we have $xg_\la(x)\in
L^2(\mu_t)$ and $D(\la,x)=(\la-x)g_\la(x)=0$ in $L^2(\mu_t)$. By discreteness of $\mu_t$ we get $D(\la,x)=0$ for all $x\in\supp(\mu_t)$. The last equivalence of (<ref>) follows from Remark <ref>. On the other hand, it
is easy to see that the last two equivalent conditions of (<ref>) imply $\mathfrak{p}_\la\in D(T_t)$, because the zero-function $x\mapsto (\la-x)g_\la(x)$ as well as $\la g_\la(x)$ are in $L^2(\mu_t)$, hence also $xg_\la(x)\in L^2(\mu_t)$. Therefore $g_\la\in D(M_{\mu_t})$ and finally
$\mathfrak{p}_\la\in D(T_t)$.

If $\mathfrak{q}_\la\in D(T_t)$ then $(T_t-\la I)\mathfrak{q}_\la=e_0$ by (<ref>), because $T_t$ is a restriction of $T^*$. This shows that $(x-\la)h_\la(x)=1$ in $L^2(\mu_t)$ and hence $C(\la,x)=0$ for all $x\in\supp(\mu_t)$. On the other hand, if $C(\la,x)=0$ for $x\in \supp(\mu_t)$, we conclude that $xh_\la(x)\in L^2(\mu_t)$, hence $\mathfrak{q}_\la\in D(T_t)$. This establishes (<ref>).

We know from Proposition <ref> that $\supp(\mu_t)$ is the the zero set
of the entire function $B(z)+tD(z)$ understood as $D(z)$ if $t=\infty$. Using this we get the following Corollary about $\mathfrak{p}_\la$. We get a similar result about $\mathfrak{q}_\la$ from (<ref>).

For $t\in\R$ and $t=\infty$ we have

\begin{eqnarray*}
\mathfrak{p}_\la \in  D(T_t) &\iff &  B(\la)+tD(\la)=0,\\
\mathfrak{q}_\la \in  D(T_t) &\iff & A(\la)+tC(\la)=0.\\
\mathfrak{p}_\la \in  D(T_\infty) &\iff &  D(\la)=0,\\
\mathfrak{q}_\la \in  D(T_\infty) &\iff & C(\la)=0.
\end{eqnarray*}

In particular $\mathfrak{p}_\la$ and $\mathfrak{q}_\la$ only belong to $D(T_t)$ if
$\la\in\R$, and for $\la\in\R$ they belong to a unique $D(T_t)$. Furthermore, they never belong to the same domain $D(T_t)$.

Since $D(T)\subset D(T_t)$ for all $t\in\R^*$, it is clear that Corollary <ref> implies that $\mathfrak{p}_\la, \mathfrak{q}_\la\notin D(T)$ as stated in Corollary <ref>.

We also have a kind of converse to Theorem <ref>.

Assume that $\la,\tau\in\C$ are such that $\tau\mathfrak{p}_\la +\mathfrak{q}_\la\in D(T_t)$ for some $t\in\R^*$. Then $\la\notin \supp(\mu_t)$ and
$\tau=w_{\mu_t}(\la)$ given by (<ref>).

Assume that $\la\in\supp(\mu_t)$. By Theorem <ref> we know that $\mathfrak{p}_\la\in D(T_t)$ and hence $\mathfrak{q}_\la\in D(T_t)$, contradicting Corollary <ref>.

Having established $\la\notin\supp(\mu_t)$, we get by Theorem <ref> that $(\tau-w_{\mu_t}(\la))\mathfrak{p}_\la\in D(T_t)$, but since
$\mathfrak{p}_\la\notin D(T_t)$, we get

Let $t\in\R^*$ and $\la\in \C\setminus \supp(\mu_t)$ be given.
Then there exists a unique pair $(s,c)\in D(T)\times \C$ depending on $t,\la$ such that

\begin{equation*}
s +c(\mathfrak{q}_0+t\mathfrak{p}_0), &   t\in\R\\
s +c\mathfrak{p}_0, & t=\infty.
\end{array}
\right.
\end{equation*}

We have

\begin{equation*}
-1/(B(\la)+tD(\la)), & t\in\R \\
-1/D(\la), & t=\infty,
\end{array}
\right.
\end{equation*}

and $s$ is given by inserting the value of $c$.

Recall that $\la\in\C\setminus \supp(\mu_t)$ if and only if $B(\la)+tD(\la)\neq 0$ when $t\in\R$, and that $\la\in\C\setminus \supp(\mu_\infty)$ if and only if
$D(\la)\neq 0$. The existence and uniqueness of $(s,c)$ follow from Theorem <ref> and formula (<ref>).

In case $t\in\R$ we have

w_{\mu_t}(\lambda)\mathfrak{p}_\lambda+\mathfrak{q}_\lambda-c(\mathfrak{q}_0+t\mathfrak{p}_0)\in D(T),

and fixing $z_0$ in the open upper half-plane we have by Proposition 2.1

w_{\mu_t}(\la)s_\la^{+} +r_\la^{+} - c(r_0^{+}+ts_0^{+})= w_{\mu_t}(\la)s_\la^{-} +r_\la^{-} - c(r_0^{-}+ ts_0^{-})=0,

or equivalently by (<ref>) and (<ref>)

\begin{equation}\label{eq:overlinez_0}
w_{\mu_t}(\la) D(\la,\overline{z_0})+C(\la,\overline{z_0})-c(C(0,\overline{z_0})+tD(0,\overline{z_0}))=0,
\end{equation}

\begin{equation}\label{eq:z_0}
w_{\mu_t}(\la) D(\la,z_0)+C(\la,z_0)-c(C(0,z_0)+tD(0,z_0))=0.
\end{equation}

From (<ref>) and (<ref>) we get for $x\in\supp(\mu_t)$

\begin{eqnarray*}
c &=& -\frac{w_{\mu_t}(\la) D(\la,z_0)+C(\la,z_0)}{B(z_0)+tD(z_0)}\\
&=&  \frac{C(\la,x)D(\la,z_0)-D(\la,x)C(\la,z_0)}{(B(z_0)+tD(z_0))D(\la,x)}\\
\end{eqnarray*}

Here we have first used (<ref>) and next used (<ref>)  twice. Finally we recall that $t=-B(x)/D(x)$ for $x\in\supp(\mu_t)$, cf. Proposition <ref>.

Note that (<ref>) leads to the same expression for $c$.

The case $t=\infty$ is treated in the same way.

§ PARAMETRIZATIONS OF THE DOMAIN OF THE JACOBI OPERATOR

The Jacobi operator $(T,D(T))$ in the indeterminate case is regular in the sense of <cit.>, i.e., for any $z\in\C$ there exists $d(z)>0$ such that

\begin{equation}\label{eq:reg}
||(T-zI)c||\ge d(z)||c||,\quad c\in D(T).
\end{equation}

For $z\in\C\setminus\R$ this is true with $d(z)=|\Im(z)|$, and for $z\in\R$ let $t_0\in\R^*$ be such that $z\in\supp(\mu_{t_0})$. For $t\in\R^*\setminus\{t_0\}$ the distance

d_t(z):=\min\{|z-x|\mid x\in \supp(\mu_t)\}>0,

can be used in (<ref>), since we have

||(T-zI)c||^2= \int \left|(x-z)F_c(x)\right|^2\,d\mu_t(x)\ge d_t(z)^2||c||^2,

where $F_c$ is given in (<ref>).

We have the orthogonal decomposition in closed subspaces

\begin{equation}\label{eq:regor}
\ell^2=(T-zI)(D(T))\oplus \C\mathfrak{p}_{\overline{z}},\quad z\in\C.
\end{equation}

The operator $(T,D(T))$ has no eigenvalues, has empty continuous spectrum, and the spectrum $\sigma(T)=\C$ is equal to the residual  spectrum, cf. <cit.>.

For $z_0\in\C$ we have the orthogonal expansion

\begin{equation}\label{eq:1}
\frac{p_n(z)-p_n(z_0)}{z-z_0}=\sum_{k=0}^{n-1} a_{n,k}(z_0)p_k(z), \quad z\in\C
\end{equation}

of the polynomial $(p_n(z)-p_n(z_0)/(z-z_0)$ of degree $n-1$, and it is easy to see that

\begin{equation}\label{eq:2}
a_{n,k}(z_0)=\int \frac{p_n(x)-p_n(z_0)}{x-z_0}p_k(x)\,d\mu(x)=  q_n(z_0)p_k(z_0)-p_n(z_0)q_k(z_0),
\end{equation}

where $\mu\in V$ is an arbitrary solution to (<ref>), cf. <cit.>.

The coefficients $a_{n,k}(z_0)$ from (<ref>)

\begin{equation}\label{eq:3}
|a_{n,k}(z_0)|^2\le \left(|p_n(z_0)|^2+|q_n(z_0)|^2\right)\left(|p_k(z_0)|^2+|q_k(z_0)|^2\right).
\end{equation}

\begin{equation}\label{eq:3bis}
\sum_{n=k+1}^\infty |a_{n,k}(z_0)|^2\le (||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2)(|p_k(z_0)|^2+|q_k(z_0)|^2).
\end{equation}

\begin{equation}\label{eq:dif}
\sum_{n=0}^\infty\left|\frac{p_n(z)-p_n(z_0)}{z-z_0}\right|^2\le ||\mathfrak{p}_z||^2\left(||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2\right)^2.
\end{equation}

In particular, for $z\to z_0$

\begin{equation*}\label{eq:diff}
\sum_{n=0}^\infty|p_n'(z_0)|^2\le ||\mathfrak{p}_{z_0}||^2\left(||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2\right)^2.
\end{equation*}

Formula (<ref>) is a consequence of the Cauchy-Schwarz inequality.

From (<ref>) and (<ref>) we get

\begin{eqnarray*}
\left|\frac{p_n(z)-p_n(z_0)}{z-z_0}\right|^2 &\le& \sum_{k=0}^{n-1} |a_{n,k}(z_0)|^2\sum_{k=0}^{n-1} |p_k(z)|^2\\
&\le& ||\mathfrak{p}_z||^2\sum_{k=0}^{n-1} \left(|p_n(z_0)|^2+|q_n(z_0)|^2\right) \left(|p_k(z_0)|^2+|q_k(z_0)|^2\right),
\end{eqnarray*}

and finally

\begin{eqnarray*}
\lefteqn{\sum_{n=0}^\infty \left|\frac{p_n(z)-p_n(z_0)}{z-z_0}\right|^2 }\\
&\le & ||\mathfrak{p}_z||^2
\sum_{k=0}^\infty \left[\left(|p_k(z_0)|^2+|q_k(z_0)|^2\right) \sum_{n=k+1}^\infty
\left(|p_n(z_0)|^2+|q_n(z_0)|^2\right)\right]\\
&\le& ||\mathfrak{p}_z||^2 \left(||\mathfrak{p}_{z_0}||^2+\mathfrak{p}_{z_0}||^2\right)^2,
\end{eqnarray*}

which yields (<ref>).

We shall now show that the Hilbert space $\mathcal E=\{F_c(z)\}$ defined in (<ref>) is stable under difference quotients:

For $c\in\ell^2$  and $z_0\in\C$ there exists $\xi(c,z_0)\in\ell^2$ such that

\begin{equation}\label{eq:Fcq}
\frac{F_c(z)-F_c(z_0)}{z-z_0}=F_{\xi(c,z_0)}(z) \in\mathcal E,
\end{equation}

and the coordinates of $\xi(c,z_0)$ are defined by

\begin{equation}\label{eq:xi}
\xi_k(c,z_0)=\sum_{n=k+1}^\infty c_n a_{n,k}(z_0),\quad k\ge 0.
\end{equation}

\begin{equation}\label{eq:normxi}
||\xi(c,z_0)|| \le ||c||\left(||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2\right).
\end{equation}

The series in (<ref>) is absolutely convergent being the product of two $\ell^2$ sequences. Furthermore, by the Cauchy-Schwarz inequality and (<ref>) we get

|\xi_k(c,z_0)|^2\le ||c||^2 (||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2)\left(|p_k(z_0)|^2+|q_k(z_0)|^2\right),

and therefore $(\xi_k(c,z_0))\in\ell^2$ and (<ref>) holds.

We next find

\frac{F_c(z)-F_c(z_0)}{z-z_0}=\sum_{n=0}^\infty c_n\frac{p_n(z)-p_n(z_0)}{z-z_0},\quad z\neq z_0.

Inserting the expression
(<ref>) on the right-hand side, we get for $z\neq z_0$

\begin{eqnarray*}
\frac{F_c(z)-F_c(z_0)}{z-z_0}&=&\sum_{n=1}^\infty c_n\sum_{k=0}^{n-1} a_{n,k}(z_0)p_k(z)\\
&=&\sum_{k=0}^\infty p_k(z) \sum_{n=k+1}^\infty c_n a_{n,k}(z_0)\\
&=&\sum_{k=0}^\infty\xi_k(c,z_0) p_k(z),
\end{eqnarray*}

where the rearrangement is possible due to absolute convergence:

\begin{eqnarray*}
\lefteqn{\left|\frac{F_c(z)-F_c(z_0)}{z-z_0}\right| }\\
&\le& \sum_{n=1}^\infty |c_n|\sum_{k=0}^{n-1} |a_{n,k}(z_0)|\,|p_k(z)|\\
&=&\sum_{k=0}^\infty |p_k(z)| \sum_{n=k+1}^\infty |c_n|\,|a_{n,k}(z_0)| \\
&\le& ||c||\sum_{k=0}^\infty |p_k(z)|\left[\sum_{n=k+1}^\infty |a_{n,k}(z_0)|^2\right]^{1/2}\\
&\le& ||c||\sum_{k=0}^\infty |p_k(z)|\left[\left(||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2\right)\left(|p_k(z_0)|^2+|q_k(z_0)|^2\right)\right]^{1/2}\\
&\le& ||c||\left(||\mathfrak{p}_{z_0}||^2+||\mathfrak{q}_{z_0}||^2\right)^{1/2}||\mathfrak{p}_z||\left(\sum_{k=0}^\infty\left(|p_k(z_0)|^2+|q_k(z_0)|^2\right)\right)^{1/2}\\
\end{eqnarray*}

where we have used (<ref>).

It is now clear that the entire functions
$z\mapsto (F_c(z)-F_c(z_0))/(z-z_0)$, with value $F'_c(z_0)$ for $z=z_0$, and $F_{\xi(c,z_0)}(z)$ agree.

Let $\Xi_{z_0}$ denote the bounded operator in $\ell^2$ defined by

\begin{equation}\label{eq:para}
\Xi_{z_0}(c):=\xi(c,z_0),\quad z_0\in\C,\;c\in\ell^2,
\end{equation}

where $\xi(c,z_0)$ is defined in Theorem <ref>.
We have $\Xi_{z_0}(\ell^2)=D(T)$  and $\ker(\Xi_{z_0})=\C e_0$ for each $z_0\in\C$.

Furthermore, for $z_0\in\C$

\begin{equation}\label{eq:split}
(T-z_0 I)\Xi_{z_0}(c)+F_c(z_0)e_0=c, \quad c\in\ell^2.
\end{equation}

The restriction of $\Xi_{z_0}$ to $(T-z_0I)(D(T))$ is a bijection onto $D(T)$ equal to $(T-z_0I)^{-1}$.

Let $U:\ell^2\to \mathcal E$ denote the unitary mapping given by  $U(c)=F_c$. Then

\begin{equation*}
U(Jc)(z)=z\sum_{k=0}^\infty c_kp_k(z),\quad c\in\mathcal F, z\in \C,
\end{equation*}

i.e., $U$ is the intertwining operator between $J$ and the densely defined operator of multiplication with $z$ on $\C[z]\subset \mathcal E$. Therefore

\begin{equation}\label{eq:unit}
U(Tc)(z)=zF_c(z),\quad c\in D(T), z\in\C.
\end{equation}

For $c\in\ell^2$ and $z_0\in\C$ we have

c-F_c(z_0)e_0 \perp \mathfrak{p}_{\overline{z_0}},

so by (<ref>) $c-F_c(z_0)e_0$ belongs to $(T-z_0 I)(D(T))$. Therefore, there exists a unique vector $v\in D(T)$ such that

\begin{equation}\label{eq:v}
c-F_c(z_0)e_0 =(T-z_0 I)(v),
\end{equation}

and applying $U$ to (<ref>) we get by (<ref>)

F_c(z)-F_c(z_0)=(z-z_0)F_v(z),\quad z\in\C.

Now (<ref>) shows that $F_v(z)=F_{\xi(c,z_0)}(z)$  for $z\neq z_0$, hence for all $z$, and finally  $v=\xi(c,z_0)$, showing that $\Xi_{z_0}(c)\in D(T)$. Inserting $v=\xi(c,z_0)$ in (<ref>) yields (<ref>).

For $v\in D(T)$ we define $c=(T-z_0 I)(v)$. Then $F_c(z_0)=0$ as $c\perp \mathfrak{p}_{\overline{z_0}}$ by (<ref>), and then (<ref>) gives $(T-z_0I)\Xi_{z_0}(c)=c$. By injectivity of $T-z_0I$ we get $v=\Xi_{z_0}(c)=(T-z_0 I)^{-1}(c)$.

It is easy to see that $\xi(e_0,z_0)=0$, hence $\C e_0\subseteq \ker(\Xi_{z_0})$, and from (<ref>) the converse inclusion follows.

The operator $\Xi_{z_0}$ defined in (<ref>) is  seen to satisfy

\Xi_{z_0}(e_n)=\sum_{k=0}^{n-1} a_{n,k}(z_0) e_k,\quad n\ge 1,

and it follows easily that $\Xi_{z_0}(\mathcal F)=\mathcal F$.

Moreover, since $(\C e_0)^\perp=\{c\in\ell^2 \mid c_0=0\}$, we
have the following parametrizations of $D(T)$

D(T)=\{\Xi_{z_0}(c) \mid c\in\ell^2, c_0=0\},\quad z_0\in\C.

§ APPENDIX

We need the following result about the Nevanlinna functions defined in the Introduction.

For $u,v,w\in\C$ we have

\begin{eqnarray}
A(u,v)&=& C(u,w)A(w,v)-A(u,w)B(w,v)\label{eq:A3}\\
B(u,v)&=& D(u,w)A(w,v)-B(u,w)B(w,v)\label{eq:B3}\\
C(u,v)&=& C(u,w)C(w,v)-A(u,w)D(w,v)\label{eq:C3}\\
D(u,v)&=& D(u,w)C(w,v)-B(u,w)D(w,v)\label{eq:D3}.
\end{eqnarray}

From the obvious relations

A(u,v)=-A(v,u),\; B(u,v)=-C(v,u),\; D(u,v)=-D(v,u)

and putting $w=0$ in the formulas of Theorem <ref>, we get the following formulas in terms of the one variable functions (<ref>):

For $u, v\in\C$ we have

\begin{eqnarray}
A(u,v)&=& \left|\begin{array}{cc}
B(u,v)&=& \left|\begin{array}{cc}
C(u,v)&=& \left|\begin{array}{cc}
D(u,v)&=& \left|\begin{array}{cc}
\end{eqnarray}

We have not been able to find the formulas of Theorem <ref> in the literature, so we indicate a proof.  The formulas of Corollary <ref> expressing the two variable functions in terms of the one variable functions were, as far as we know,  first  given in [11] and included in <cit.>.  (Unfortunately there is a misprint in the exercise: $B$ and $C$ are interchanged.)

We begin by introducing polynomial approximations to the Nevanlinna functions.

<cit.> For $u,v\in\C$ and $n\ge 0$ we have

\begin{eqnarray*}
B_n(u,v)&:=&-1+(u-v)\sum_{k=0}^n p_k(u)q_k(v)=
C_n(u,v)&:=&1+(u-v)\sum_{k=0}^n q_k(u)p_k(v)=
\end{eqnarray*}

It is important to notice that

\begin{equation*}\label{eq:det4}
\left|\begin{array}{cc}\ A_n(u,v)&\;B_n(u,v)\\
C_n(u,v)&\; D_n(u,v)\end{array}\right|=1\textrm{ for }(u,v)\in\mathbb
\end{equation*}

cf. <cit.>.

For later use we introduce the transfer matrix with determinant 1

\begin{equation}\label{eq:transfer1}
h_n(u,v)=\left(\begin{array}{cc} C_n(u,v)&\; A_n(u,v)\\
-D_n(u,v)&\; -B_n(u,v)\end{array}\right),\quad u,v\in\mathbb
C,\;n\ge 0.
\end{equation}

The name transfer matrix is motivated by

For $u,v\in\mathbb C,\;n\ge 0$ we have

\begin{equation}\label{eq:transfer2}
\left(\begin{array}{cc}
\end{equation}

The four formulas of Proposition <ref> can be expressed
as the matrix equation

\left(\begin{array}{cc}

However, by <cit.>

\left(\begin{array}{cc}
\left(\begin{array}{cc}

and (<ref>) follows.

By the uniqueness of a matrix $h_n(u,v)$ satisfying (<ref>) we get:
For $u,v,w\in\mathbb C,\;n\ge 0$ we

\begin{equation*}\label{eq:transfer3}
\end{equation*}

\begin{equation*}\label{eq:transfer4}
\end{equation*}

Proof of Theorem <ref>. Letting $n$ tend to infinity in (<ref>) we obtain the entire matrix function

h(u,v)=\left(\begin{array}{cc} C(u,v)&\; A(u,v)\\
-D(u,v)&\; -B(u,v)\end{array}\right),\quad u,v\in\mathbb

with determinant 1 satisfying $h(u,w)h(w,v)=h(u,v)$, which is equivalent to the formulas
(<ref>),(<ref>),(<ref>) and (<ref>) of the theorem.$\quad\square$

The Möbius transformation $M(u,v):\C^*\to \C^*$ defined by

\begin{equation*}\label{eq:mob}
M(u,v)(z):= \frac{C(u,v) z+A(u,v)}{-D(u,v)z-B(u,v)},\quad z\in\C^*,
\end{equation*}

maps the Weyl circle $K_v$ onto the Weyl circle $K_u$, where $K_u=\R^*$ if $u\in\R$. For $u\in\C\setminus\R$ the Weyl circle is defined in <cit.>.

The following Lemma unifies some calculations:
For vectors $x,y,z,w\in\mathbb C^2$ we have
the following determinant equation

|x\;\;y| & |x\;\;z|\\
|w\;\;y| & |w\;\;z|\end{array}\right|=

\textrm{ etc.}

First method: Direct computation.

Second method: Define

where it should be noticed that $y,z,w$ appear in its cyclic
permutations. Clearly $M$ is a $4$-linear form on $\mathbb C^2$,
and it is alternating, i.e., is zero, if any two arguments agree. An
alternating $4$-linear form on a vector space of dimension $\leq 3$
is identically zero, hence $M\equiv0$.

In various proofs we need that certain functions are Pick function, i.e., holomorphic functions in the cut plane $\C\setminus\R$ with certain properties, see [12].

The meromorphic functions $B/D$ and $A/C$ are Pick functions, i.e., they map the upper (resp. lower) open half-plane into itself.

The result about $B/D$ is in <cit.>. The result about $A/C$ can be deduced from the previous result by considering the indeterminate moment problem corresponding to the truncated Jacobi matrix $J^{(1)}$ considered in Remark <ref>. There are simple relations between the Nevanlinna functions $\widetilde{A},\ldots,\widetilde{D}$ of the truncated problem and those of the original moment problem, see [14]:

\begin{equation*}\label{eq:netr}
A(z)=a_0^{-2}\widetilde{D}(z),\quad C(z)=-b_0 a_0^{-2}\widetilde{D}(z)-\widetilde{B}(z),\quad z\in\C.
\end{equation*}

Therefore $-C/A=b_0+a_0^2(\widetilde{B}/\widetilde{D})$, which shows that $-C/A$ is a Pick function, and so is $A/C$.

By a famous Theorem of M. Riesz each of the functions $F_c$ defined in (<ref>) are of minimal exponential type meaning that for each $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that

|F_c(z)|\le C_\varepsilon e^{\varepsilon |z|},\quad z\in\C.

This  follows from the Cauchy-Schwarz inequality because the norm $||\mathfrak{p}_z||$ satisfies the same inequality by <cit.>. Using that the polynomials $q_{n+1}/q_1$ are the orthonormal polynomials for the indeterminate truncated Jacobi matrix $J^{(1)}$, cf. Remark <ref>, we also get that the functions $G_c$ from (<ref>) are of minimal exponential type.

We next recall an important property of these functions in case they are not polynomials.

For each $c\in \ell^2\setminus\mathcal F$ the functions
$F_c, G_c$ are transcendental  and have a countably infinite set of zeros.

In particular, for each $v\in\C$ the functions of the variable $u$, $A(u,v), B(u,v)$, $C(u,v), D(u,v)$ have a countably infinite set of zeros.

An entire transcendental function $f$ of minimal exponential type has a countably infinite set of zeros. In fact, the order $\rho$ of $f$ is either strictly less than 1 or equal to 1, and in the latter case the type of $f$ is zero. In the first case the result follows from the Hadamard factorization Theorem, cf. <cit.>. In the second case the result follows from a Theorem of Lindelöf, see <cit.>.

For $v\in\C$ and $F$ being one of the functions $A,\ldots,D$, we see that $u\mapsto F(u,v)$ is an entire transcendental function of minimal exponential type.

Proof of Theorem <ref>: Case 1: Let us first consider the case of $D$ with
$Z(D)_v=\{u\in\C\mid D(u,v)=0\}$ for given $v\in\C$, cf. (<ref>).

If $v\in\R$ then $Z(D)_v\subset\R$ by <cit.>, and furthermore $Z(D)_v$ equals the support of the unique N-extremal measure which contains $v$ in the support.

If $v\in\C\setminus\R$, then $D(v)\neq 0$, and using (<ref>) we get

D(u,v)=D(v)[B(u)-\rho D(u)],\;\rho:=B(v)/D(v),

so $u\in Z(D)_v$ iff $B(u)=\rho D(u)$. For such $u$ we must have $D(u)\neq 0$ for otherwise $B(u)=D(u)=0$ contradicting (<ref>). This gives $u\in Z(D)_v$ iff
$B(u)/D(u)=\rho$. Using that $B/D$ is a Pick function by  Proposition <ref>,  we see that $u,v$ belong to the same half-plane.

Case 2: We consider $Z(B)_v$ and  use (<ref>), viz.

Let first $v\in\R$. If $C(v)=0$ then $A(v)\neq 0$ by (<ref>), so $B(u,v)=0$ iff $D(u)=0$, hence $Z(B)_v\subset\R$. If $C(v)\neq 0$ then

B(u,v)=C(v)[B(u)-\tau D(u)],\;\tau:=A(v)/C(v)\in\R,

so $Z(B)_v$ is the zero set of $B-\tau D$, hence real  by Proposition <ref>.

Let next $v\in\C\setminus \R$. Then $C(v)\neq 0$, so $u\in Z(B)_v$ iff $B(u)=\tau D(u)$ with $\tau$ as above, but this is only possible if $D(u)\neq 0$ and hence $B(u)/D(u)=\tau$.
Using that both  $A/C$ and $B/D$ are Pick functions, cf. Proposition <ref>, we see that $u,v$ belong to the same half-plane.

Case 3: We consider $Z(C)_v$ and  use (<ref>), viz.

By considering the cases $v\in\R$ and $v\in\C\setminus \R$ separately  and factor out $D(v)$ in case it is non-zero, we may
proceed as in case 2.

Finally, the case of $Z(A)_v$  follows from the case 1 by considering the truncated case as in Remark <ref>. $\square$

As noticed in the proof above one has for $v\in\R$:

D(u,v)=0 \iff u\in\supp(\mu),

where $\mu$ is the N-extremal measure such that $v\in\supp(\mu)$.

Compare also with Remark <ref>.

Proof of Proposition <ref>:

Case (i): From the proof of Theorem <ref> (i) we know that
$B(u,v)=-D(u,z)/D(v,z)$ for all $z\in\C\setminus\R$ since $D(v,z)\neq 0$ for these $z$. By assumption $D(x,v)\neq 0$ for $u<x<v$, so by continuity
$B(u,v)=-D(u,x)/D(v,x)$ for these $x$. We next observe that

\begin{equation}\label{eq:Bhelp}
B(u,v)\frac{v-x}{x-u}=\frac{v-x}{D(v,x)}\frac{D(u,x)}{u-x} \neq 0,\quad u<x<v,
\end{equation}

\lim_{x\to u^+}\frac{D(u,x)}{u-x}=||\mathfrak{p}_u||^2,\quad
\lim_{x\to v^-}\frac{D(v,x)}{v-x}=||\mathfrak{p}_v||^2,

so the function in (<ref>) is positive for $u<x<v$, hence $B(u,v)>0$.

Case (ii): This case is reduced to case (i) for the truncated Jacobi matrix from Remark <ref>. If $\widetilde{A},\ldots,\widetilde{D}$ denote the Nevalinna functions of two variables for the truncated case, the following formulas can be found in [14]. (The reader is warned that this reference follows the normalization of [11].)

\begin{equation*}\label{eq:tilde}
\widetilde{B}(u,v)=(v-b_0)A(u,v)-C(u,v),\quad \widetilde{D}(u,v)=a_0^2A(u,v),
\end{equation*}

and since we assume that $A(u,v)=0$, we have $-C(u,v)=\widetilde{B}(u,v)>0$.

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Christian Berg

Department of Mathematical Sciences, University of Copenhagen

Universitetsparken 5, DK-2100 Copenhagen, Denmark

e-mail<EMAIL_ADDRESS>
Ryszard Szwarc

Institute of Mathematics, University of Wrocław

pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

e-mail<EMAIL_ADDRESS>
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Integr. Equ. Oper. Theory 92(1) (2020), 1–9.

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# On the Relationship Between Network Topology and Throughput in Mesh Optical
Networks

Daniel Semrau1  Shahzaib Durrani1  Georgios Zervas1  Robert I. Killey1  Polina
Bayvel1<EMAIL_ADDRESS>1Optical Networks Group, Department of Electronic
and Electrical Engineering, UCL (University College London), Torrington Place,
London WC1E 7JE, United Kingdom

###### Abstract

The relationship between topology and network throughput of arbitrarily-
connected mesh networks is studied. Taking into account nonlinear channel
properties, it is shown that throughput decreases logarithmically with
physical network size with minor dependence on network ellipticity.

###### keywords:

Wavelength routed optical networks, Fibre based networks, physical layer aware
networking, maximising network throughput

##  1 Introduction

Optical networks underpin the infrastructure of the modern Internet. To
maximise the throughput of these networks, physical propagation effects must
be taken into account. Thus, the network throughput becomes a nonlinear
function of fibre parameters, link distances and lightpath configurations, and
networks can no longer be only characterised only by logical topologies, i.e.
a set of vertices and edges.

The development of low complexity transmission models of nonlinear fibre
propagation enabled their incorporation in network algorithms. As a
consequence, various studies have suggested new optimisation strategies for
wavelength and route assignments nw1 ; nw2 ; nw4 ; nw7 , launch power
optimisations and the use of various modulation formats and code rates Ives ;
nw3  within the same network, tailored to the lightpath quality.

While many works analyse specific topologies, often derived from
existing/published networks, more general relationships between physical
network topology and throughput are largely nonexistant. Most work Baroni ;
nw5  analysed network performance as a function of node, link number and
connectivity matrix in the absence of physical transmission properties.
However, physical transmission properties must now be included in order to
calculate both throughput as well as derive scaling laws that lead to a more
intelligent use of current optical networks.

In this work, arbitrarily-connected mesh optical networks (ACMN) were analysed
to analyse the relationship between physical topology properties and network
throughput. The focus of the work was two network properties - scaling the
network size and paramterising network ellipticity.

Topology generation:Generate ACMNsAssign linkdistances$\times 100000$$\times
1000$$\times 100$For each topology:Compute NSR weightsfrom link distances
using GN modelFor each node-pair: compute paths with capacity $C_{i}\geq
x\cdot\underset{\forall i}{\max}\left(C_{i}\right)$ using k-shortest
pathsAssign $N_{\lambda}$ ligth paths to each node-pair such thatcongestion is
minimisedC-band reached on any link?Compute light pathcapacities based on RWA
using GN modelprovides ‘equal-cost‘ paths for each node-
pairyesno$N_{\lambda}$++10003000$0$$0.05$$0.1$$0.15$Link distance $d$
[km]ProbabiltyNSFNETACMN Figure 1: Flow chart of ACMN generation and
throughput estimation. 1000 distinct topologies were generated where each was
assigned 100 different link distance realisations. The throughput relaxation
$x\in\left[0,1\right]$ was swept to maximise total network throughput and to
trade-off individual lightpath capacity for decreased network congestion (i.e.
increased $N_{\lambda}$).

##  2 Methodology and throughput estimation

To study the relationship between topology and network throughput, 1000
distinct logical topologies were generated with 14 nodes and 21 links each, as
variants of the NSFNET topology NSFNET , chosen as the reference network.
Physical link lengths were randomly assigned to each network link (edge)
according to a probabilty density function (PDF), as shown in Fig. 1. The link
distances of NSFNET in Ives  were converted to a continuous PDF by means of
kernel density estimation. Using this approach, the generated ACMNs had
similar number of nodes, links as well as physical footprint to the NSFNET
topology. Each logical topology was assigned 100 different distance
realisations.

50017253000$9$$10$$11$$12$Average link distance $\bar{d}$ [km]Lightpaths per
node-pair $N_{\lambda}$average68% range50017253000$200$$250$$300$$350$Average
link distance $\bar{d}$ [km]Average throughput [Gb/s]average68%
range50017253000150200250300350400Average link distance $\bar{d}$ [km]Total
throughput [Tb/s]average68% range Figure 2: Established lightpaths per node-
pair, their average throughput and total network throughput as a function of
average network link distance $\bar{d}$.

All links were assumed to be multiple of 80 km standard single mode fibre
spans with parameters $\alpha=0.2\frac{\text{dB}}{\text{km}}$,
$D=18\frac{\text{ps}}{\text{nm}\cdot\text{km}}$ and
$\gamma=1.2\frac{\text{1}}{\text{W}\cdot\text{km}}$. For this work, the
optical bandwidth was constrained to C-band (5 THz). Colourless, directionless
and contentionless reconfigurable optical add-drop multiplexers (ROADM) and
Erbium-doped fibre amplifiers with 4 dB noise figure were assumed. $32$ GBd,
Nyquist-spaced WDM channels were considered and filtering effects and
insertion loss of ROADMs were neglected.

For each physical topology, the throughput was calculated following in Fig. 1.
The overall objective was to maximise the total network throughput. It was
assumed that all 91 node-pairs are assigned the same number of lightpaths
$N_{\lambda}$, which is then maximised within C-band. First, a noise-to-signal
ratio (NSR), defined as inverse signal-to-noise ratio
$\text{NSR}=\frac{1}{\text{SNR}}$, was assigned to each network link. The
advantage of working with NSR is that, the total NSR of a lightpath is simply
the sum of its link NSRs. This allows the computation of the k-highest SNR
paths for a given node-pair using a weighted k-shortest path algorithm. The
NSR was calculated using a closed-form approximation of the Gaussian Noise
(GN) model cfJLT , modelling nonlinear interference as additive white Gaussian
noise (AWGN) and capable of accounting for variably loaded network links. For
the initial link NSR assignment, however, uniform signal spectra were assumed.
The launch power was optimised for every channel in each link following the
LOGON strategy LOGON .

Based on the link NSR, lightpath capacities $C_{i}$ were estimated using
Shannon’s formula shannon , yielding the maximised lightpath throughput for a
given SNR. For each node-pair, NSR weighted k-shortest paths were used to
determine all possible lightpaths within a certain capacity range. In
particular, only lightpaths that satisfied $C_{i}\geq x\cdot\underset{\forall
i}{\max}\left(C_{i}\right)$ were kept for a throughput relaxation
$x\in\left[0,1\right]$ and where $i$ ranges over all possible lightpaths for a
given node-pair. Increasing throughput relaxation $x$ generates a larger
optimisation space, i.e more ’equal-cost’ paths, for the routing and
wavelength assignment (RWA) stage. For larger $x$, lightpaths can be rerouted
over longer routes leading to lower network congestion and to more established
lightpaths across C-band. However, longer routes exhibit lower throughput per
lightpath. For this reason, $x$ was swept and optimised for each physical
topology to maximise total network throughput.

The set of ’equal-cost’ paths for each node-pair is then passed to the RWA.
The objective of the RWA is to find a lightpath configuration that minimises
the network congestion and in turn maximises the number of established
lightpaths per node-pair $N_{\lambda}$ within C-band. This is done by
rerouting lightpaths over longer paths such that physical wavelengths can be
reused over physically diverse paths, without collision. A heuristic, which
maximises $N_{\lambda}$, within a constrained optical bandwidth, was proposed
in Baroni . For each node pair, an arbitrary set of ’equal-cost’ paths is
required and the heuristic returns a wavelength and route configuration which
minimises spectral occupation across all network links for a given
$N_{\lambda}$. The heuristic has been shown to match results of ILP
formulations for logical topologies, without taking physical transmission
properties into account. Here, the heuristic is extended to include physical
properties by carefully selecting the set of ’equal-cost’ paths according to
their physical transmission capacities. After each iteration of the RWA,
$N_{\lambda}$ is incremented until the entire C-band is filled. In summary,
the proposed approach finds the maximum number of lightpaths for each node-
pair $N_{\lambda}$ that can be assigned over C-band, for a given optimisation
space of ’equal-cost’ paths, determined by the throughput relaxation $x$.

Finally, the throughput of each lightpath is computed based on the exact
lightpath configuration returned by the RWA algorithm. The throughput of a
particular lightpath is, hence, dependent on the transmitted distance, fibre
parameters as well as number and channel spacing of co-propagating lightpaths
in each link.

$1.7$$2.3$$3$$9$$10$$11$$12$Normalized network diameter $D$Lightpaths per
node-pair $N_{\lambda}$average68%
range$1.7$$2.3$$3$$270$$280$$290$$300$$310$Normalized network diameter
$D$Average throughput [Gb/s]average68%
range$1.7$$2.3$$3$240260280300320Normalized network diameter $D$Total
throughput [Tb/s]average68% range Figure 3: Established lightpaths per node-
pair, their average throughput and the total network throughput as a function
of normalised network diameter $D$. The normalised network diameter is a
measure for the ellipticity of a physical network topology.

##  3 Scaling the size of optical networks

To study the impact of physical size on network throughput, the average link
distance of each ACMN was scaled from 640 km to 3040 km, by scaling the
assigned link distances by the factor $\frac{\bar{d}}{1463\text{km}}$, where
$\bar{d}$ is the average link distance. The used average link distance of
NSFNET is 1463 km. For comparison the average link distances, estimated as in
Ives , of the DTAG/T-Systems dtag  and the Google B4 google  topology are 236
km and 3910 km, respectively.

The assigned lightpaths per node pair $N_{\lambda}$, the average throughput of
all lightpaths and the total network throughput are shown Fig. 2. As 100,000
physical topologies were studied, only the average and the range of 68% of all
realisations are shown. Transmission in mesh optical networks is different to
point-to-point transmission, due to variably loaded links. However, the
throughput $C_{i}$ in a single point-to-point link decreases approximately
logarithmically with the number of identical spans $n_{s}$ as
$C_{i}\propto\log_{2}\left(1+\text{SNR}_{n_{s}}\right)\approx\log_{2}\left(\text{SNR}_{1}\right)-\log_{2}\left({n_{s}}\right)$.
Fig 2a) shows that, as the physical network size increases, the congestion
increases and, therefore, the number of lightpaths per node pair $N_{\lambda}$
is reduced. This is because as the overall link distance increases, the number
of ’equal-cost’ paths is reduced, due to the logarithmic throughput scaling.
Thus the RWA heuristic must optimise over fewer ’equal-cost’ paths, resulting
in worse load balancing and fewer lightpaths. The average throughput of all
lightpaths established decreases (approximately) logarithmically with average
link distance, as in Fig. 2b), similar to the scaling in a point-to-point
transmission. As $N_{\lambda}$ decreases only marginally with average link
distance, in terms of the average throughput, Fig. 2c), too, shows an
approximate logarithmic decrease of total network throughput with physical
network size.

Based on this analysis, it can be concluded that the precise routing only
marginally depends on the (uniform) physical network size, but the average
throughput per lightpath, as well as the total throughput, decrease
logarithmically with the physical network footprint.

##  4 Scaling the ellipticity in optical networks

Another important property of a physical network is its shape. In this work we
denote it by the term ’ellipticity’ and define a normalised network diameter
as $D=\frac{\text{network diameter }\left[\text{km}\right]}{\text{average
node-pair distance}\left[\text{km}\right]}$, where the network diameter is the
maximum shortest path distance in km over all node-pairs. If the diameter is
then normalised by the average node-pair distance, it can serve as way of
parameterising the impact of network shape and defines the ellipticity of a
topology. In this work, for each logical topology, link distances were
randomly assigned and only kept if they satisfied a fixed normalised network
diameter $D$. The average node-pair distance was fixed to 3070 km, which is
that of the used NSFNET topology. For comparison, NSFNET has $D=2.13$, the
DTAG/T-systems topology and the Google B4 topologies have $D=2.04$ and
$D=2.79$.

Fig. 3 shows the results for the number of established lightpaths per node
pair, average lightpath throughput and total network throughput as a function
of ellipticity. It can be seen that Fig. 3a) shows a marginal decrease in the
number of the established lightpaths, thus increasing congestion, as the
ellipticity increases. Fig. 3b) shows an increase of the average lightpath
throughput. Increasing ellipticity increases the distances for some
lightpaths, and decreases the distances of the remaining ones. To aid the
interpretation of the results, we looked at the scaling of the average
throughput for two point-to-point links, where one is increased and the other
decreased by $\Delta n_{s}$ fibre spans. The average throughput of two point-
to-point links scales as $\bar{C}_{i}\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}2\log_{2}\left(\text{SNR}_{1}\right)-\log_{2}\left(n_{s}^{2}-\Delta
n_{s}^{2}\right)$ which, indeed, predicts an increase in average lightpath
throughput as $\Delta n$ (the ellipticity) is increased. The marginally
decreased $N_{\lambda}$ is balancing the effect of increased average
throughput, as ellipticity grows. Thus, the total throughput only marginally
depends on ellipticity, as shown in Fig. 3c).

The analysis shows that throughout in optical networks is only weakly
dependent on the topology ellipticity, for the ranges studied in this paper.

##  5 Conclusion

The relationship between physical network topology and throughput was studied
by parametrising and analysing arbitrarily-connected mesh optical networks. It
was found that, similarly to point-to-point links, network throughput
decreases approximately logarithmically with physical network size while there
is only a minor throughput dependence on the ellipticity of a topology. The
study further suggests that network congestion only marginally increases with
a uniform increase of physical network size and ellipticity.

Support for this work is from UK EPSRC under DTG PhD studentship to D. Semrau
and EPSRC TRANSNET Programme Grant.

## References

## References

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  * [12] Shannon, C. E.: "A mathematical theory of communication," Bell System Technical Journal, 1948, 27, pp. 379–423
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# A note on the strong convergence of partial sums with respect to Vilenkin
system

G. Tutberidze  G.Tutberidze, The University of Georgia, school of Informatics,
Engineering and Mathematics, IV, 77a Merab Kostava St, Tbilisi, 0128, Georgia,
& Department of Engineering Sciences and Mathematics, LuleåUniversity of
Technology, SE-971 87 Luleå, Sweden<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate some strong convergence theorems for partial sums
with respect to Vilenkin system.

The research was supported by a Swedish Institute scholarship and by Shota
Rustaveli National Science Foundation grant YS15-2.1.1-47.

2010 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, partial sums, Fejér means, Hardy
space, strong convergence.

##  1\. Introduction

It is well-known (for details see e.g. [8] and [14]) that the Vilenkin system
does not form a basis in the space $L_{1}\left(G_{m}\right).$ Moreover, there
is a function in the Hardy space $H_{1}\left(G_{m}\right),$ (for details see
[12, 13, 21, 22]) such that the partial sums of $f$ are not bounded in
$L_{1}$-norm. However, (see e.g. [2, 23]) the subsequence $S_{M_{n}}$ of
partial sums are bounded from the Hardy space $H_{1}\left(G_{m}\right)$ to the
Lebesgue space $L_{1}\left(G_{m}\right):$

(1) $\left\|S_{M_{k}}f\right\|_{H_{1}}\leq c\left\|f\right\|_{H_{1}}\text{ \ \
\ }(k\in\mathbb{N}).$

Moreover, in Gát [7] (see also Simon [18, 19]) it was proved the following
strong convergence result for all $f\in H_{1}:$

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f-f\right\|_{1}}{k}=0,$

where $S_{k}f$ denotes the $k$-th partial sum of the Vilenkin-Fourier series
of $f.$

It follows that there exists an absolute constant $c,$ such that

(2) $\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{1}}{k}\leq
c\left\|f\right\|_{H_{1}}\text{ \ }\left(n=2,3...\right)$

and

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{1}}{k}=\left\|f\right\|_{H_{1}},$

for all $f\in H_{1}.$

Analogical result for the trigonometric system was proved by Smith [20], and
for the Walsh-Paley system by Simon [17].

If the partial sums of Vilenkin-Fourier series was bounded from $H_{1}$ to
$L_{1}$ we also would have:

(3)
$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}\left\|S_{m}f\right\|_{1}\leq
c\left\|f\right\|_{H_{1}},$

but as it was presented above the boundednes of the partial sums does not hold
from $H_{1}$ to $L_{1}$, However, we have inequality (2).

On the other hand, in the one-dimensional case, Fujji [6] and Simon [16]
proved that maximal operator Fejér means are bounded from $H_{1}$ to $L_{1}$,
that is

(4)
$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\left\|\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}S_{m}f\right\|_{1}<c\left\|f\right\|_{H_{1}}.$

So, natural question has arised that if inequality (3) holds true, which would
be generalization of inequality (4) or is we have negative answer on this
problem.

In this paper we prove that there exists a function $f\in H_{1}$ such that

$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}\left\|S_{m}f\right\|_{1}=\infty.$

This paper is organized as follows: in order not to disturb our discussions
later on some definitions and notations are presented in Section 2. For the
proofs of the main results we need some auxiliary Lemmas. These results are
presented in Section 3. The formulation and detailed proof of main results can
be found in Section 4.

##  2\. Definitions and Notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$

Let $m:=(m_{0,}m_{1},\dots)$ denote a sequence of positive integers not less
than 2.

Denote by

$Z_{m_{k}}:=\\{0,1,\dots,m_{k}-1\\}$

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group
$Z_{m_{j}}$ with the product of the discrete topologies of $Z_{m_{j}}$ ,s.

The direct product $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \qquad}(j\in Z_{m_{k}})$

is the Haar measure on $G_{m_{\text{ }}}$with $\mu\left(G_{m}\right)=1.$

If $\sup_{n\in\mathbb{N}}m_{n}<\infty$, then we call $G_{m}$ a bounded
Vilenkin group. If the generating sequence $m$ is not bounded, then $G_{m}$ is
said to be an unbounded Vilenkin group.

The elements of $G_{m}$ are represented by sequences

$x:=(x_{0},x_{1},\dots,x_{k},\dots)\qquad\left(\text{ }x_{k}\in
Z_{m_{k}}\right).$

It is easy to give a base for the neighbourhood of $G_{m}$ namely

$\displaystyle I_{0}\left(x\right)$ $\displaystyle:=$ $\displaystyle G_{m},$
$\displaystyle I_{n}(x)$ $\displaystyle:=$ $\displaystyle\\{y\in G_{m}\mid
y_{0}=x_{0},\dots,y_{n-1}=x_{n-1}\\}\text{ }(x\in G_{m},\text{
}n\in\mathbb{N})$

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$ .

Let

$e_{n}:=\left(0,\dots,0,x_{n}=1,0,\dots\right)\in
G_{m}\qquad\left(n\in\mathbb{N}\right).$

If we define the so-called generalized number system based on $m$ in the
following way:

$M_{0}:=1,\text{\qquad}M_{k+1}:=m_{k}M_{k\text{ }},\ \qquad(k\in\mathbb{N})$

then every $n\in\mathbb{N}$ can be uniquely expressed as
$n=\sum_{k=0}^{\infty}n_{j}M_{j}$ where $n_{j}\in Z_{m_{j}}$
$~{}(j\in\mathbb{N})$ and only a finite number of $n_{j}`$s differ from zero.
Let $\left|n\right|:=\max$ $\\{j\in\mathbb{N};$ $n_{j}\neq 0\\}.$

Next, we introduce on $G_{m}$ an orthonormal system, which is called the
Vilenkin system.

At first define the complex valued function
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions as

$r_{k}\left(x\right):=\exp\left(2\pi\imath x_{k}/m_{k}\right)\text{
\qquad}\left(\imath^{2}=-1,\text{ }x\in G_{m},\text{ }k\in\mathbb{N}\right).$

Now define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
as:

$\psi_{n}\left(x\right):=\prod_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right)\text{
\qquad}\left(n\in\mathbb{N}\right).$

Specially, we call this system the Walsh-Paley one if $m\equiv 2.$

The norm (or quasi norm) of the space $L_{p}(G_{m})$ is defined by

$\left\|f\right\|_{p}:=\left(\int_{G_{m}}\left|f(x)\right|^{p}d\mu(x)\right)^{1/p}\qquad\left(0<p<\infty\right).$

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)\,$
(for details see e.g. [1, 25]).

If $\ f\in L_{1}\left(G_{m}\right)$ we can define Fourier coefficients,
partial sums of the Fourier series, Fejér means, Dirichlet kernels with
respect to the Vilenkin system in the usual manner:

$\displaystyle\widehat{f}(k)$ $\displaystyle:$
$\displaystyle=\int_{G_{m}}f\overline{\psi}_{k}d\mu\text{\thinspace\qquad\ \ \
\ }\left(\text{ }k\in\mathbb{N}\text{ }\right)$ $\displaystyle S_{n}f$
$\displaystyle:$
$\displaystyle=\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k}\
\text{\qquad\ \ }\left(\text{ }n\in\mathbb{N}_{+},\text{ }S_{0}f:=0\right)$
$\displaystyle\sigma_{n}f$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\sum_{k=0}^{n-1}S_{k}f\text{ \qquad\ \ \ \ \
}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right)$ $\displaystyle D_{n}$
$\displaystyle:$ $\displaystyle=\sum_{k=0}^{n-1}\psi_{k\text{ }}\text{ \qquad\
\ \qquad}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right).$

Recall that

(5) $\quad D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{l}\text{
}M_{n},\text{ \ \ \ }x\in I_{n}\\\ \text{ }0,\text{ \qquad}x\notin
I_{n}\end{array}\right.$

and

(6)
$D_{s_{n}M_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}\psi_{kM_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}r_{n}^{k},\text{
\qquad}1\leq s_{n}\leq m_{n}-1.$

The $n$-th Lebesgue constant is defined in the following way

$L_{n}=\left\|D_{n}\right\|_{1}.$

It is well-known [25] that

(7) $L_{n}=O(\log n),\ \ n\rightarrow\infty.$

Moreover, (for unbounded Vilenkin systems it can be found in [5], for bounded
Vilenkin systems see e.g. [9] and [11, 24]) there exist absolute constant
$c_{1}$ and $c_{2}$ such that

(8) $c_{1}\log
n\leq\frac{1}{n}\underset{k=1}{\overset{n}{\sum}}L\left(k\right)\leq c_{2}\log
n,\ \ (n=2,3,...).$

The concept of the Hardy space [4] can be defined in various manners, e.g. by
a maximal function

$f^{\ast}:=\sup_{n}\left|S_{M_{n}}f\right|\text{ \ \ }\left(f\in
G_{m}\right),$

saying that $f$ belongs to the Hardy space if $f^{\ast}\in
L^{1}\left(G_{m}\right).$ This definition is suitable if the sequence $m$ is
bounded. In this case a good property of the space $\left\\{f\in
L^{1}\left(G_{m}\right):\text{ }f^{\ast}\in L^{1}\left(G_{m}\right)\right\\}$
is the atomic structure [4]. To the definition of space of Hardy type for an
arbitrary $m$, first we give the concept of the atoms [16]. A set $I\subset
G_{m}$ is called an interval if for some $x\in G_{m}$ and $n\in N$, $I$ is of
the form $I=\bigcup\limits_{k\in U}I_{n}\left(x,k\right)$, where $U$ is
obtained from $Z_{m_{n}}$ by dyadic partition.

The sets $U_{1},U_{2},...\subset Z_{m_{n}}$, are obtained by means of such a
partition if

$U_{1}=\left\\{0,...,\left[\frac{m_{n}}{2}\right]-1\right\\},\text{ \
}U_{2}=\left\\{\left[\frac{m_{n}}{2}\right],...,m_{n}-1\right\\},$
$U_{3}=\left\\{0,...,\left[\frac{\left[m_{n}/2\right]-1}{2}\right]-1\right\\},\text{
\
}U_{4}=\left\\{\left[\frac{[m_{n}/2]-1}{2}\right],...,\left[\frac{m_{n}}{2}\right]-1\right\\},...$

etc.; $[\ \ ]$ denotes the entire part. We define the atoms as follows: the
function $a\in L^{\infty}\left(G_{m}\right)$ is called an atom if eather
$a\equiv 1$ or there exists an interval $I$ for which $\sup a\subset I,$
$\left|a\right|\leq\left|I\right|^{-1}$ and $\int_{I}a=0$ hold.
$\left(\left|I\right|\text{ denotes the Haar measure of }I\text{ }\right).$

Now we can define the space $H_{1}\left(G_{m}\right)$ (for details see e.g
[26, 27]) as the set of all functions
$f=\sum\limits_{i=0}^{\infty}\lambda_{i}a_{i}$,where $a_{i}$’s are atoms and
for the coefficients $\lambda_{i}$ we have
$\sum\limits_{i=0}^{\infty}\left|\lambda_{i}\right|<\infty.$
$H_{1}\left(G_{m}\right)$ is a Banach space with respect to the norm

(9)
$\left\|f\right\|_{H_{1}}:=\inf\sum\limits_{k=0}^{\infty}\left|\lambda_{k}\right|<\infty.$

The infimum is taken over all decompositions

$f=\sum\limits_{i=0}^{\infty}\lambda_{i}a_{i}.$

It is known [7] that $\left\|f\right\|_{H_{1}}$ is equivalent to
$\left\|f^{\ast\ast}\right\|_{1}\left(f\in L^{1}\left(G_{m}\right)\right)$,
where
$f$**$\left(x\right):=\sup_{I}\left|I\right|^{-1}\left|\int_{I}f\right|$,
$\left(x\in G_{m}\text{, }x\in I\text{ and }I\text{ is interval}\right)$.
Since by (5)

$f^{*}\left(x\right)=\sup_{n\in\mathbb{N}}\frac{1}{\left|I_{n}\left(x\right)\right|}\left|\int_{I_{n}\left(x\right)}f\left(u\right)\mu\left(u\right)\right|$

we have $f^{\ast}\leq f^{\ast\ast}$ and, thus,
$H\left(G_{m}\right)\subset\left\\{f\in L^{1}\left(G_{m}\right):\text{
}f^{\ast}\in L^{1}\left(G_{m}\right)\right\\}.$ Moreover these spaces coincide
if the sequence $m$ is bounded.

##  3\. Main Result

Our main result reads:

######  Theorem 1.

a) Let $f\in H_{1}.$ Then there exists an absolute constant $c,$ such that

$\sup_{n\in\mathbb{N}}\frac{1}{n\log
n}\overset{n}{\underset{k=1}{\sum}}\left\|S_{k}f\right\|_{1}\leq\left\|f\right\|_{H_{1}}.$

b) Let $\varphi:N_{+}\rightarrow[1,$ $\infty)$ be a nondecreasing function
satisfying the condition

(10) $\overline{\lim_{n\rightarrow\infty}}\frac{\log n}{\varphi_{n}}=+\infty.$

Then there exists a function $f\in H_{1},$ such that

$\sup_{n\in\mathbb{N}}\frac{1}{n\varphi_{n}}\overset{n}{\underset{k=1}{\sum}}\left\|S_{k}f\right\|_{1}=\infty.$

######  Corollary 1.

(see e.g. [10, 16, 18])There exists a function $f\in H_{1},$ such that

$\sup_{n\in\mathbb{N}}\frac{1}{n}\overset{n}{\underset{k=1}{\sum}}\left\|S_{k}f\right\|_{1}=\infty.$

##  4\. Proof of Theorem 1

###### Proof.

By using (7) we can conclude that

$\frac{1}{n\log
n}\overset{n}{\underset{k=1}{\sum}}\left\|S_{k}f\right\|_{1}\leq\frac{c\left\|f\right\|_{H_{1}}}{n\log
n}\overset{n}{\underset{k=1}{\sum}}{\log k}\leq c\left\|f\right\|_{H_{1}}.$

The proof of part a) is complete.

Under the condition (10) there exists an increasing sequence of the positive
integers $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ such that

$\overline{\lim_{k\rightarrow\infty}}\frac{\log
M_{{\alpha_{k}}}}{\varphi_{2M_{\alpha_{k}}}}=+\infty$

and

(11)
$\sum_{k=0}^{\infty}\frac{\varphi_{2M_{\alpha_{k}}}^{1/2}}{\log^{1/2}M_{{\alpha_{k}}}}<c<\infty.$

Let

$f=\sum_{k=1}^{\infty}{\lambda_{k}}{a_{k}},$

where

$a_{k}=r_{\alpha_{k}}D_{M_{{}_{\alpha_{k}}}}=D_{2M_{{}_{\alpha_{k}}}}-D_{M_{{}_{\alpha_{k}}}}$

and

$\lambda_{k}=\frac{\varphi_{2M_{\alpha_{k}}}^{1/2}}{\log^{1/2}M_{{\alpha_{k}}}}.$

By the definition of $H_{1}$ and (9), if we apply (11) we can conclude that
$f\in H_{1}.$ Moreover,

(12) $\widehat{f}(j)=\left\\{\begin{array}[]{l}{\lambda_{k}},\,\,\text{\ \ \ \
\thinspace\thinspace}j\in\left\\{M_{\alpha_{k}},...,2M_{\alpha_{k}}-1\right\\},\text{
}k\in\mathbb{N}\\\ 0\text{ },\text{
\thinspace\qquad\thinspace\thinspace\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{M_{\alpha_{k}},...,2M_{\alpha_{k}}-1\right\\}.\text{
}\end{array}\right.$

Since

$D_{j+M_{\alpha_{k}}}=D_{M_{\alpha_{k}}}+\psi_{{}_{M_{\alpha_{k}}}}D_{j},\text{
\qquad when \thinspace\thinspace}j\leq M_{\alpha_{k}},$

if we apply (12) we obtain that

$\displaystyle S_{j}f$ $\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\sum_{v=M_{\alpha_{k}}}^{j-1}\widehat{f}(v)\psi_{v}$
$\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+{\lambda_{k}}\sum_{v=M_{\alpha_{k}}}^{j-1}\psi_{v}$
$\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+{\lambda_{k}}\left(D_{j}-D_{M_{\alpha_{k}}}\right)$
$\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+{\lambda_{k}}\psi_{M_{\alpha_{k}}}D_{j-M_{\alpha_{k}}}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}.$

In view of (1) we can write that

(14)
$\displaystyle\left\|I_{1}\right\|_{1}\leq\left\|S_{M_{\alpha_{k}}}f\right\|_{1}\leq
c\left\|f\right\|_{H_{1}}.$

By combining (8) and (14) we get that

$\displaystyle\left\|S_{n}f\right\|_{1}\geq\left\|I_{2}\right\|_{1}-\left\|I_{1}\right\|_{1}\geq{\lambda_{k}}{L\left({n-M_{\alpha_{k}}}\right)}-c\left\|f\right\|_{H_{1}}.$

Hence,

$\displaystyle\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n\varphi_{n}}\underset{k=1}{\overset{n}{\sum}}\left\|S_{k}f\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2M_{\alpha_{k}}\varphi_{2M_{\alpha_{k}}}}\underset{\left\\{M_{\alpha_{k}}\leq
l\leq 2M_{\alpha_{k}}\right\\}}{\sum}\left\|S_{l}f\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2M_{\alpha_{k}}\varphi_{2M_{\alpha_{k}}}}\underset{\left\\{M_{\alpha_{k}}\leq
l\leq
2M_{\alpha_{k}}\right\\}}{\sum}\left(\frac{L\left(l-M_{\alpha_{k}}\right)\varphi_{2M{\alpha_{k}}}^{1/2}}{\log^{1/2}M_{\alpha_{k}}}-c\left\|f\right\|_{H_{1}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{c\varphi_{2M{\alpha_{k}}}^{1/2}}{2M_{\alpha_{k}}\log^{1/2}M_{\alpha_{k}}\varphi_{2M_{\alpha_{k}}}}\underset{l=1}{\overset{M_{\alpha_{k}}-1}{\sum}}L\left(l\right)-c\left\|f\right\|_{H_{1}}^{1/2}$
$\displaystyle\geq$ $\displaystyle\frac{c\varphi_{2M{\alpha_{k}}}^{1/2}\log
M_{\alpha_{k}}}{\log^{1/2}M_{\alpha_{k}}\varphi_{2M_{\alpha_{k}}}}\geq\frac{c\log^{1/2}M_{\alpha_{k}}}{\varphi_{2M_{\alpha_{k}}}^{1/2}}\rightarrow\infty,\text{
as \ }k\rightarrow\infty.$

The proof is complete. ∎

Acknowledgment: The author would like to thank the referee for helpful
suggestions, which improved the final version of the paper.

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