One Hat Cyber Team

View File Name : Extrload.tex

%\input{deflat.tex}
\input{def12.tex}
\begin{document}
{\bf\bc ON  EXTREMAL PROBLEM FOR ALGEBRAIC POLYNOMIALS\\ IN LOADING SPACES \ec}
{\bf\bc
                  B.P.Osilenker \\
\ec}
\noindent Verkhnyaya Pervomajskaya,\\
dom 59/35,\, korp.2,\, kv.10,\\
Moscow, \,105264,\\
Russia\\
e-mail: b\_osilenker@mail.ru\\


\begin{abstract}
We consider discrete loading spaces under the inner product
\begin{multline*}
\langle f,g \rangle =\int_{\mathbb R}f(x)g(x)d{\mu}(x)+\\
\sum_{j=1}^{m}M_j f(x_j)g(x_j)\enskip(M_j\geq 0,\enskip x_j\in \mathbb R\enskip
(j=1,2,...,m)),
\end{multline*}
where $\mu$ be a finite positive Borel measure such that the moments are finite
and support is an infinite set. In this spaces we study the problem of finding
$$
\inf_{a_0,a_1,...,a_{N-r}}\{\langle {\Pi}^{(r)}_N,{\Pi}^{(r)}_N\rangle;
{\Pi}^{(r)}_N(x)=\sum_{k=0}^{N-r}a_kx^k+\sum_{k=N-r+1}^{N}a^0_k x^k\},
$$
where $a^0_N>0,a^0_{N-1},...,a^0_{N-r+1}$ are fixed real numbers. The extremal
polynomials are also constructed\footnote{AMS classification: Primary 41A10;
Secondary 33C45.\newline\phantom{aaai}Keywords: Extremal problems, Orthogonal
polynomials, Loading spaces.}.
\end{abstract}

{\bf\bc
                    \S 1. Introduction\\
\ec}
Let $\Phi(x)$ be a positive linear functional on the linear space $\mathbb P$
of polynomials with real coefficients.
Define by\\
$$
\hat q_n(x)=\hat k^{(n)}_nx^n+\hat k^{(n)}_{n-1} x^{n-1}+...,
\hat k^{(n)}_n>0 \qquad (n\in\mathbb Z_{+})
$$
 polynomials orthonormal with respect to (w.r.t.) the linear functional $\Phi$:\\
$$
                    \Phi(\hat q_m,\hat q_n)={\delta}_{m,n}(m,n\in\mathbb Z_{+}).\\
$$
%Let $\mathbb P_{+}^N$ be a linear space of polynomials with real coefficients of degree
%not greater  than $N$.\\
 Define by ${\Re}^{(r)}_N$    a class of all polynomials ${\Pi}_N(x)\in\mathbb P$
of degree $N$ with $r$-fixed coefficients:\\
$$
{\Re}^{(r)}_N  =\{{\Pi}^{(r)}_N, {\Pi}^{(r)}_N(x)=\sum_{j=N-r+1}^{N}a^{0}_jx^j+
\sum_{j=0}^{N-r}a_jx^j,a^{0}_N>0\},\\
$$
where $a^{0}_N,a^{0}_{N-1},...,a^{0}_{N-r+1}$ are fixed real numbers.\\
We expand polynomial ${\Pi}^{(r)}_N(x)$ by the system $\{\hat q_n\}(n\in\mathbb Z_{+}):$\\
$$
                   {\Pi}^{(r)}_N(x)=\sum_{j=0}^N{\alpha}^{(N)}_j \hat q_j(x).\\
$$
As in \cite{[1]}, one gets the following statement.\\
{\bf Theorem 1.}  i) \textit{The following representation}
$$
\inf_{{\Pi}^{(r)}_N\in {\Re}^{(r)}_N}\Phi\{|{\Pi}^{(r)}_N(x)|^2\}=
\sum_{j=N-r+1}^{N}[{\alpha}^{(N),0}_j]^2\\
$$
\textit{holds,where the coefficients}
 ${\alpha}^{(N),0}_s(s=N-r+1,N-r+2,...,N-1,N)$
\textit{are the solutions of the system}\\
$$
\sum_{j=s}^{N}\hat k^{(j)}_s {\alpha}^{(N)}_j=a^{0}_s (s=N-r+1,N-r+2,...,N-1,N); \\
$$
ii)\textit{extremal polynomials is}\\
$$
{\Pi}^{(r),extr}_N(x)=\sum_{j=N-r+1}^N {\alpha}^{(N),0}_j\hat q_j(x).\\
$$
We consider this extremal problem in a discrete loading space.\\
On the space $\mathbb P$ we introduce a discrete loading  inner product \\
$$
\langle p,q\rangle=\int_{\mathbb R}p(x)q(x)d{\mu}(x)+
\sum_{j=1}^{m}M_j p(x_j)q(x_j) (p,q\in \mathbb P),\\
$$
where $\mu$ be a finite positive Borel measure on $\mathbb R$ whose moments
are finite and whose support is an infinite set, $M_j\geq 0, x_j\in \mathbb R\,
(j=1,2,...,m)$.\\
Completion of $\mathbb P$  w.r.t. the norm
$\|f\|^2=\langle f,f \rangle$
we will call a discrete loading space $S$ w.r.t. the inner product\\
$$
\langle f,g\rangle=\int_{\mathbb R}f(x)g(x)d{\mu}(x)+
\sum_{j=1}^{m}M_jf(x_j)g(x_j).\qquad (1.1)\\
$$
Let $q_n(x)(n\in\mathbb Z_{+})$
be the polynomials orthonormal w.r.t. the inner product (1.1 ):\\
$$
                \langle q_m,q_n\rangle={\delta}_{m,n}.\\
$$
We will call their a discrete loading orthonormal polynomials.\\
Space $S$ and polynomial system $\{q_n\}(n\in\mathbb Z_{+})$ have attracted
the interest of many researches.
They play an important role in some problems of functional analysis, function
 theory and mathematical physics \cite{[2]}--\cite{[6]}.\\
Polynomial system $\{q_n\}(n\in\mathbb Z_{+})$ is used in some investigations of
boundary problems with spectral parameter in the boundary conditions; differential
equation with discontinuous coefficients; the loading integral equations.\\

{\bf Example} \textit{A discrete loading Jacobi  polynomials}.\\
First, we remind some properties of the classical Jacobi polynomials
$P^{(\alpha,\beta)}_n(x)$ orthogonal on interval (-1,1) w.r.t the Jacobi measure\\
$$
d{\mu}_{\alpha,\beta}(x)=\frac {\Gamma(\alpha+\beta+2)}
{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}(1-x)^{\alpha}(1+x)^{\beta}dx
(\alpha,\beta>-1).\qquad (1.2)
$$
(see \cite {[7]}).\\
By the Rodrigues formula\\
$$
P^{(\alpha,\beta)}_n(x)=\frac {(-1)^n}{n!2^n}(1-x)^{-\alpha}(1+x)^{-\beta}
[(1-x)^{n+\alpha}(1+x)^{n+\beta}]^{(n)}(x\in (-1,1);n\in\mathbb Z_{+}).\\
$$
One has\\
$$
  P^{(\alpha,\beta)}_n(x)=
\frac 1{n!2^n}\frac {\Gamma(2n+\alpha+\beta+1)}{\Gamma(n+\alpha+\beta+1)}x^n+...\\
$$
For classical orthogonal Jacobi polynomials the values at the points $\pm 1$
$$
P^{(\alpha,\beta)}_n(1)=\frac {(\alpha+1)_n}{n!},
P^{(\alpha,\beta)}_n(-1)=(-1)^n \frac {(\beta+1)_n}{n!},\\
$$
where "shifted factorial"(Pochhammer symbol) is defined by\\
$$
(a)_n=a(a+1)(a+2)\cdots (a+n-1)=\frac {\Gamma(n+a)}{\Gamma(a)}(n=1,2,...),(a)_0=1.\\
$$
The squard norm of classical Jacobi orthogonal polynomials is\\
\begin{multline*}
||P^{(\alpha,\beta)}_n||^{2}_{d{\mu}_{\alpha,\beta}}
:=\int_{-1}^{1}[P^{(\alpha,\beta)}_n(x)]^2d{\mu}_{\alpha,\beta}(x)=\\
\frac 1{2n+\alpha+\beta+1}\frac {\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}
\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{n!\Gamma(n+\alpha+\beta+1)}.\\
\end{multline*}
Let $\widehat P^{(\alpha,\beta)}_n(x) $ be polynomials orthonormal
on the interval (-1,1) w.r.t. the measure $d{\mu}_{\alpha,\beta}(x)$(see (1.2)).
Then\\
\begin{multline*}
\widehat P^{(\alpha,\beta)}_n(x)=\sqrt {\frac {\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}}*\\
\frac 1{2^n}\sqrt{\frac{2n+\alpha+\beta+1}{n!}}
\frac{\Gamma(2n+\alpha+\beta+1)}
{\sqrt{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)\Gamma(n+\alpha+\beta+1)}}x^n+...
\qquad(1.3)\\
\end{multline*}
We consider the inner product\\
$$
{\langle f,g\rangle}_{\alpha,\beta}=\int_{-1}^{1}f(x)g(x)d{\mu}_{\alpha,\beta}(x)+
Lf(1)g(1)+Mf(-1)g(-1), L,M\geq 0  \qquad (1.4)\\
$$
and define by $\widehat P^{\alpha,\beta;L,M}_n(x)(n\in\mathbb Z_{+})$
 polynomial system orthonormal w.r.t.the inner product (1.4). These polynomials
were introduced by T.Koornwinder \cite{[8]}.
It is called the loading Jacobi polynomials (or generalized Jacobi polynomials).
It should be noted that the loading Jacobi polynomials have some properties
other than the classical Jacobi polynomials (behaviour at the points $\pm 1$;
linear differential operator, for which polynomials
$\widehat P^{(\alpha,\beta;L,M)}_n(x)$ are eigenfunctions and so on;
see \cite{[9]}-\cite {[12]}).\\

{\bf\bc
\S2. Extremal problem for algebraic polynomials in a discrete loading space\\
\ec}
Let\\
$$
p_n(x)=\sum_{i=0}^{n}l^{(n)}_i x^i,\quad l^{(n)}_n>0(n\in\mathbb Z_{+})\qquad (2.1)
$$
be the polynomials orthonormal w.r.t. the measure $\mu$:\\
$$
(p_m,p_n)=\int_{\mathbb R}p_m(x)p_n(x)d{\mu}(x)={\delta}_{m,n}(m,n\in\mathbb Z_{+}).\qquad (2.2)\\
$$
and let\\
$$
q_n(x)=\sum_{s=0}^{n}k^{(n)}_sx^s,\quad  k^{(n)}_n>0(n\in\mathbb Z_{+}) \qquad (2.3)\\
$$
be the polynomials orthonormal w.r.t the inner product (1.1).\\
We expand polynomial $q_n(x)$ by basis $\{p_n\}(n\in\mathbb Z_{+})$:\\
$$
          q_n(x)=\sum_{s=0}^{n} a^{(n)}_s p_s(x).\qquad (2.4)
$$
Using (2.1)--(2.3) and comparing the coefficients at $x^n$ in the relation (2.4),
 one obtains\\
$$
   a^{(n)}_s=\int_{\mathbb R}q_n(x)p_s(x)d{\mu}(x) (s=0,1,...,n-1) \qquad (2.5)
$$
and
$$
  a^{(n)}_n=\frac {k^{(n)}_n}{l^{(n)}_n}=\int_{\mathbb R}q_n(x)p_n(x)d{\mu}(x)
  (n\in\mathbb Z_{+}).\qquad (2.6)
$$
Substituting relations (2.5) and (2.6) in (2.4) ,one gets\\
$$
q_n(x)=\frac {k^{(n)}_n}{l^{(n)}_n}p_n(x)-
\sum_{j=1}^{m}M_jq_n(x_j)D_{n-1}(x_j,x) ,\qquad (2.7)
$$
where\\
$$
 D_n(t,x)=\sum_{i=0}^{n}p_i(t)p_i(x)(t,x\in \mathbb R;n\in\mathbb Z_{+})\\
$$
is the Dirichlet kernel of the system $\{p_n\}(n\in\mathbb Z_{+}).$\\
Taking into account (2.7) at the points $x_i(i=1,2,...,m)$,one has\\
\begin{multline*}
[1+M_iD_{n-1}(x_i,x_i)]q_n(x_i)+
\sum_{j=1,j\ne i}^m M_j D_{n-1}(x_j,x_i)q_n(x_j)=\\
\frac {k^{(n)}_n}{l^{(n)}_n}p_n(x_i)(i=1,2,...,m). \qquad (2.8)
\end{multline*}
Denote by \\
\begin{multline*}
           \mathbb D_{n-1}:= \mathbb D_{n-1}$$\begin{pmatrix}
                                            x_1,&x_2,&...,&x_m \\
                                            x_1,&x_2,&...,&x_m
                                            $$\end{pmatrix}=
           $$\begin{pmatrix}
          D_{n-1}(x_1,x_1) & D_{n-1}(x_2,x_1)&...&D_{n-1}(x_m,x_1)\\
          D_{n-1}(x_1,x_2) & D_{n-1}(x_2,x_2)&...&D_{n-1}(x_m,x_2)\\
          \hdotsfor{3}\\
          D_{n-1}(x_1,x_m) & D_{n-1}(x_2,x_m)&...&D_{n-1}(x_m,x_m)\\
          \end{pmatrix} $$
\end{multline*}
Note that the $m\star m$ matrix $\mathbb D_{n-1}$ is symmetric and
positive definite.\\
Similarly we put\\
$$
P_n=(p_n(x_1),p_n(x_2),...,p_n(x_m))^T,\\
Q_n=(q_n(x_1),q_n(x_2),...,q_n(x_m))^T,\\
$$
where $T$ is transposition, and\\
$$
M=diag (M_1,M_2,...,M_m).\\
$$
Then the system (2.8) one can rewrite in the form\\
\begin{multline*}
\Bigl[I+\mathbb D_{n-1}$$\begin{pmatrix}
                          x_1,&x_2,&...,&x_m\\
                          x_1.&x_2,&...,&x_m
                       $$\end{pmatrix}
M\Bigr]Q_n=\frac {k^{(n)}_n}{l^{(n)}_n} P_n,\qquad (2.9)\\
\end{multline*}
where $I$ is the identity matrix of order $m$.\\
Denote by\\
\begin{multline*}
\Delta_{n-1}=|I+\mathbb D_{n-1}$$\begin{pmatrix}
                                 x_1,&x_2,&...,&x_m\\
                                 x_1,&x_2,&...,&x_m
                               $$\end{pmatrix}
M|.         \qquad (2.10)\\
\end{multline*}
It follows from (2.9) by Cramer's rule\\
$$
q_n(x_j)=\frac {k^{(n)}_n}{l^{(n)}_n}
\frac {\tilde {\Delta}_{n-1}^{(j)}}{{\Delta}_{n-1}},\qquad (2.11)
$$
where $\tilde {\Delta}_{n-1}^{(j)} $ is a determinant was obtained from
${\Delta}_{n-1}$ by substitution $j$--column by column matrix $P_n$.\\

{\bf Theorem 2} \textit{For a loading orthonormal polynomials}  $q_n(x)$
\textit{the following representation}\\
$$
q_n(x)=\sqrt {\frac {{\Delta}_{n-1}}{ {\Delta}_n}}p_n(x)-
\frac {1}{\sqrt {\Delta_{n-1}\Delta_n}}
\sum_{s=1}^{n-1}M_s{\tilde {\Delta}^{(s)}_{n-1}}D_{n-1}(x_s,x).\qquad (2.12)\\
$$
holds.\\
{\bf Proof.} We expand polynomial $p_n(x)$ by basis $\{q_n\}(n\in\mathbb Z_{+}):$\\
$$
           p_n(x)=\sum_{j=0}^{n}b^{(n)}_j q_j(x).\quad(2.13)\\
$$
Then,as above\\
$$
  b^{(n)}_n=\frac {l^{(n)}_n}{k^{(n)}_n}=
\langle p_n,q_n \rangle (n\in\mathbb Z_{+}) \qquad(2.14)\\
$$
and\\
$$
p_n(x)=\frac {l^{(n)}_n}{k^{(n)}_n}q_n(x)+\sum_{j=0}^{n-1}b^{(n)}_j q_j(x).\\
$$
Taking into account (2.6) and by definition of the inner product (1.1), one has\\
$$
\frac {k^{(n)}_n}{l^{(n)}_n}=\int_{\mathbb R}q_n(x)p_n(x)d{\mu}(x)=
\langle p_n,q_n \rangle - \sum_{j=1}^{m}M_jq_n(x_j)p_n(x_j).\\
$$
It follows from (2.14) that\\
$$
\frac {k^{(n)}_n}{l^{(n)}_n}=\frac {l^{(n)}_n}{k^{(n)}_n} -
\sum_{j=1}^{m} M_jq_n(x_j)p_n(x_j).\\
$$
By (2.11) one obtains\\
$$
\frac {l^{(n)}_n}{k^{(n)}_n}=\frac 1{{\Delta}_{n-1}}\frac {k^{(n)}_n}{l^{(n)}_n}
\Bigl[{\Delta}_{n-1}+
\sum_{j=1}^{m}M_j p_n(x_j){\tilde {\Delta}^{(j)}_{n-1}}\Bigr].\qquad (2.15)\\
$$
We show that\\
$$
{\Delta}_{n-1}+\sum_{j=1}^{m}M_j p_n(x_j){\tilde {\Delta}^{(j)}_{n-1}}=
\Delta_n. \qquad (2.16)
$$
It is not difficult to see  from  (2.10) that\\
$$
\Delta_n=|I+\mathbb D_{n-1}M+\mathbb H_{n-1}M|,\\
$$
where\\
\begin{multline*}
\mathbb H_{n-1}:=\mathbb H_{n-1}$$\begin{pmatrix}
                                  x_1,&x_2,&...&,x_m\\
                                  x_1,&x_2,&...&,x_m\\
                                  \end{pmatrix}$$=
            $$\begin{pmatrix}
            p^2_n(x_1) & p_n(x_1)p_n(x_2)&...&p_n(x_1)p_n(x_m)\\
            p_n(x_1)p_n(x_2) &p_n^2(x_2)&...&p_n(x_2)p_n(x_m)\\
            \hdotsfor{3}\\
            p_n(x_1)p_n(x_m)& p_n(x_2)p_n(x_m)&...&p^2_n(x_m).\\
            \end{pmatrix}$$
\end{multline*}
Consequently, using properties of the determinant, one gets (2.16).\\
Taking into account (2.15), one obtains\\
$$
   \frac {l^{(n)}_n}{k^{(n)}_n}=
\frac {{\Delta}_n}{{\Delta}_{n-1}}
\frac {k^{(n)}_n}{l^{(n)}_n}(n\in\mathbb Z_{+}). \\
$$
Then\\
$$
k^{(n)}_n=
\sqrt {\frac {{\Delta}_{n-1}}{\Delta_n}}l^{(n)}_n
\quad (n\in\mathbb Z_{+})\qquad (2.17)\\
$$
Finally, substituting expression (2.11) and (2.17) in (2.7), one obtains (2.12).\\
Theorem 2 is completely proved.\\
{\bf Corollary 1.} \textit{For the coefficients of a loading orthonormal
polynomials} $q_n(x)(n\in\mathbb Z_{+})$ \textit{(see (2.3)) the following
 representation}\\
$$
k^{(n)}_i=\sqrt{\frac {{\Delta}_{n-1}}{\Delta_n}}l^{(n)}_i-
\frac {1}{\sqrt {{\Delta}_{n-1}{\Delta}_n}}
\sum_{j=1}^{n-1}M_j {\tilde {\Delta}}^{(j)}_{n-1}
\sum_{s=i}^{n-1}p_s(x_j)l^{(s)}_i(i=0,1,...,n-1) (2.18)\\
$$
\textit{holds, where }$l^{(n)}_i(i=0,1,...,n-1)$\textit{are the coefficients of
polynomials} $p_n(x)$ \textit{(see (2.1)).} \\
Corollary 1 follows from (2.1), (2.3) and (2.12)  by comparing the
coefficients at\\
 $x^i(i=0,1,2,...,n-1).$\\
In particular, one obtains\\
$$
k^{(n)}_{n-1}=\sqrt{\frac {{\Delta}_{n-1}}{{\Delta}_n}} l^{(n)}_{n-1}-
\frac {1}{\sqrt{{\Delta}_{n-1}{\Delta}_n}}
\Bigl[\sum_{j=1}^{n-1}M_j {\tilde {\Delta}}^{(j)}_{n-1}p_{n-1}(x_j)\Bigr]
 l^{(n-1)}_{n-1} \qquad (2.19)\\
$$
and\\
$$
\frac {k^{(n)}_{n-1}}{k^{(n)}_n}=\frac {l^{(n)}_{n-1}}{l^{(n)}_n}
-\frac 1{{\Delta}_{n-1}}\Bigl[\sum_{j=1}^{n-1}M_j {\tilde \Delta}^{(j)}_{n-1}
p_{n-1}(x_j)\Bigr]\frac {l^{(n-1)}_{n-1}}{l^{(n)}_n}.\qquad (2.20)
$$
Using Theorem 1 and (2.7),(2.19), one gets\\
{\bf Corollary 2.}\textit{For the monic polynomials}
$$
{\Pi}^{(1)}_N(x)=x^N+\sum_{s=0}^{N-1}a^{(N-1)}_s x^s;
{\Pi}^{(2)}_N(x)=x^N-\sigma x^{N-1}+\sum_{s=0}^{N-2}b^{(N-2)}_s x^s\\
$$
($\sigma$ \textit{is a fixed real number}) \textit{the following
assertions are valid:}\\
i) $$
{\kappa}^{(1)}_N:=inf_{{\Pi}^{(1)}_N\in {\Re}^{(1)}_N}
\langle {\Pi}^{(1)}_N,{\Pi}^{(1)}_N \rangle=\frac {\Delta_N}{\Delta_{N-1}}
\frac 1{(l^{(N)}_N)^2};
{\Pi}^{(1),extr}_N(x)=
\sqrt {\frac{\Delta_N}{{\Delta}_{N-1}}}\frac 1{l^{(N)}_N} q_N(x);\qquad (2.21)\\
$$
ii)\textit{(Zolotarev's problem in the metric of a discrete loading space S):}\\
\begin{multline*}
{\kappa}^{(2)}_N:=inf_{{\Pi}^{(2)}_N\in {\Re}^{(2)}_N}
\langle {\Pi}^{(2)}_N,{\Pi}^{(2)}_N\rangle=\\
\frac {\Delta_N}{{\Delta}_{N-1}}\frac 1{(l^{(N)}_N)^2}+
\frac {\Delta_{N-1}}{\Delta_{N-2}}\frac 1{(l^{(N-1)}_{N-1})^2}
{\Bigl[\sigma+\frac {k^{(N)}_{N-1}}{k^{(N)}_N}\Bigr]}^2\\
\end{multline*}
\textit{and}\\
$$
{\Pi}^{(2),extr}_N(x)=
\sqrt {\frac {{\Delta}_{N}}{{\Delta}_{N-1}}}
\frac 1{l^{(N)}_N}q_N(x)-\\
\sqrt{\frac {\Delta_{N-1}}{\Delta_{N-2}}}\frac 1{l^{(N-1)}_{N-1}}
\Bigl[\sigma+\frac {k^{(N)}_{N-1}}{k^{(N)}_N}\Bigr]q_{N-1}(x).\\
$$
Using relations (2.19) and (2.20) one can calculate ${\kappa}^{(1)}_N$,
${\kappa}^{(2)}_N$ and ${\Pi}^{(1),extr}_N(x)$, ${\Pi}^{(2),extr}_N(x)$.\\
{\bf Remark.}\textit{It should be noted that one can calculate the detrminants using
the following formula}\cite{[13]}:\textit{determinant}\\
            $$\begin{vmatrix}
             a_{11}+1& a_{12}   & \hdotsfor{2}...    &a_{1n}\\
             a_{21}  &  a_{22}+1&\hdotsfor{2}...     &a_{2n}\\
             \hdotsfor {5}\\
             a_{n1}  &a_{n2}    &\hdotsfor{2}...      &a_{nn}+1\\
             \end{vmatrix}$$
\textit{is equal to}\\
$$
                  1+\sum_{k=1}^n S_k,\\
$$
\textit{where} $S_k(k=1,2,...,S_n)$\textit{are sum of main minors
of order k  for } \\
$$\begin{vmatrix}
a_{11}& a_{12}&\hdotsfor{2}... & a_{1n}\\
a_{21}& a_{22}&\hdotsfor{2}... &a_{2n}\\
\hdotsfor{5}\\
a_{n1}&a_{n2} &\hdotsfor{2}... &a_{nn}\\
\end{vmatrix}$$

{\bf Example} \textit{A discrete loading Jacobi polynomials.} \\
We put $x_1=1, x_2=-1.$ Then for Dirichlet kernel $D_n(t,x)$ we have the following
representation (see \cite{[7]}):\\
$$
\left\{
\begin{aligned}
D^{(\alpha,\beta)}_{n-1}(1,1)=&
\frac {\lambda^2}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\alpha+2)}
\frac {\Gamma(n+\alpha+1)\Gamma(n+\alpha+\beta+1)}{\Gamma(n)\Gamma(n+\beta)};\\
D^{(\alpha,\beta)}_{n-1}(-1,1)=&
\frac {\lambda^2}{2^{\alpha+\beta+1}\Gamma(\beta+1)\Gamma(\beta+2)}
\frac {\Gamma(n+\beta+1)\Gamma(n+\alpha+\beta+1)}{\Gamma(n)\Gamma(n+\alpha)};\\
D^{(\alpha,\beta)}_{n-1}(1,-1)=D^{(\alpha,\beta)}_{n-1}(-1,1)=&
\frac {(-1)^{n-1}{\lambda}^2}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}
\frac {\Gamma(n+\alpha+\beta+1)}{\Gamma(n)}.\\
\end{aligned}
\right.
$$
where\\
$$
 {\lambda}^2=\frac {2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}
{\Gamma(\alpha+\beta+2)}.\\
$$
Substituting last formulas in (2.10) for\\
$$
\Delta_{n-1}=\begin{vmatrix}
1+D^{(\alpha,\beta)}_{n-1}(1,1)&D^{(\alpha,\beta)}_{n-1}(-1,1)\\
D^{(\alpha,\beta)}_{n-1}(1,-1)&1+D^{(\alpha,\beta)}_{n-1}(-1,-1)
\end{vmatrix}
$$
we obtain\\
\begin{multline*}
\Delta_{n-1}={\Delta}^{\alpha,\beta;L,M}_{n-1}=\\
1+\frac {\Gamma(\beta+1)}
{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\frac{\Gamma(n+\alpha+1)\Gamma(n+\alpha+\beta+1)}
{\Gamma(n)\Gamma(n+\beta)}L+\\
\frac {\Gamma(\alpha+1)}{\Gamma(\beta+2)\Gamma(\alpha+\beta+2)}
\frac {\Gamma(n+\beta+1)\Gamma(n+\alpha+\beta+1)}{\Gamma(n)\Gamma(n+\alpha)}M+\\
\frac {1}{(\alpha+1)(\beta+1)\Gamma^2(\alpha+\beta+2)}
\frac {\Gamma(n+\alpha+\beta+1)\Gamma(n+\alpha+\beta+2)}{\Gamma(n-1)\Gamma(n)}LM.\\
\end{multline*}
It follows from (1.3) and (2.17) that\\
\begin{multline*}
\widehat P^{\alpha,\beta;L,M}_n(x)=\frac 1{2^n}
\sqrt{\frac {\Gamma(\alpha+1)\Gamma(\beta+1)(2n+\alpha+\beta+1)}
{\Gamma(\alpha+\beta+2)\Gamma(n+1)}}\\
\frac {\Gamma(2n+\alpha+\beta+1)}
{\sqrt {\Gamma(n+\alpha+1)\Gamma(n+\beta+1)\Gamma(n+\alpha+\beta+1)}}
\sqrt {\frac {\Delta^{\alpha,\beta;L,M}_n}{{\Delta}^{\alpha,\beta;L,M}_{n-1}}}
x^n+...
\end{multline*}
By  (2.21) we obtain the following solution of extremal problem
for a discrete loading Jacobi space.\\
{\bf Theorem 3.} \textit{The following representation}
\begin{multline*}
\inf_{{\Pi}^{(1)}_N\in {\Re}^{(1)}_N}{\langle {\Pi}^{(1)}_N, {\Pi}^{(1)}_N \rangle}_{\alpha,\beta}=\\
=\frac {\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}
\frac {2^{2N}N!}{2N+\alpha+\beta+1}\\
\frac {\Gamma(N+\alpha+\beta+1)\Gamma(N+\alpha+1)\Gamma(N+\beta+1)}{{\Gamma}^2(2N+\alpha+\beta+1)}
\frac {{\Delta}^{\alpha,\beta;L,M}_N}{{\Delta}^{\alpha,\beta;L,M}_{N-1}}.\\
\end{multline*}
\textit{holds. In addition, equality in the last relation is realized on the
following polynomial}\\
\begin{multline*}
{\Pi}^{(1),extr}_N=
\sqrt {\frac{\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}}
\frac {2^N \sqrt {N!}}{\sqrt {2N+\alpha+\beta+1}}\\
\frac {\sqrt {\Gamma(N+\alpha+1)\Gamma(N+\beta+1)\Gamma(N+\alpha+\beta+1)}}
{\Gamma(2N+\alpha+\beta+1)}
\sqrt {\frac {{\Delta}^{\alpha,\beta;L,M}_N}{{\Delta}^{\alpha,\beta;L,M}_{N-1}}}
\widehat P^{\alpha,\beta;L,M}_N(x),\\
\end{multline*}
\textit{where} $\widehat P^{\alpha,\beta;L,M}_n(x)$ is a polynomial system
orthonormal w.r.t. the inner product (1.4).\\

{\bf\bc
                 Acknowledgements
\ec}
The research of the author was supported by the Russian Foundation for Basic
Research (Grant 00-01-00286)\\

\begin{thebibliography}{20}
\bibitem{[1]} Osilenker B.P. On extremal problem for algebraic polynomials in a symmetric
discrete Gegenbauer--Sobolev space. Matem Zametki 82(2007), 411--425 [in russian].\\
\bibitem{[2]} F.V.Atkinson,Discrete and continuous boundary problems.Academic
Press,1964.\\
\bibitem{[3]}  Arvesu J., Marcellan F.,Alvarez--Nodarse R. On a modification
of the Jacobi linear functional:properties and zeros the corresponding orthogonal
 polynomials. J. Comput. Appl. Math. 71(2002), 127--158.\\
\bibitem{[4]} Courant R., Hilbert D., Methoden der mathematischen Physik.Vol.1.
Berlin, 1931.\\
\bibitem{[5]} Krall A. Hilbert space, Boundary value problems and Orthogonal polynomials.
Birkh\"{a}user, 2002.\\
\bibitem{[6]} Marcellan F.,Alfaro M., Rezola M.L. Orthogonal polynomials:old
and new directions.J. Comput. Appl. Math.48(1993) ,113--132.\\
\bibitem{[7]} Szego G. Orthogonal Polynomials. 4rd Edition, Amer.
Math. Soc., Providence,1975.\\

\bibitem{[8]} Koorwinder T.H. Orthogonal polynomials with the weight function\\
$M\delta(x+1)+N\delta(x-1)$, Canad. Math. Bull. 27(2)(1984), 205--214.\\

\bibitem{[9]} Koekoek R. Differential equations for symmetric generalized
 ultraspherical polynomials. Trans. Amer. Math. Soc. 345(1994), 47--72.\\
\bibitem{[10]} Koekoek J., Koekoek R. Differential equations for
generalized Jacobi polynomials. J. Comput. Appl. Math.126(2000), 1--31.\\
\bibitem{[11]}Littlejohn L.L. The Krall polynomials: A new class of
orthogonal polynomials. Quaest. Math. 5(1982), 255--265.\\
\bibitem{[12]} Osilenker B.P. Generalized trace formula and asymptotics
 of the averaged Turan determinant for polynomials orthogonal with
a discrete Sobolev inner product. J.Approx. Theory 141(2006), 70--97.\\
\bibitem{[13]} Guntmukher F.R. Matrix Theory. Moscow, Nauka, 1966.\\
\end{thebibliography}





Moscow State Civil Engineering University\\
Yaroslavskoe Shosse 26\\
Moscow 129336\\
Russia\\
Mathematical Department\\
Professor\\













\end{document}