\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 80, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/80\hfil Uniform convergence of spectral expansions]
{Uniform convergence of the spectral expansions in terms
 of root functions for a spectral problem}

\author[N. B. Kerimov, S. Goktas, E. A. Maris \hfil EJDE-2016/80\hfilneg]
{Nazim B. Kerimov, Sertac Goktas, Emir A. Maris}

\address{Nazim B. Kerimov \newline
Department of Mathematics,
Mersin University,
33343, Mersin, Turkey}
\email{nazimkerimov@yahoo.com}

\address{Sertac Goktas \newline
Department of Mathematics,
Mersin University,
33343, Mersin, Turkey}
\email{srtcgoktas@gmail.com}

\address{Emir A. Maris \newline
Department of Mathematics,
Mersin University,
33343, Mersin, Turkey}
\email{ealimaris@gmail.com}

\thanks{Submitted February 22, 2016. Published March 18, 2016.}
\subjclass[2010]{34B05, 34B24, 34L10, 34L20}
\keywords{Differential Operator; eigenvalues; root functions;
\hfill\break\indent uniform convergence of spectral expansion}

\begin{abstract}
 In this article, we consider the spectral problem
 \begin{gather*}
 -y''+q(x)y=\lambda y,\quad 0<x<1,\\
 y'(0)\sin \beta =y(0)\cos \beta , \quad
  0\le \beta <\pi ; \quad y'(1)=(a\lambda +b)y(1)
 \end{gather*}
 where $\lambda $ is a spectral parameter, $a$ and $b$ are real
 constants and $a<0$, $q(x)$ is a real-valued continuous
 function on the interval $[0,1]$. The root function
 system of this problem can also consist of associated functions.
 We investigate the uniform convergence of the spectral expansions in terms
 of root functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Consider the spectral problem
\begin{gather}\label{e1}
 -y''+q(x)y=\lambda y, \quad 0<x<1, \\
\label{e2}
 y'(0)\sin \beta =y(0)\cos \beta ,\quad 0\le \beta <\pi , \\
\label{e3}
 y'(1)=(a\lambda +b)y(1),
\end{gather}
where  $ \lambda$ is a spectral parameter, $ a $ and $ b $ are 
real constants and $ a<0 $, $ q(x) $ is real-valued continuous function 
on the interval $[0,1]$.

In this article, we study the uniform convergence of the expansions 
in terms of root functions of the boundary value problem 
\eqref{e1}--\eqref{e3} for the functions which belong to $ C[0,1] $. 
There are many articles which investigate the uniform convergence of the 
expansions for the functions in terms of root functions of some 
differential operators with a spectral parameter in the boundary conditions 
(see, for example,
 \cite{Gulyaev1,Gulyaev2,Kapustin1,Kapustin2,Kapustin3,Kapustin4,Kapustin5,
Kerimov1,Kerimov2,Marchenkov}).

Especially, the spectral problems which investigated the uniform convergence 
of the spectral expansions underlie an important class of the mathematical 
physics problems. For example, the problem
\begin{gather*}
u''(x)+\lambda u(x)=0 \quad (0<x<1),\\
u(1)=0, \quad (a-\lambda )u'(0)+b\lambda u(0)=0, \quad  a,b>0
\end{gather*}
appears in a model of transrelaxation heat process and in the mathematical 
description of vibrations of a loaded string (see \cite{Kapustin2}), 
and the problems on vibrations of a homogeneous loaded string, 
torsional vibrations of a rod with a pulley at one end, heat propagation 
in a rod with lumped heat capacity at one end, the current in a cable 
ground at one end through a concentrated capacitance or inductance 
lead to the spectral problem
\begin{gather*}
u''(x)+\lambda u(x)=0 \, (0<x<1),\\
u(0)=0, \quad  u'(1)=d\lambda u(1), \quad d>0
\end{gather*}
(see \cite{Kapustin2,Kapustin3}).

In  \cite{Kerimov2}, it has been investigated the uniform convergence 
of the Fourier series expansions in terms of eigenfunctions for the spectral 
problem
\begin{gather}\label{e0.1}
 -y''+q(x)y=\lambda y, \quad 0<x<1, \\
\label{e0.2}
 {{b}_0}y(0)={{d}_0}y'(0), \quad
({{a}_1}\lambda +{{b}_1})y(1)=({{c}_1}\lambda +{{d}_1})y'(1),
\end{gather}
where  $ \lambda  $ is a spectral parameter, $ q(x) $ is a real-valued 
continuous function on the interval $[0,1]$, and 
$ {{a}_1},{{b}_0},{{b}_1},{{c}_1},{{d}_0} $ and $ {{d}_1} $ 
are real constants that satisfy the conditions
\begin{equation}\label{e0.3}
 | {{b}_0} |+| {{d}_0} |\ne 0, \quad
 \sigma ={{a}_1}{{d}_1}-{{b}_1}{{c}_1}>0.
\end{equation}
Note that all the eigenvalues of problem \eqref{e0.1}, \eqref{e0.2} are real and simple, 
hence the root functions system of this problem consists of 
only eigenfunctions. Problem \eqref{e1}--\eqref{e3} does not satisfy 
the condition \eqref{e0.3}, because $ \sigma =a<0 $.

It was proved \cite{Binding} that the eigenvalues of  \eqref{e1}--\eqref{e3} 
form an infinite sequence $ {{\lambda }_n}$, ($n=0,1,2,\dots$)
 without finite limit points and only the following cases are possible:
\begin{itemize}
\item[(i)] all the eigenvalues are real and simple.

\item[(ii)] all the eigenvalues are real and all, except one double, are simple.

\item[(iii)] all the eigenvalues are real and all, except one triple, are simple.

\item[(iv)] all the eigenvalues are simple and all, except a conjugate pair 
of non-real, are real.
\end{itemize}
Note that the eigenvalues $ {{\lambda }_n}$  ($n=0,1,2,\dots$)  were considered 
to be listed according to non-decreasing real part and repeated according 
to algebraic multiplicity.
Therefore, the results of the article \cite{Kerimov2} cannot be applied directly
to the problem \eqref{e1}--\eqref{e3}.

We need some properties of eigenvalues, eigenfunctions and associated 
functions of problem \eqref{e1}--\eqref{e3}, for the uniform convergence 
of the spectral expansions in terms of root functions of this problem.

Let $  \varphi (x,\lambda ) $ and $ \psi (x,\lambda ) $ 
denote the solutions of  \eqref{e1} which satisfy the initial conditions
\begin{gather}\label{e1'}
 \varphi (0,\lambda )=1, \quad {\varphi }'(0,\lambda )=h, \\
\label{e2'}
 \psi (0,\lambda )=0, \quad {\psi }'(0,\lambda )=1,
\end{gather}
where $ h=\cot \beta$,  $(0<\beta <\pi )$.

It is easy to see by the same method as in \cite[theorem 2.1]{Kerimov2} 
that the following asymptotic formulae are valid for sufficiently large $n$:
\begin{itemize}

\item[(i)]  If $ \beta =0 $ and $ {{\lambda }_n}=\mu _n^2$,
 ($\operatorname{Re} \mu _n\ge 0$),  then
\begin{gather}\label{e4}
 {\mu }_n=n\pi +\frac{{{A}_1}}{n\pi }+O\big(\frac{\delta _n^{(1)}}{n} \big), \\
\label{e5}
\begin{aligned}
y_n(x)& =\psi (x,\lambda _n) \\
&=\frac{\sin n\pi x}{n\pi }+\frac{\cos n\pi x}{{{( n\pi  )}^2}}
\big[{{A}_1}x-\frac{1}{2}\int_0^{x}{q(\tau )d\tau }
+\frac{1}{2}\int_0^{x}{q(\tau )\cos 2n\pi \tau d\tau }\big]\\
&\quad+\frac{\sin n\pi x}{2{{( n\pi  )}^2}}
\int_0^{x}{q(\tau )\sin (2n\pi \tau) d\tau }
+O( \frac{\delta _n^{(1)}}{{{n}^2}} ),
\end{aligned}
\end{gather}
where 
\[
  {{A}_1}=\frac{1}{a}+\frac{1}{2}\int_0^{1}{q(\tau )d\tau } , \quad
 \delta _n^{(1)}=| \int_0^{1}{q(\tau )\cos (2n\pi \tau )d\tau } |+\frac{1}{n}.
\]

\item[(ii)]  If $  0<\beta <\pi $  and 
$ {{\lambda }_n}=\mu _n^2 (\operatorname{Re}{{\mu }_n}\ge 0) $ then
\begin{gather}\label{e6}
 {{\mu }_n}=\big( n-\frac{1}{2} \big)\pi
+\frac{{{A}_{2}}}{( n-\frac{1}{2} )\pi }+O\big(\frac{\delta _n^{(2)}}{n} \big), \\
\label{e7}
 \begin{aligned}
y_n(x)&=\varphi (x,{{\lambda }_n})\\
&=\cos ( n-\frac{1}{2} )\pi x+\frac{\sin ( n-\frac{1}{2} )\pi x}
{( n-\frac{1}{2} )\pi }
\Big[h-{{A}_{2}}x+\frac{1}{2}\int_0^{x}{q(\tau )d\tau }\\
&\quad +\frac{1}{2}\int_0^{x}{q(\tau )\cos ( 2n-1 )\pi \tau d\tau }\Big]\\
&\quad -\frac{\cos ( n-\frac{1}{2} )\pi x}{( 2n-1 )\pi }
\int_0^{x}{q(\tau )\sin( 2n-1 )\pi \tau d\tau }
+O\big( \frac{\delta _n^{(2)}}{n} \big),
 \end{aligned}
\end{gather}
where 
\[
 {{A}_{2}}=h+\frac{1}{a}+\frac{1}{2}\int_0^{1}{q(\tau )d\tau } ,\quad
\delta _n^{(2)}=\big| \int_0^{1}{q(\tau )\cos ( 2n-1 )\pi \tau d\tau } 
\big|+\frac{1}{n}.
\]
\end{itemize}

Let $ {{\lambda }_{k}} $ be a multiple eigenvalue 
$ ( {{\lambda }_{k}}={{\lambda }_{k+1}} ) $. 
Then for the first order associated function $y_{k+1}$
corresponding to the eigenfunction $y_k$,  the
following relations hold \cite[p. 28]{Naimark}
\begin{gather*}
-{{y''}_{k+1}}+q(x){{y}_{k+1}}={{\lambda }_{k}}{{y}_{k+1}}+{{y}_{k}},\\
{{y'}_{k+1}}(0)\sin \beta ={{y}_{k+1}}(0)\cos \beta ,\\
{{y'}_{k+1}}(1)=(a{{\lambda }_{k}}+b){{y}_{k+1}}(1)+a{{y}_{k}}(1).
\end{gather*}

Let $ {{\lambda }_{k}} $ be a triple eigenvalue 
$ ( {{\lambda }_{k}}={{\lambda }_{k+1}}={{\lambda }_{k+2}} ) $. 
Then for  the first order associated function $ {{y}_{k+1}} $ 
there exist the second order associated function $ {{y}_{k+2}} $ 
for which the following relations hold
\begin{gather*}
-{{y''}_{k+2}}+q(x){{y}_{k+2}}={{\lambda }_{k}}{{y}_{k+2}}+{{y}_{k+1}},\\
{{y'}_{k+2}}(0)\sin \beta ={{y}_{k+2}}(0)\cos \beta ,\\
{{y'}_{k+2}}(1)=(a{{\lambda }_{k}}+b){{y}_{k+2}}(1)+a{{y}_{k+1}}(1).
\end{gather*}

Note that the functions $ {{y}_{k+1}}+c{{y}_{k}} $ and $ {{y}_{k+2}}+d{{y}_{k}} $, 
where $c$ and $d$ are arbitrary constants, are also associated functions of the 
first and second order respectively.

Let $ y(x,\lambda ) $ denote the solution of the equation \eqref{e1} which 
satisfy the initial condition \eqref{e1'} if $ 0<\beta <\pi $  or \eqref{e2'} 
if $ \beta =0 $. Then, the eigenvalues of  \eqref{e1}--\eqref{e3} 
are the roots of the characteristic equation
\begin{equation}\label{e8}
 \omega (\lambda )=y'(1,\lambda )-(a\lambda +b)y(1,\lambda ).
\end{equation}

It was proven in  \cite{Aliyev} that if $ {{\lambda }_{k}} $ is a multiple 
(double or triple) eigenvalue of  \eqref{e1}--\eqref{e3}, then
\begin{equation}\label{e0.8}
\begin{gathered}
y(x,\lambda )\to {{y}_{k}}(x), \quad y'(x,\lambda )\to {{y'}_{k}}(x),\\
{{y}_{\lambda }}(x,\lambda )\to {{\tilde{y}}_{k+1}}(x),\quad
{{y'}_{\lambda }}(x,\lambda )\to {{{\tilde{y}}'}_{k+1}}(x)
\end{gathered}
\end{equation}
uniformly according to $ x\in [0,1] $, as $ \lambda \to {{\lambda }_{k}} $ 
(see also \cite{Ince}), where $ {{\tilde{y}}_{k+1}} $ is one of the 
associated functions of the first order. It is obvious that 
$ {{\tilde{y}}_{k+1}}={{y}_{k+1}}+\tilde{c}{{y}_{k}}$.

Furthermore, if $ {{\lambda }_{k}} $ is a triple eigenvalue of \eqref{e1}--\eqref{e3},
then 
\begin{equation}\label{e0.9}
 {{y}_{\lambda \lambda }}(x,\lambda )\to 2{{\tilde{y}}_{k+2}}(x),\quad
{{y'}_{\lambda \lambda }}(x,\lambda )\to 2{{{\tilde{y}}'}_{k+2}}(x)
\end{equation}
uniformly according to $ x\in [0,1] $, as $ \lambda \to {{\lambda }_{k}} $, 
where $ {{\tilde{y}}_{k+2}} $ is one of the associated functions of the 
second order corresponding to the first associated function 
$ {{\tilde{y}}_{k+1}} $. It is obvious that 
$ {{\tilde{y}}_{k+2}}={{y}_{k+2}}+\tilde{c}{{y}_{k+1}}+\tilde{d}{{y}_{k}} $ 
\cite{Aliyev,Ince}.

It is easily seen from \eqref{e0.8} and \eqref{e0.9} that
\begin{gather}\label{e8'}
 \tilde{c}= \begin{cases}
  -{{{y'}}_{k+1}}(0), & \text{if }  \beta =0,\\
  -{{y}_{k+1}}(0), & \text{if } 0<\beta <\pi,
 \end{cases}\\
\label{e9'}
 \tilde{d}=\begin{cases}
  {{( {{{y'}}_{k+1}}(0) )}^2}-{{{y'}}_{k+2}}(0),
& \text{if } \beta =0 , \\
y_{k+1}^2(0)-{{y}_{k+2}}(0), & \text{if }  0<\beta <\pi. 
 \end{cases}
\end{gather}

The following systems were investigated in \cite{Aliyev}:
\begin{itemize}
\item[(a)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne l) $, if all of eigenvalues of 
 \eqref{e1}--\eqref{e3} are real and simple, where $l$ is an 
arbitrary non-negative integer.

\item[(b)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne k+1) $, if $ {{\lambda }_{k}} $ 
is double eigenvalue $ ( {{\lambda }_{k}}={{\lambda }_{k+1}} ) $ of the problem 
\eqref{e1}--\eqref{e3}.

\item[(c)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne k) $, if 
$ {{\lambda }_{k}} $ is double eigenvalue $ ( {{\lambda }_{k}}={{\lambda }_{k+1}} ) $
 of  \eqref{e1}--\eqref{e3} and
\begin{equation}\label{e10'}
 {\omega }'''({{\lambda }_{k}})\ne 3\tilde{c}{\omega }''({{\lambda }_{k}}).
\end{equation}

\item[(d)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne l) $, if 
$ {{\lambda }_{k}} $ is double eigenvalue $ ( {{\lambda }_{k}}={{\lambda }_{k+1}} ) $
 of  \eqref{e1}--\eqref{e3}, where $ l\ne k,k+1 $ is an arbitrary non-negative 
integer.

\item[(e)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne k+2) $, if $ {{\lambda }_{k}} $ 
is triple eigenvalues $ ( {{\lambda }_{k}}={{\lambda }_{k+1}}={{\lambda }_{k+2}} ) $
 of \eqref{e1}--\eqref{e3}.

\item[(f)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne k+1) $, if 
$ {{\lambda }_{k}} $ is triple eigenvalues 
$ ( {{\lambda }_{k}}={{\lambda }_{k+1}}={{\lambda }_{k+2}} ) $ of 
 \eqref{e1}--\eqref{e3} and
\begin{equation}\label{e11'}
 {{\omega }^{IV}}({{\lambda }_{k}})\ne 4\tilde{c}{\omega }'''({{\lambda }_{k}}).
\end{equation}

\item[(h)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne k) $, if $ {{\lambda }_{k}} $ 
is triple eigenvalues $ ( {{\lambda }_{k}}={{\lambda }_{k+1}}={{\lambda }_{k+2}} ) $
 of  \eqref{e1}--\eqref{e3} and
\begin{equation}\label{e12'}
\frac{{{\omega }^{IV}}({{\lambda }_{k}})}{4!}
\Big(\frac{{{\omega }^{IV}}({{\lambda }_{k}})}{4!}
-\tilde{c}\frac{{\omega }'''({{\lambda }_{k}})}{3!} \Big)
\neq \frac{{\omega }'''({{\lambda }_{k}})}{3!}
\big(\frac{{{\omega }^{V}}({{\lambda }_{k}})}{5!}-\tilde{d}
\frac{{\omega }'''({{\lambda }_{k}})}{3!} \Big).
\end{equation}

\item[(h)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne l) $, if 
$ {{\lambda }_{k}} $ is triple eigenvalues 
$ ( {{\lambda }_{k}}={{\lambda }_{k+1}}={{\lambda }_{k+2}} ) $ of 
 \eqref{e1}--\eqref{e3}, where $ l\ne k,k+1,k+2 $ is an arbitrary non-negative 
integer.

\item[(i)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne r) $, if 
$ {{\lambda }_{r}} $ and $ {{\lambda }_{s}} $ are conjugate of non-real 
eigenvalues $ ( {{\lambda }_{s}}={{\overline{\lambda }}_{r}} ) $ of 
 \eqref{e1}--\eqref{e3}.

\item[(j)] $ {{y}_n}(x)$ $(n=0,1,\dots;n\ne l) $, if 
$ {{\lambda }_{r}} $ and $ {{\lambda }_{s}} $ are conjugate of non-real
 eigenvalues $ ( {{\lambda }_{s}}={{\overline{\lambda }}_{r}} ) $ 
of  \eqref{e1}--\eqref{e3}, where $l\ne r,s$ is an arbitrary non-negative integer.

\end{itemize}
It was proven in \cite{Aliyev} that each of the systems 
(a)-(j) is a basis of $ {{L}_{p}}( 0,1 ),1<p<\infty $; moreover, 
if $ p=2 $, then this basis is unconditional.

We denote by $\{ {{u}_n}( x )\} $ corresponding biorhogonally conjugate 
to each of the systems (a)-(j). For example, the system 
$ {{u}_n}(x) \, (n=0,1,\dots;n\ne k) $ is biorhogonally conjugate to 
system (c).

The following auxiliary associated functions were considered in \cite{Aliyev}:
\begin{gather}\label{e3.1}
 y_{k+1}^{*}={{y}_{k+1}}+{{c}_1}{{y}_{k}}, \\
\label{e3.2}
 y_{k+1}^{**}={{y}_{k+1}}+{{c}_{2}}{{y}_{k}}, \\
\label{e3.3}
 y_{k+2}^{\#\#}={{y}_{k+2}}+{{c}_{2}}{{y}_{k+1}}+{{d}_{2}}{{y}_{k}}
\end{gather}
where
\begin{gather}\label{e3.4}
 {{c}_1}=-\frac{{\omega }'''({{\lambda }_{k}})}{3{\omega }''({{\lambda }_{k}})}
-\frac{{{y}_{k+1}}(1)}{{{y}_{k}}(1)}+\tilde{c}, \\
\label{e3.5}
 {{c}_{2}}=-\frac{{{\omega }^{IV}}({{\lambda }_{k}})}{4{\omega }'''
({{\lambda }_{k}})}-\frac{{{y}_{k+1}}(1)}{{{y}_{k}}(1)}+\tilde{c}, \\
\label{e3.6}
\begin{aligned}
{{d}_{2}}
&=-\frac{{{\omega }^{V}}({{\lambda }_{k}})}{20{\omega }'''
({{\lambda }_{k}})}+{{\Big( \frac{{{\omega }^{IV}}({{\lambda }_{k}})}{4{\omega }'''
({{\lambda }_{k}})} \Big)}^2} \\
&\quad +\Big( \frac{{{\omega }^{IV}}({{\lambda }_{k}})}{4{\omega }'''({{\lambda }_{k}})}
+\frac{{{y}_{k+1}}(1)}{{{y}_{k}}(1)} \Big)
\Big( \frac{{{y}_{k+1}}(1)}{{{y}_{k}}(1)}-\tilde{c} \Big)
-\frac{{{y}_{k+2}}(1)}{{{y}_{k}}(1)}+\tilde{d}.
\end{aligned}
\end{gather}
These auxiliary associated functions were studied for the basis properties 
of  systems (c), (f) and (g) respectively. We will use them for 
the uniform convergence of the spectral expansions in  systems (c), (f) and (g).


It is verified in \cite{Aliyev} that if $ {{\lambda }_{k}} $ is double 
eigenvalue of the problem \eqref{e1}--\eqref{e3}, the condition \eqref{e10'} 
is equivalent to the condition $ y_{k+1}^{*}( 1 )\ne 0 $; 
if $ {{\lambda }_{k}} $ is triple eigenvalue of  \eqref{e1}--\eqref{e3}, 
the conditions \eqref{e11'} and \eqref{e12'} are equivalent to the
 conditions $ y_{k+1}^{**}( 1 )\ne 0 $ and $ y_{k+2}^{\#\#}( 1 )\ne 0 $ 
respectively.

\section{Uniform convergence of the spectral expansions for the boundary value 
problem \eqref{e1}--\eqref{e3}}

In this section, we give uniformly convergent spectral expansions 
in terms of root functions of the problem \eqref{e1}--\eqref{e3}.
We define the trigonometric system
 $\{ {{\theta }_n}(x)\}_{n=1}^{\infty }$ as follows:
\[
 {{\theta }_n}(x)= \begin{cases}
  \sqrt{2}\sin n\pi x , & \text{if } \beta=0 , \\
  \sqrt{2}\cos ( n-\frac{1}{2} )\pi x , & \text{if } 0<\beta<\pi .
 \end{cases}
\]

\begin{theorem}\label{thm1}
 Suppose that $f\in C[0,1]$ and $f( x )$ has a uniformly convergent Fourier
expansions in the system $\left\{ {{\theta }_n}(x) \right\}_{n=1}^{\infty }$ 
on the interval $[0,1]$. Then, the function $f( x )$ can be expanded in 
Fourier series in each of the systems (a)-(j) and these expansions are uniformly
 convergent on every interval $ [0,b],0<b<1 $. Moreover, the Fourier series 
of $f( x )$ in  systems (a)-(j) are uniformly convergent on $[0,1]$ if 
and only if $( f,{{y}_{l}} )=0$ for  systems (a), (d), (h) and (j); 
$( f,{{y}_{k}} )=0$ for the systems (b) and (e); $( f,y_{k+1}^{*} )=0$ 
for  system (c); $( f,y_{k+1}^{**} )=0$ for system (f); 
$( f,y_{k+2}^{\#\#} )=0$ for  system (g) and $( f,{{y}_{s}} )=0$ for  system (i).
\end{theorem}

\begin{proof}
We only prove theorem \ref{thm1} for  system (c). The proof of the theorem 
for other systems is similar.

Let $\beta =0$. Consider the Fourier series $f( x )$ on the interval $[0,1]$ 
in system (c):
 \begin{equation}\label{e19}
  F(x)=\sum_{n=0,n\ne k}^{\infty }{( f,{{u}_n} ){{y}_n}(x)},
 \end{equation}
where the system ${{u}_n}(x)$ $(n=0,1,\dots;n\ne k)$ is defined by 
(see \cite{Aliyev})
\begin{equation}\label{e2.1}
  {{u}_n}(x)=\frac{{{y}_n}(x)-\frac{{{y}_n}(1)}{y_{k+1}^{*}(1)}y_{k+1}^{*}(x)}{{{\| {{y}_n} \|}^2}+ay_n^2(1)},{{u}_{k+1}}(x)=\frac{{{y}_{k}}(x)-\frac{{{y}_{k}}(1)}{y_{k+1}^{*}(1)}y_{k+1}^{*}(x)}{-{{y}_{k}}(1)\frac{{\omega }''({{\lambda }_{k}})}{2}}
 \end{equation}
where $ y_{k+1}^{*} $ is defined by \eqref{e3.1}.
Let

 \begin{equation}\label{e20}
  {{g}_n}={{\Big( {{\| {{y}_n} \|}^2}+ay_n^2(1) \Big)}^{-1}}.
 \end{equation}
Then according to \eqref{e2.1}, we obtain
\begin{equation}\label{e21}
  {{u}_n}(x)={{g}_n}\Big( {{y}_n}(x)-\frac{{{y}_n}(1)}{y_{k+1}^{*}(1)}y_{k+1}^{*}(x) 
\Big).
 \end{equation}
By  \eqref{e5}, we have the estimates
\begin{gather}\label{e22}
  {{y}_n}(1)=\frac{{{(-1)}^{n}}}{a{{(n\pi )}^2}}
+O\big( \frac{\delta _n^{(1)}}{{{n}^2}} \big), \\
\label{e23}
  {{\| {{y}_n} \|}^2}=\frac{1}{2{{(n\pi )}^2}}+O( {{n}^{-3}} )\,.
\end{gather}
By \eqref{e22} and \eqref{e23},  equality \eqref{e20} can be written as
\begin{equation}\label{e24}
  {{g}_n}=2{{(n\pi )}^2}+O(n).
 \end{equation}
Note that the series \eqref{e19} is uniformly convergent on $[0,1]$ 
if and only if the series
\begin{equation}\label{e25}
  {{F}_1}(x)=\sum_{n=k+2}^{\infty }{( f,{{u}_n} ){{y}_n}(x)}
 \end{equation}
is uniformly convergent on $[0,1]$. Suppose that the sequence 
$\{ {{S}_{m}}( x )\}_{m=k+2}^{\infty } $ is the partial sum of the series 
\eqref{e25}. By using \eqref{e21}, the equality
\[
{{S}_{m}}(x)={{S}_{m,1}}(x)+{{S}_{m,2}}(x)
\]
 holds, where
\begin{equation}\label{e26}
  \begin{gathered}
{{S}_{m,1}}(x)=\sum_{n=k+2}^{m}{{{g}_n}( f,{{y}_n} ){{y}_n}(x)}, \\
   {{S}_{m,2}}(x)=-\frac{( f,y_{k+1}^{*} )}{y_{k+1}^{*}(1)}
\sum_{n=k+2}^{m}{{{g}_n}{{y}_n}( 1 ){{y}_n}(x)}.
  \end{gathered}
 \end{equation}
Firstly, we analyze the uniform convergence of the first sequence 
in \eqref{e26}. Using \eqref{e24}, we obtain
\begin{equation}\label{e28}
  {{g}_n}(f,{{y}_n}){{y}_n}(x)=2(f,n\pi {{y}_n})n\pi {{y}_n}(x)
+(f,n\pi {{y}_n}){{y}_n}(x)O(1).
 \end{equation}
By \eqref{e5}, the estimate
\begin{equation}\label{e29}
\begin{aligned}
  n\pi {{y}_n}(x)
&=\sin n\pi x
+\frac{\alpha ( x )\cos n\pi x}{n\pi }+\frac{{{\alpha }_n}( x )\cos n\pi x}{2n\pi }\\
&\quad + \frac{{{\beta }_n}( x )\sin n\pi x}{2n\pi }
+O\big( \frac{\delta _n^{(1)}}{n} \big)
\end{aligned}
 \end{equation}
holds, where
\begin{gather}\label{e31}
  \alpha (x)={{A}_1}x-\frac{1}{2}\int_0^{x}{q( \tau  )}d\tau, \\
\label{e32}
  {{\alpha }_n}(x)=\int_0^{x}{q( \tau  )}\cos 2n\pi \tau d\tau, \\
\label{e33}
  {{\beta }_n}(x)=\int_0^{x}{q( \tau  )\sin n\pi \tau }d\tau.
 \end{gather}
Note that $ \alpha (x)\in C[0,1] $ and the functional sequences 
$ \{ {{\alpha }_n}(x) \}_{n=k+2}^{\infty } $, 
$ \{ {{\beta }_n}(x) \}_{n=k+2}^{\infty } $ are uniformly bounded. 
Hence, by \eqref{e29}, the equality \eqref{e28} can be written as
\[
{{g}_n}(f,{{y}_n}){{y}_n}(x)=2(f,\sin n\pi x)\sin n\pi x+{{B}_n}(x),
\]
 where
\begin{equation}\label{e30}
\begin{aligned}
   {{B}_n}(x)
&=(f,\sin n\pi x)O( \frac{1}{n} )+(\alpha (x)f(x),\cos n\pi x)O( \frac{1}{n} ) \\
&\quad +(f,{{\alpha }_n}(x)\cos n\pi x)O( \frac{1}{n} )
+(f,{{\beta }_n}(x)\sin n\pi x)O( \frac{1}{n} )
+O\big(\frac{\delta _n^{(1)}}{n}\big).
\end{aligned}
 \end{equation}
Therefore
 \[
{{S}_{m,1}}(x)=\sum_{n=k+2}^{m}{( f,\sqrt{2}\sin n\pi x )
\sqrt{2}\sin n\pi x+}\sum_{n=k+2}^{m}{{{B}_n}(x)}\,.
\]
The series
\begin{equation}\label{e34}
  \sum_{n=k+2}^{\infty }{{{B}_n}(x)}
 \end{equation}
is absolutely and uniformly convergent on $[0,1]$. 
Indeed, by \eqref{e30} we have
\begin{align*}
 | {{B}_n}(x) |
&\le \frac{{{C}_1}}{n}\big\{ | ( f,\sin n\pi x ) |
 +| ( \alpha (x)f(x),\cos n\pi x ) |  \\
&\quad  +| ( f,{{\alpha }_n}(x)\cos n\pi x ) |
 +| ( f,{{\beta }_n}(x)\sin n\pi x ) |+\delta _n^{(1)} \big\}  \\
& \le {{C}_{2}}\Big\{ {{| ( f,\sin n\pi x ) |}^2}
 +{{| ( \alpha (x)f(x),\cos n\pi x ) |}^2}  \\
&\quad +{{\Big( \int_0^{1}{| f(x){{\alpha }_n}(x) |dx} \Big)}^2}
 +{{\Big(\int_0^{1}{| f(x){{\beta }_n}(x) |dx} \Big)}^2}
 +\frac{\delta _n^{(1)}}{n} \Big\}, 
 \end{align*}
where $ {{C}_1} $ and $ {{C}_{2}} $ are certain positive constants.
 By the Bessel inequality for the Fourier coefficients, the numerical series
 \[
\sum_{n=k+2}^{\infty }{{{| ( f,\sin n\pi x ) |}^2},} \quad
\sum_{n=k+2}^{\infty }{{{| ( \alpha (x)f(x),\cos n\pi x ) |}^2},\quad
\sum_{n=k+2}^{\infty }{\frac{\delta _n^{(1)}}{n}}}
\]
 are convergent. By using Bessel inequality again and by \eqref{e32}, we obtain
\begin{align*}
 \sum_{n=k+2}^{\infty }{{{( \int_0^{1}{| f(x){{\alpha }_n}(x) |}dx )}^2}}
&\le {{\| f \|}^2}\sum_{n=k+2}^{\infty }
 {\int_0^{1}{{{| {{\alpha }_n}(x) |}^2}}}dx  \\
 &\le {{\| f \|}^2}\int_0^{1}{\Big[\sum_{n=k+2}^{\infty }{{{
\big| \int_0^{x}{q(\tau )\cos 2n\pi \tau d\tau } \big|}^2}}\Big]}dx   \\
 &\le {{C}_3}{{\| f \|}^2}\int_0^{1}{\int_0^{x}{{{| q(\tau ) |}^2}d\tau }}dx
\le {{C}_3}{{\| f \|}^2}{{\| q \|}^2},
 \end{align*}
where $ {{C}_3} $ is a certain positive constant. Similarly, by \eqref{e33}
 we obtain the estimate
\[
\sum_{n=k+2}^{\infty }{{{\Big( \int_0^{1}{| f(x){{\beta }_n}(x) |dx} \Big)}^2}}
\le {{C}_4}{{\| f \|}^2}{{\| q \|}^2},
\]
where $ {{C}_4} $ is a certain positive constant. Thus, 
the functional series \eqref{e34} is absolutely and uniformly convergent. 
Since the series
\[
\sum_{n=k+2}^{\infty }{( f,\sqrt{2}\sin n\pi x )\sqrt{2}\sin n\pi x}
\]
is uniformly convergent on the interval $[0,1]$. The sequence 
$ \left\{ {{S}_{m,1}}(x) \right\}_{m=k+2}^{\infty } $ is also uniformly 
convergent on this interval.

 If $ ( f,y_{k+1}^{*} )=0 $, then the equality $ {{S}_{m}}(x)={{S}_{m,1}}(x) $ holds. 
Hence, the functional sequence $ \left\{ {{S}_{m}}(x) \right\}_{m=k+2}^{\infty } $ 
is uniformly convergent on the interval $[0,1]$. Consequently, 
 in the case $\beta =0 $, the second part of the Theorem \ref{thm1} is proven.

 Suppose that $ ( f,y_{k+1}^{*} )\ne 0 $. We now analyze the uniform convergence 
of the second functional sequence in \eqref{e26}. By using \eqref{e5}, \eqref{e22} 
and \eqref{e24}, we obtain
\[
\sum_{n=k+2}^{m}{{{g}_n}{{y}_n}(1)}{{y}_n}(x)
=\frac{2}{a\pi }\sum_{n=k+2}^{m}{\frac{\sin n\pi (1+x)}{n}}
+\sum_{n=k+2}^{m}{O({{n}^{-2}})}.
\]
Note that the series
 \[
\sum_{n=k+2}^{\infty }{\frac{\sin nt}{n}}
\]
 is uniformly convergent on every closed interval $[\delta ,2\pi -\delta] $, 
where $  0<\delta <\pi $ \cite[Chapter I, \S30, Theorem I]{Bary}. So, the series
 \[
\sum_{n=k+2}^{\infty }{\frac{\sin n\pi (1+x)}{n}}
\]
 is uniformly convergent on the interval $ [0,b] $, $ 0<b<1 $. Hence, the 
functional sequence $ \left\{ {{S}_{m,2}}(x) \right\}_{m=r+1}^{\infty } $ 
is uniformly convergent on $ [0,b] $, $ 0<b<1 $.

Let $  0<\beta <\pi  $. Consider the Fourier series $ f(x) $ on the interval 
$ [0,1] $ in  system (c):
\begin{equation}\label{e35}
  G(x)=\sum_{n=0,n\ne k}^{\infty }{( f,{{u}_n} )}{{y}_n}(x),
 \end{equation}
where the system $ {{u}_n}(x) \, (n=0,1,\dots;n\ne k) $ 
is defined by \eqref{e2.1}.

 Note that the series \eqref{e35} is uniformly convergent on 
$[0,1]$ if and only if the series
\begin{equation}\label{e36}
  {{G}_1}(x)=\sum_{n=k+2}^{\infty }{( f,{{u}_n} )}{{y}_n}(x),
 \end{equation}
is uniformly convergent on $[0,1]$.

 Suppose that the sequence $ \left\{ {{G}_{m}}(x) \right\}_{m=k+2}^{\infty } $ 
is the partial sum of the series \eqref{e36}. By using \eqref{e2.1}, the equality
 \[
{{G}_{m}}(x)={{G}_{m,1}}(x)+{{G}_{m,2}}(x)
\]
 holds, where
\begin{gather*}
{{G}_{m,1}}(x)=\sum_{n=k+2}^{m}{{{h}_n}( f,{{y}_n} ){{y}_n}(x)},\\
{{G}_{m,2}}(x)=-\frac{( f,y_{k+1}^{*} )}{y_{k+1}^{*}(1)}
\sum_{n=k+2}^{m}{{{h}_n}{{y}_n}( 1 ){{y}_n}(x)},\\
{{h}_n}={{( {{\| {{y}_n} \|}^2}+ay_n^2(1) )}^{-1}}.
\end{gather*}
 By using \eqref{e7}, we obtain the estimates
\begin{gather}\label{e37}
  {{y}_n}( 1 )=\frac{2{{(-1)}^{n}}}{a(2n-1)\pi }
 +O\Big( \frac{\delta _n^{(2)}}{n} \Big), \\
\label{e38}
  {{h}_n}=2+O({{n}^{-1}}).
 \end{gather}
From \eqref{e7}, \eqref{e37} and \eqref{e38},
 \[
{{h}_n}{{y}_n}(1){{y}_n}(x)
=-\frac{4}{a\pi }\frac{\sin ( n-\frac{1}{2} )\pi ( 1+x )}{2n-1}
+O\Big( \frac{\delta _n^{(2)}}{n} \Big)
\]
Since
\begin{align*}
 \big| \sum_{n=k+2}^{m}{\sin ( n-\frac{1}{2} )\pi (1+x)} \big|
&=\frac{| \cos (k+1)\pi ( 1+x )-\cos m\pi ( 1+x ) |}{2\sin \frac{\pi ( 1+x )}{2}}\\
&\le \frac{1}{\sin \frac{\pi ( 1+x )}{2}}
 \le \frac{1}{\sin \frac{\pi ( 1+b )}{2}}, 
\end{align*}
for $0\le x\le b<1$
 and the numerical series $ \sum_{n=k+2}^{\infty } \delta _n^{(2)}/n $ 
is convergent, then the sequence $ \left\{ {{G}_{m,2}}(x) \right\}_{m=k+2}^{\infty }$
 is absolutely and uniformly convergent on the interval  $ [0,b] $,
 $ 0<b<1 $ \cite[Introductory material, \S1 , Abel's Lemma]{Bary}.

 Note that the sequence $\{ {{G}_{m,1}}(x)\}_{m=k+2}^{\infty } $ is uniformly 
convergent on the interval $ [0,1] $. This can be seen by the method of the 
case $\beta =0$.
The proof of the theorem \ref{thm1} is complete.
\end{proof}

\begin{theorem}\label{thm2}
 Suppose that $f\in C[0,1]$ and $f( x )$ has a uniformly convergent Fourier 
expansions in the system $\{ {{\theta }_n}(x) \}_{n=1}^{\infty }$ on the 
interval $[0,1]$, then this function can be expanded in Fourier series 
in each of the systems $\{ {{u}_n}( x )\}$ which are biorthogonally 
conjugates to  systems {\rm (a)-(j)} and these expansions are uniformly 
convergent on the interval $[0,1]$.
\end{theorem}

\begin{proof}
 We only prove theorem \ref{thm2} for system \eqref{e2.1} which is 
biorthogonally conjugate to  system (c). The proof of the theorem for
 other systems is similar.

Let $\beta =0$. Consider the Fourier series $ f(x) $ on the interval $[0,1]$
in  \eqref{e2.1}:
\begin{equation}\label{e39}
  T(x)=\sum_{n=0,n\ne k}^{\infty }{( f,{{y}_n} )}{{u}_n}(x).
 \end{equation}
Note that the series \eqref{e39} is uniformly convergent on $[0,1]$ 
if and only if the series
\begin{equation}\label{e40}
  {{T}_1}(x)=\sum_{n=k+2}^{\infty }{( f,{{y}_n} )}{{u}_n}(x)
 \end{equation}
 is uniformly convergent on $[0,1]$.

Suppose that the sequence $\{ {{T}_{m}}(x)\}_{m=k+2}^{\infty } $ 
is the partial sum of the series \eqref{e40}. By using \eqref{e2.1}, the equality
\[
{{T}_{m}}(x)={{T}_{m,1}}(x)+{{T}_{m,2}}(x)
\]
holds, where
\begin{gather*}
{{T}_{m,1}}(x)=\sum_{n=k+2}^{m}{{{g}_n}( f,{{y}_n} ){{y}_n}(x)}, \\
{{T}_{m,2}}(x)=-\frac{y_{k+1}^{*}(x)}{y_{k+1}^{*}(1)}
\sum_{n=k+2}^{m}{{{g}_n}{{y}_n}( 1 )(f,{{y}_n})}.
\end{gather*}
The sequences $\{ {{S}_{m,1}}(x)\}_{m=k+2}^{\infty } $ and 
$\{ {{T}_{m,1}}(x)\}_{m=k+2}^{\infty }$ are the same. 
Therefore, the sequence $\{ {{T}_{m,1}}(x)\}_{m=k+2}^{\infty } $ is uniformly 
convergent on the interval $[0,1]$.

Using \eqref{e5}, \eqref{e22} and \eqref{e24} we obtain
\[
{{g}_n}{{y}_n}(1)(f,{{y}_n})=\frac{2{{(-1)}^{n}}}{an\pi }( f,\sin n\pi x )
+O\Big( \frac{\delta _n^{(1)}}{n} \Big).
\]
 From here, the estimate
\[
 | {{g}_n}{{y}_n}(1)(f,{{y}_n}) |
 \le \frac{{{C}_{5}}}{n} \big\{ | ( f,\sin n\pi x ) |+\delta _n^{(1)} \big\} 
 \le {{C}_{6}}\big\{ {{| ( f,\sin n\pi x ) |}^2}
 +\frac{\delta _n^{(1)}}{n} \big\} 
\]
holds, where $ {C}_{5} $ and $ {C}_{6} $ are certain positive number. 
The numerical series
\[
\sum_{n=k+2}^{\infty }{{{| ( f,\sin n\pi x ) |}^2}}, \quad
 \sum_{n=k+2}^{\infty }{\frac{\delta _n^{(1)}}{n}}
\]
are convergent. Consequently, the sequence $\{ {{T}_{m,2}}(x)\}_{m=k+2}^{\infty } $
 is absolutely and uniformly convergent on $[0,1]$.

In the case $ 0<\beta <\pi $, the proof is similar.
Theorem \ref{thm2} is proven.
\end{proof}

\section{Examples}

\begin{example} \label{examp1} \rm
Consider the spectral problem
\begin{gather}\label{e3.1'}
 -y''=\lambda y, \quad  0<x<1, \\
\label{e3.2'}
 y(0)=0, \, y'(1)=( -\frac{\lambda }{3}+1 )y(1)
\end{gather}
where $ \lambda $  is a spectral parameter.

The eigenvalues of problem \eqref{e3.1'}--\eqref{e3.2'} are the root
the equation $ \omega ( \lambda  )=0 $, where 
$  \omega ( \lambda  )=( \frac{\lambda }{3}-1 )
\frac{\sin \sqrt{\lambda }}{\sqrt{\lambda }}+\cos \sqrt{\lambda } $
 and $ \operatorname{Re}\sqrt{\lambda }\ge 0 $. It is easy to see that
\begin{equation}\label{e3.3'}
 \omega ( \lambda  )=-{{\lambda }^2}\sum_{n=0}^{\infty }{\frac{4{{( -1 )}^{n}}( n+1 )
( n+2 )}{( 2n+5 )!}{{\lambda }^{n}}}.
\end{equation}
Therefore, $ \lambda =0 $ is double eigenvalue of  \eqref{e3.1'}-\eqref{e3.2'}. 
Hence, all the eigenvalues of  \eqref{e3.1'}-\eqref{e3.2'} are real and all, 
except one double, are simple. 
Further, by \eqref{e3.3'},  if $\lambda <0$, then $w( \lambda  )<0$.  
Then, $\lambda =0$ is the first eigenvalue of  \eqref{e3.1'}--\eqref{e3.2'}
 and ${{\lambda }_0}={{\lambda }_1}=0$.

From \eqref{e3.3'}, $ \omega ( 0 )={\omega }'( 0 )=0 $, 
$ {\omega }''( 0 )=-\frac{2}{45} $ and $ {\omega }'''( 0 )=\frac{1}{105} $. 
Eigenfunctions corresponding to $ {{\lambda }_n}( 0,2,3,\dots ) $ are 
$ {{y}_0}( x )=x $ and 
$ {{y}_n}( x )=\frac{\sin \sqrt{{{\lambda }_n}}x}{\sqrt{{{\lambda }_n}}}$
$( n\ge 2 ) $, associated function corresponding to eigenfunction $ {{y}_0} $ 
is $ {{y}_1}( x )=-\frac{{{x}^{3}}}{6}+cx $, where $ c $ is an arbitrary constant. 
From \eqref{e8'}, $ \tilde{c}=-c $. By \eqref{e3.4}, 
\[
{{c}_1}=-\frac{{\omega }'''( 0 )}{3{\omega }''( 0 )}
-\frac{{{y}_{k+1}}( 1 )}{{{y}_{k}}( 1 )}+\tilde{c}=\frac{5}{21}-2c . 
\]
Note that $ y_1^{*}={{y}_1}+{{c}_1}{{y}_0} $ and 
$ y_1^{*}( 1 )\ne 0( \text{or} \, \, {\omega }'''({{\lambda }_0})
\ne 3\tilde{c}{\omega }''({{\lambda }_0}) ) $, hence $ c\ne 1/14$.
Therefore, if $ c\ne 1/14 $, then the system 
$ {{y}_n}(x)$ $(n=1,2,\dots)$ is a basis in 
$ {{L}_{p}}( 0,1 ),1<p<\infty $ (see, \cite{Aliyev}).

Let $ f( x )={{x}^2}-x $. Since 
\[
( f,\sin n\pi x )= \begin{cases}
 0 , & \text{if $n$ is even} \\
-\frac{4}{{{n}^{3}}{{\pi }^{3}}}, & \text{if $n$ is odd}, 
\end{cases} 
\]
the function $f( x )$  can be expanded uniformly convergent 
Fourier series in the system  $\{ \sqrt{2}\sin n\pi x \}_{n=1}^{\infty }$. 
Further, $( f,{{y}_1}^{*} )=\frac{8}{315}-\frac{c}{3}$. Consequently, if 
$ c=\frac{8}{105} $, then the Fourier series of $ f( x ) $ 
in the system $ {{y}_n}(x) \, (n=1,2,\dots) $ is uniformly convergent on 
$[0,1]$; if $ c\ne \frac{8}{105},\frac{1}{4} $, then the Fourier series of 
$ f( x ) $ in the system $ {{y}_n}(x) \, (n=1,2,\dots) $ is uniformly 
convergent on $ [0,b] $, $ 0<b<1 $.
\end{example}


\begin{example} \label{examp2}\rm
 Consider the spectral problem
\begin{gather}\label{e4.1}
 -y''=\lambda y, \quad  0<x<1, \\
\label{e4.2}
 y'( 0 )=\alpha y(0), \quad y'(1)=( a\lambda +b )y(1)
\end{gather}
where $ \lambda $ is a spectral parameter, $ \alpha $ is unique real 
root of the equation
\begin{equation}\label{e4.3}
 {{\alpha }^{3}}+6{{\alpha }^2}+15\alpha +15=0
\end{equation}
(verify that $\alpha =\sqrt[3]{\frac{2}{1+\sqrt{5}}}-\sqrt[3]{\frac{1+\sqrt{5}}{2}}-2$)
 and
\begin{equation}\label{e4.4}
 a=-\frac{{{\alpha }^2}+3\alpha +3}{3{{( \alpha +1 )}^2}}, \quad
 b=\frac{\alpha }{\alpha +1}.
\end{equation}
The eigenvalues of \eqref{e4.1}--\eqref{e4.2} are the roots of the function
\begin{equation}\label{e4.5}
 \omega ( \lambda  )=( -a\lambda +\alpha -b )\cos \sqrt{\lambda }
-( ( \alpha a+1 )\lambda +\alpha b )\frac{\sin \sqrt{\lambda }}{\sqrt{\lambda }}
\end{equation}
where $ \operatorname{Re}\sqrt{\lambda }\ge 0 $.

Note that by \eqref{e4.3} and \eqref{e4.4}, the equalities 
$ \alpha -b=\alpha b=15a$, $\alpha a+1=-6a $ hold. 
Therefore, the equality \eqref{e4.5} can be written as 
$ \omega ( \lambda  )=( 15a-a\lambda  )\cos \sqrt{\lambda }
+( 6a\lambda -15a )\frac{\sin \sqrt{\lambda }}{\sqrt{\lambda }} $.
 Hence, the Maclaurin series of $\omega ( \lambda  )$ forms 
\begin{equation}\label{e4.6}
 \omega ( \lambda  )=-a{{\lambda }^{3}}\sum_{n=0}^{\infty }{\frac{2{{( -1 )}^{n}}
( n+2 )( n+3 )( 4n+19 )}{( 2n+7 )!}}{{\lambda }^{n}}.
\end{equation}
Therefore, $ \lambda =0 $ is triple eigenvalue of  \eqref{e4.1}--\eqref{e4.2}. 
Hence, all the eigenvalues of \eqref{e4.1}--\eqref{e4.2} are real and all, 
except one triple, are simple. Further, by \eqref{e4.6},  if $\lambda <0$, 
then $w( \lambda  )<0$.  Then, $\lambda =0$ is the first eigenvalue of 
\eqref{e4.1}--\eqref{e4.2} and ${{\lambda }_0}={{\lambda }_1}={{\lambda }_{2}}=0$.

From \eqref{e4.6}, we obtain
 $ \omega ( 0 )={\omega }'( 0 )={\omega }''( 0 )=0$, 
${\omega }'''( 0 )=-\frac{288a}{7!} $,  
$ {{\omega }^{IV}}( 0 )=\frac{4608a}{9!} $ and 
$ {{\omega }^{V}}( 0 )=-\frac{57600a}{11!} $. 
Eigenfunctions corresponding to $ {{\lambda }_n} \, ( n=0,3,4,\dots ) $ are 
$ {{y}_0}( x )=\alpha x+1 $ and 
$ {{y}_n}( x )=\cos \sqrt{{{\lambda }_n}}x
+\alpha \frac{\sin \sqrt{{{\lambda }_n}}x}{\sqrt{{{\lambda }_n}}}( n\ge 3 ) $. 
The first and the second order associated functions corresponding to
 $ {{y}_0} $ are 
$ {{y}_1}( x )=-\frac{\alpha {{x}^{3}}}{3!}-\frac{{{x}^2}}{2!}+\alpha Ax+A  $ 
and  $ {{y}_{2}}( x )
=\frac{\alpha {{x}^{5}}}{5!}+\frac{{{x}^{4}}}{4!}
-\frac{\alpha A{{x}^{3}}}{3!}-\frac{A{{x}^2}}{2!}+\alpha Bx+B $  
respectively, where $A$ and $B$ are arbitrary constants.

Note that $ 0<\beta <\pi $  for  problem \eqref{e4.1}--\eqref{e4.2}.
 From here, \eqref{e8'} and \eqref{e9'}, $\tilde{c}=-A  $ and
 $ \tilde{d}={{A}^2}-B $. According to above calculations, the 
condition \eqref{e12'} can be written as
\begin{equation}\label{e13'}
 B\ne {{A}^2}-\frac{A}{18}+\frac{13}{7128}.
\end{equation}
Therefore, if  condition \eqref{e13'} is satisfied, then
 the system $ {{y}_n}(x)$ $(n=1,2,\dots) $ is a basis 
in $ {{L}_{p}}(0,1) \, (1<p<\infty ) $.

Let $ {{F}_{s}}(x) ={{\text{P}}_{s}}( 2x-1 )( {{x}^2}-x ) $, where 
$ {{\text{P}}_{s}}( t )( s=0,1,2,\dots  ) $ are Legendre polynomials 
\cite[p.162]{Rainville}:
\[
{{\text{P}}_{s}}( t )=\frac{1}{{{2}^{s}}s!}
\frac{{{d}^{s}}}{{{x}^{s}}}[{{( {{x}^2}-1 )}^{s}}].
\]
Since $ {{F}_{s}}(0)={{F}_{s}}(1)=0 \, ,( {{F}_{s}},
\cos ( n-\frac{1}{2} )\pi x )=O( {{n}^{-2}} ) $. 
It means that this function can be expanded uniformly convergent 
Fourier series in the system 
$\{ \sqrt{2}\cos ( n-\frac{1}{2} )\pi x\}_{n=1}^{\infty } $ 
on the interval $[0,1]$.

Note that the equalities
\[
\int_{-1}^{1}{{{t}^{k}}{{\text{P}}_{s}}( t )dt=0,}
\int_{-1}^{1}{{{t}^{7}}{{\text{P}}_{7}}( t )dt=}\frac{{{2}^{8}}{{( 7! )}^2}}{15!}
\]
hold, where $k=0,1,\dots ,s-1$ \cite[p.174 and 175]{Rainville}.
 From here, since the functions $ ( {{t}^2}-1 ){{y}_{j}}( \frac{t+1}{2} )$
$( j=0,1,2 ) $ are polynomials of degree seven or less than seven, we obtain
\begin{align*}
\int_0^{1}{{F}_{s}}( x ){{y}_{j}}( x )dx
&=\int_0^{1}{{{P}_{s}}( 2x-1 )( {{x}^2}-x ){{y}_{j}}( x )dx}\\
&=\frac{1}{8}\int_{-1}^{1}{{{P}_{s}}( t )( {{t}^2}-1 ){{y}_{j}}( \frac{t+1}{2} )dt} \\
&=\begin{cases}
0 , & \text{if $s\ge 8$ and $ j=0,1,2$,} \\
\frac{\alpha {{( 7! )}^2}}{5!15!}, & \text{if $s=7$ and $j=2$}.
\end{cases}
\end{align*}
Hence, the condition 
\[
( {{F}_{s}},y_{2}^{\#\#} )=\begin{cases}
0 , & \text{if } s\ge 8 \\
\frac{\alpha {{( 7! )}^2}}{5!15!}, & \text{if } s=7 
\end{cases} 
\]
is satisfied. Consequently, from theorem \ref{thm1}, the function
 $ {{F}_{s}}( x ) $ can be 
expanded uniformly convergent Fourier series in the system
 $ {{y}_n}(x)$ $(n=1,2,\dots) $ on $[0,1]$ for 
$ s\ge 8 $, on $ [0,b] $ $  (0<b<1)  $ for $ s=7 $.
\end{example}

\begin{thebibliography}{99}

\bibitem{Aliyev} Y. N. Aliyev;
\emph{On the basis properties of Sturm-Liouville problems with decreasing
affine boundary conditions}, Proc. IMM of NAS, \textbf{24}(2006), 35--52.

\bibitem{Bary} N. K. Bary;
\emph{A Treatise on Trigonometric Series Vol I}, Pergamon Press, (1964).

\bibitem{Binding} P. A. Binding, P. J. Browne, W. J. Code, B. A. Watson;
\emph{Transformation of Sturm- Liouville problems with decreasing affine
boundary conditions}, Proc. Edinb. Math. Soc., \textbf{47}(2004), 533--552.

\bibitem{Gulyaev1} D. A. Gulyaev;
\emph{On the uniform convergence of spectral expansions for a spectral problem
with boundary conditions of the third kind one of which contains the spectral
parameter}, Differential Equations, \textbf{47}(10) (2011), 1503--1507.

\bibitem{Gulyaev2} D. A. Gulyaev;
\emph{On the uniform convergence in $W_{2}^{m}$ of spectral expansions
for a spectral problem with boundary conditions of the third kind one
of which contains the spectral parameter}, Differential Equations,
\textbf{48}(10) (2012), 1450--1453.

\bibitem{Ince} E. L. Ince;
\emph{Ordinary Differential Equations Vol I}, Dover, New York, (1956).

\bibitem{Kapustin1} N. Yu. Kapustin;
\emph{On the uniform convergence of  the Fourier Series for a spectral problem 
with squared spectral parameter in a boundary condition}, 
Differential Equations, \textbf{46}(10) (2010) 1504--1507.

\bibitem{Kapustin2} N. Yu. Kapustin, E. I. Moiseev;
\emph{A remark on the convergence problem for spectral expansions corresponding 
to a classical problem with spectral parameter in the boundary condition}, 
Differential Equations, \textbf{37}(12) (2001), 1677--1683.

\bibitem{Kapustin3} N. Yu. Kapustin, E. I. Moiseev;
\emph{Convergence of spectral expansions for functions of the Hölder 
Class for two problems with spectral parameter in the boundary condition}, 
Differential Equations, \textbf{36}(8) (2000), 1182--1188.

\bibitem{Kapustin4} N. Yu. Kapustin;
\emph{On the uniform convergence in ${{C}^{1}}$ of Fourier Series for a spectral 
problem with squared spectral parameter in a boundary condition}, 
Differential Equations, \textbf{47}(10) (2011), 1394--1399.

\bibitem{Kapustin5} N. Yu. Kapustin;
\emph{On the spectral problem arising in the solution of a mixed problem for 
the heat equation with a mixed derivative in the boundary condition},
 Differential Equations, \textbf{45}(5) (2012), 694--699.

\bibitem{Kerimov1} N. B. Kerimov, E. A. Maris;
\emph{On the basis properties  and convergence of expansions in terms 
of eigenfunctions for a spectral problem with a spectral parameter 
in the boundary condition}, Proc. IMM of NAS, Sp. Issue (2014), 1245--1258.

\bibitem{Kerimov2} N. B. Kerimov, E. A. Maris;
\emph{On the uniform convergence of the Fourier series for one spectral 
problem with a spectral parameter in a boundary condition}, 
Math. Meth. Appl. Sci., DOI:10.1002/mma.3640 (2015).

\bibitem{Marchenkov} D. B. Marchenkov;
\emph{On the convergence of spectral expansions of functions for problems
with a spectral parameter in a boundary condition},
Differential Equations, \textbf{41}(10) (2005), 1419--1422.

\bibitem{Naimark} M. A. Naimark;
\emph{Linear Differential Operators Vol I}, Nauka, Moscow, (1969).

\bibitem{Rainville} E. D. Rainville;
\emph{Special Functions}, The Macmillan Company, New York, (1965).

\end{thebibliography}

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