THE UNIFORM CONVERGENCE OF FOURIER SERIES EXPANSIONS OF A STURM-LIOUVILLE PROBLEM WITH BOUNDARY CONDITION WHICH CONTAINS THE EIGENPARAMETER

This paper is devoted to investigating the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of a Sturm-Liouville problem with eigenparameter in one of the boundary conditions on a closed interval. Such problems are quite common in mathematical physics problems. 1. Introduction In many mathematical physics problems, partial di¤erential equations are encountered. The most common of these are di¤erential equations of the second order. Partial di¤erential equations of the second order of the hyperbolic type occur most frequently in physical problems with vibration processes such as the transverse or longitudinal vibrations of a string, a membrane and rod, the vibrational energy of a string, hydrodynamics and acoustics, etc. [16]. On the other hand, di¤erential equations often give an innite number of solutions. However, for the mathematical description of a physical process, su¢ cient conditions must be determined to uniquely determine the process. That is, solutions should be limited under certain conditions. Whence, it is necessary to supplement that equation with certain additional equations such as initial or boundary conditions in order to specify the process uniquely when the physical problem is led to a partial di¤erential equation. In this way, a physical model corresponds to a boundary value problem. 2020 Mathematics Subject Classication. 34L05, 34L20, 34L10, 34B05. Keywords and phrases. Eigenvalues, eigenfunctions, uniform convergence of spectral expansion. srtcgoktas@gmail.com-Corresponding author; e.ali.maris@gmail.com 0000-0001-7842-6309; 0000-0001-7620-8754. c 2021 Ankara University Communications Facu lty of Sciences University of Ankara-Series A1 Mathematics and Statistics 205 206 S. GOKTAS, E. A. MARIS One of the widest methods used to solve partial di¤erential equations is Fouriers method, which is also called the method of separation of variables. Finding values , called eigenvalues for which non-trivial solutions called eigenfunctions, of the eigenvalue problem that arises with the application of this method is required, and also obtaining these solutions of it. The problem formulated in this way is also called Sturm-Liouvilles problem. It is known from Fourier series theory that the arbitrary partially continuous and partially di¤erentiable a function f(x) given in the interval [0; l] can be expanded as Fourier series in the sine and cosine functions or eigenfunctions of an eigenvalue problem. Therefore, the solutions of an eigenvalue problem are in the form of an innite series. If this series divergent or the function dened by this series is not di¤erentiable, then it cant naturally represent a solution of a di¤erential equation. Consequently, investigating the convergence conditions of these series and nding the representation of the solutions are of great importance. In this study, our aim is to investigate the uniform convergence conditions of spectral expansions of continuous functions in terms of eigenfunctions of the SturmLiouville problem: u00 + q(x)u = u; 0 < x < 1; (1) u(0) = 0; u0(0) = (au(1) + bu0(1)) (2) where is an eigenparameter, q(x) 2 L1(0; 1) is a complex-valued function, a and b are arbitrary complex numbers which satisfy the condition jaj+ jbj 6 = 0. In the spectral theory of di¤erential operators, there are many articles containing Sturm-Liouville equations with boundary conditions linearly or polynomially dependent on the spectral parameter [6, 7, 3, 4, 5, 1, 2, 9, 10, 11, 12, 17]. The convergence conditions of Fourier series expansions of functions in some functional class of Sturm-Liouville operators are investigated in [6,7,3,4,5,1,2,9,10]. For example, the convergence conditions of series expansions of the following problems are studied in [1], [6], [7], respectively: spectral problems that appear modeling heat transfer in a homogeneous rod with a linear relation between the heat ux and temperature at one endpoint and with a lumped heat capacity at the other endpoint, spectral problems that appear in a model of a transrelaxation heat process and in the mathematical description of vibrations of a loaded string and, spectral problems that appear 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, and the current in a cable grounded at one end through a concentrated capacitance or inductance. Moreover, the spectral problem u00 + q(x)u = u; 0 < x < 1; u(0) = 0; u0(0) a u(1) = 0 was considered in [14] for q(x) = 0 and a > 0; in [8] for q(x) 6= 0 and a 6= 0. The conditions of the uniform convergence of spectral expansions of continuous functions in the system of eigenfunctions in these studies were established. Note THE UNIFORM CONVERGENCE CONDITION OF EXPANSIONS OF A PROBLEM 207 that, the problems considered in these studies are special cases of the problem (1)-(2). Therefore, we will assume that b 6= 0 from now on. 2. Preliminaries In this section, some properties will be given in order to reach the desired results of the problem (1)-(2). Denote the solution of the equation (1), satisfying the initial conditions (0) = 0; 0(0) = 1 (3) by (x) = (x; ). Lemma 1 ( [13]). Let = . Then, (x; ) = sin x + 1 x Z 0 sinf (x )g q( ) ( ; )d : (4) Lemma 2 ( [13]). Let = +it. Then, there exists 0 > 0 such that, for j j > 0, the estimate (x; ) = sin x +O ejtjx j j ! (5) is valid, where the function O ejtjxj j 2 is the entire function of for any xed x in [0; 1]. Moreover, the estimate (5) is uniform with respect to x 2 [0; 1]. Theorem 3 ( [15]). All the eigenvalues of the problem (1)-(2) are simple. Moreover, they have form innite sequence n (n = 0; 1; 2; :::) which has no nite limit points. And, the following asymptotic estimate are valid for su¢ ciently large n: n = n 1 2 2 +O (1) ; (6) n(x) = (x; n) = sin n 12 x n +O 1 n2 : (7) Theorem 4 ( [15]). If (b n) 6= 1 (n = 0; 1; 2; :::) and r is an arbitrary xed non-negative integer, then the system f n(x)g (n = 0; 1; :::;n 6= r) is minimal in L2(0; 1). Corollary 5 ( [15]). If (b n) 6= 1 (n = 0; 1; 2; :::) and r is an arbitrary xed non-negative integer, then the system f'n(x)g which are biorthogonally conjugates to the system f n(x)g is given by the following formula: 'n(x) = an n(1 x) + b n n(x) n(1) r(1 x) + b r r(x) r (1) ; (8) where an = 1 n + n a (1; n) @ +b @ 0 (1; n) @ 1 : 208 S. GOKTAS, E. A. MARIS Theorem 6 ( [15]). If (b n) 6= 1 (n = 0; 1; 2; :::) and r is an arbitrary xed nonnegative integer, then the system f n(x)g (n = 0; 1; :::;n 6= r) is a basis in Lp(0; 1) (1 < p <1), and this basis is unconditional for p = 2. 3. Main Results 3.1. The Sharpened Asymptotics for Eigenparameters. The asymptotic estimates of eigenvalues (6) and eigenfunctions (7) of the problem (1)-(2) should be sharpened to give theorem about the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of this problem. In this subsection, the expression and proof of the relevant theorem will be given to nalize these asymptotic estimates. Theorem 7. Let n = n (Re n 0). The asymptotic estimates n = n 1 2 + c0 n +O n n ; (9) n(x) = sin n 12 x n 12 + (x) (n ) 2 cos n 1 2 x + n(x) 2(n ) 2 cos n 1 2 x+ n(x) 2(n ) 2 sin n 1 2 x


Introduction
In many mathematical physics problems, partial di¤erential equations are encountered. The most common of these are di¤erential equations of the second order. Partial di¤erential equations of the second order of the hyperbolic type occur most frequently in physical problems with vibration processes such as the transverse or longitudinal vibrations of a string, a membrane and rod, the vibrational energy of a string, hydrodynamics and acoustics, etc. [16]. On the other hand, di¤erential equations often give an in…nite number of solutions. However, for the mathematical description of a physical process, su¢ cient conditions must be determined to uniquely determine the process. That is, solutions should be limited under certain conditions. Whence, it is necessary to supplement that equation with certain additional equations such as initial or boundary conditions in order to specify the process uniquely when the physical problem is led to a partial di¤erential equation. In this way, a physical model corresponds to a boundary value problem. 206

S. GOKTAS, E. A. M ARIS
One of the widest methods used to solve partial di¤erential equations is Fourier's method, which is also called the method of separation of variables. Finding values , called eigenvalues for which non-trivial solutions called eigenfunctions, of the eigenvalue problem that arises with the application of this method is required, and also obtaining these solutions of it. The problem formulated in this way is also called Sturm-Liouville's problem. It is known from Fourier series theory that the arbitrary partially continuous and partially di¤erentiable a function f (x) given in the interval [0; l] can be expanded as Fourier series in the sine and cosine functions or eigenfunctions of an eigenvalue problem. Therefore, the solutions of an eigenvalue problem are in the form of an in…nite series. If this series divergent or the function de…ned by this series is not di¤erentiable, then it can't naturally represent a solution of a di¤erential equation. Consequently, investigating the convergence conditions of these series and …nding the representation of the solutions are of great importance.
In this study, our aim is to investigate the uniform convergence conditions of spectral expansions of continuous functions in terms of eigenfunctions of the Sturm-Liouville problem: where is an eigenparameter, q(x) 2 L 1 (0; 1) is a complex-valued function, a and b are arbitrary complex numbers which satisfy the condition jaj + jbj 6 = 0. In the spectral theory of di¤erential operators, there are many articles containing Sturm-Liouville equations with boundary conditions linearly or polynomially dependent on the spectral parameter [6,7,3,4,5,1,2,9,10,11,12,17]. The convergence conditions of Fourier series expansions of functions in some functional class of Sturm-Liouville operators are investigated in [6,7,3,4,5,1,2,9,10]. For example, the convergence conditions of series expansions of the following problems are studied in [1], [6], [7], respectively: spectral problems that appear modeling heat transfer in a homogeneous rod with a linear relation between the heat ‡ux and temperature at one endpoint and with a lumped heat capacity at the other endpoint, spectral problems that appear in a model of a transrelaxation heat process and in the mathematical description of vibrations of a loaded string and, spectral problems that appear 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, and the current in a cable grounded at one end through a concentrated capacitance or inductance.
Moreover, the spectral problem was considered in [14] for q(x) = 0 and a > 0; in [8] for q(x) 6 = 0 and a 6 = 0. The conditions of the uniform convergence of spectral expansions of continuous functions in the system of eigenfunctions in these studies were established. Note that, the problems considered in these studies are special cases of the problem (1)- (2). Therefore, we will assume that b 6 = 0 from now on.

Preliminaries
In this section, some properties will be given in order to reach the desired results of the problem (1)- (2).
Denote the solution of the equation (1), satisfying the initial conditions by (x) = (x; ).
is valid, where the function O e jtjx j j 2 is the entire function of for any …xed

Main Results
3.1. The Sharpened Asymptotics for Eigenparameters. The asymptotic estimates of eigenvalues (6) and eigenfunctions (7) of the problem (1)-(2) should be sharpened to give theorem about the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of this problem. In this subsection, the expression and proof of the relevant theorem will be given to …nalize these asymptotic estimates.
Theorem 7. Let n = 2 n (Re n 0). The asymptotic estimates because n (x) is the solution of the equation (1) which satis…es the conditions (3). Firstly, we now need to do the following calculations: Let n = 2 n . Then, the estimate is satis…ed from (6), where n = O n 1 . The main purpose of this proof is to write a sharper expression of the estimate n .

THE UNIFORM CONVERGENCE CONDITION OF EXPANSIONS OF A PROBLEM 209
On the other hand, it can be easily seen that the estimate is valid from (5). By using (12) and (13) in the estimate obtained as a result of some basic calculations in (4) and in the di¤erential of this estimate with respect to x, we obtain the estimates from (12), we can respectively rewrite for x = 1 the estimates (14) and (15) as follows:  (16) and (17) in the equation (11), we obtain the equation a ( 1) The last equation implies that the estimate holds. The above formal reasoning shows that the estimate (9) should hold. Furthermore, by using (9) and (18), we have the term where Consequently, the asymptotic estimate (10) follows from (14) and (19). The Theorem 7 is proved with this. Proof. Let us consider the Fourier series expansion of a continuous function f (x) in the system f n (x)g on [0; 1]: where the system ' n (x) (n = 0; 1; : : : ; n 6 = r) is de…ned by (8). In addition, the estimate a n in this system can also be written as the form a n = 2( 1)

THE UNIFORM CONVERGENCE CONDITION OF EXPANSIONS OF A PROBLEM 211
Now, we can analyze the series (20) in the form to investigate the uniform convergence of it on [0; 1], where Firstly, the series (23) is uniformly convergent on the interval [0; 1] by virtue of the estimate a n n (1) which follows from (16)  On the other hand, since ban n n (1) = 2 n + O (n) by using (26), we have the identity p 2 n n (x) = p 2sin n where (x), n (x) and n (x) are de…ned on Theorem 7 and n = 2 n . From the last identity, we obtain ba n n n (1) So, the identity holds.
The …rst sum in the last identity is the Fourier series expansion of f (x) in the system p 2 sin n 1 2 x 1 n=1 . Consequently, the uniform convergence of this sum as m ! 1 on the interval [0; 1] is stipulated in the assumptions of the Theorem 8.
On the other hand, we now will examine the uniformly convergence of the other sums as m ! 1: From (27), we have jG n (x)j c 1 n f; sin n 1 2 x + f (x) (x); cos n 1 2 x where c 3 and c 4 are real constants. Namely, the series (28) as m ! 1 is absolutely and uniformly convergent on the interval [0; 1]. Thirdly, the series (25) is uniformly convergent on the interval [0; 1], by virtue of the estimate 1 X n=r+1 a n n (x) = 1 X n=r+1 O n 2 which follows from (14) and (21).
As a result of all these calculations, the series (22), so the series (20), is uniformly convergent on the interval [0; 1]. The proof of the Theorem 8 is completed.
Corollary 9. If f (x) is a function that provides the hypothesis of the Theorem 8, then this function can also be expanded as Fourier series in the biorthogonal system ' n (x) (n = 0; 1; : : : ; n 6 = r) is de…ned by (8) a n (f; n (x)): It can be seen that the convergence of the series (29)-(31) are calculated by a similar method using in examining the convergence of the series (23)-(25).

Authors Contribution Statement
The authors contributed equally. All authors read and approved the …nal copy of the manuscript.

Declaration of Competing Interests
The authors declare that they have no competing interest.