Basis Property of Root Functions System for a Problem with Spectral Parameter in the Boundary Condition
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Basis Property of Root Functions System for a Problem with Spectral Parameter in the Boundary Condition N. Kapustin1* and A. Kholomeeva1, 2** (Submitted by A. M. Elizarov) 1
2
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991 Russia
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, 119333 Russia Received March 30, 2020; revised April 5, 2020; accepted April 20, 2020
Abstract—In this paper we study a spectral problem with spectral parameter in the boundary condition. This problem arises when solving boundary value problems for mixed type equations using the spectral method. We analyze the system of root functions of this problem and proof two theorems about basis properties of this system. The biortogonal system is constructed and uniform convergence of spectral expansions is studied. DOI: 10.1134/S1995080220100108 Keywords and phrases: mixed type equation, elliptic-hyperbolic equation, spectral problem.
1. INTRODUCTION Consider a spectral problem X (x) + λX(x) = 0, X (0) = 0,
x ∈ (0, 1),
(1)
X(1) = −dλX (1),
(2)
where d is a nonzero complex constant coefficient. The solution of problem (1), (2) is the system of eigenfunctions √ (3) Xn (x) = 2 cos( λn x), corresponding to eigenvalues λn , equation
n = 1, 2, 3, . . . which one may be found by solving a characteristic cot
√
√ λ = d( λ)3 .
(4)
√ We assume that −π/2 < arg λn π/2 and the eigenvalues are ordered in ascending order of their absolute values. * **
E-mail: [email protected] E-mail: [email protected]
2018
BASIS PROPERTY OF ROOT FUNCTIONS SYSTEM
2019
2. MAIN RESULTS Theorem 1. If d ∈ / where {z} is the set of (complex) roots of equation 3 sin z cos z = 0, 1+ z then the system {Xn (x)}, n = 1, 2, . . . , m − 1, m + 1, . . . of eigenfunctions of the spectral problem (1), (2) with one eigenfunction omitted is a basis in the functional space Lp (0, 1), p > 1 (the Riesz basis if p = 2). The biorthogonal system {Ψn (x)} for this system is constructed using the functions Ψn (x), where √ √ √ λn sin λn √ 1 √ √ Ψn (x) = 2 cos( λn x) − √ 2 cos( λm x) , 1 + 3dλn sin2 λn λm sin λm {cot z/z 3 },
n = 1, 2, . . . , m − 1, m + 1, . . . . Theorem 2. If d = cot z/z 3 , where the complex number z is an arbitrary root of equation 3 sin z cos z = 0, then system (3) of eigenfunctions of the spectral problem (1), (2) is a 1+ z basis in the functional space Lp (0, 1), p > 1 (the Riesz basis if p = 2). The biorthogonal system {Ψn (x)} for this system is constructed using the functions Ψn (x), where λl = z 2 : √ √ 3 Ψl (x) = 2 cos( λl x) − d( λl )3 x 2 sin( λl x) , 3 + 2dλl √ √ √ 1 λn sin λn √ √ √ Ψn (x) = 2 cos( λn x) − √ 2 cos( λl x) , λl sin λl 1 + 3dλn sin2 λn n = 1, 2, . . . , l − 1, l + 1, . . . . Proof of the theorems is as follows. 3 sin z cos z = 0. z Then all roots of the characteristic equation (4) are simple and we have the asymptotical expantion 1 1 λn = π(n − 2) + +O , n → ∞. (5) d(π(n − 2))3 n7 Proof. Let d = cot z/z 3
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