Riesz bases generated by the spectra of Sturm-Liouville problems (original) (raw)
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Mathematical Notes, 2009
The paper deals with the Sturm-Liouville operator Ly = −y ′′ + q(x)y, x ∈ [0, 1], generated in the space L 2 = L 2 [0, 1] by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to Sobolev space W p 1 [0, 1] with some integer p ≥ 0 and satisfy the conditions q (k) (0) = q (k) (1) = 0 for 0 ≤ k ≤ s − 1, where s≤ p. Let the functions Q and S be defined by the equalities Q(x) = x 0 q(t) dt, S(x) = Q 2 (x) and let q n , Q n , S n be the Fourier coefficients of q, Q, S with respect to the trigonometric system {e 2πinx } ∞ −∞. Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2. Then the system of eigen and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2 [0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q 2n − S 2n + Q 0 Q 2n ≍ q −2n − S −2n + 2Q 0 Q −2n , n > 1, holds.
On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator
Israel Journal of Mathematics, 2005
In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we fred the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis in L2(0, 1). Let Lt(q) be the operator generated in L2[0, 1] by the expression (1)-y" + q(x)y, and the boundary conditions (2) y(1) = eUy(O), y' (1) = eUy '(0), where q(x) is a complex-valued summable function. In this article we obtain asymptotic formulas of order O(n-l) (l = 1, 2,...,) for the n-th eigenvalue and corresponding eigenfunction of the operator Lt (q)
Sturm-Liouville systems are Riesz-spectral systems
International Journal of Applied Mathematics and Computer Science, 2003
The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L /sup 2/(a,b) and the infinitesimal generator of a C/sub 0/-semigroup of bounded linear operators.
The Arabian Journal for Science and Engineering, Volume 33, Number 1A, January, 2008
We study basisness of root functions of Sturm–Liouville problems with a boundary condition depending quadratically on the spectral parameter. We determine the explicit form of the biorthogonal system. Using this we prove that the system of root functions, with arbitrary two functions removed, form a minimal system in L2, except some cases where this system is neither complete nor minimal. For the basisness in we prove that the part of the root space is quadratically close to systems of sines and cosines. We also consider these basis properties in the context of general L_p
On the stability of basisness in L_p(1 p +\infty) of cosines and sines
Turkish Journal of Mathematics, 2011
We study the basis properties in Lp(0, π) (1 < p < ∞) of the solution system of Sturm-Liouville equations with different types of initial conditions. We first establish some results on the stability of the basis property of cosines and sines in Lp(0, π) (1 < p < ∞) and then show that the solution system above forms a basis in Lp(0, π) if and only if certain cosine system (or sine system, depending on type of initial conditions) forms a basis in Lp(0, π).
The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function
Integral Equations and Operator Theory, 2008
We consider a regular indefinite Sturm-Liouville eigenvalue problem −f ′′ + qf = λrf on [a, b] subject to general self-adjoint boundary conditions and with a weight function r which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions r for which the Riesz basis property can be completely characterized in terms of the local behavior of r in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of r in a neighborhood of the turning points, local conditions on r near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.
On the stability of basisness in L p ( 1 < p < + ∞ ) of cosines and sines
2011
We study the basis properties in Lp(0, π) (1 < p < ∞) of the solution system of Sturm–Liouville equations with different types of initial conditions. We first establish some results on the stability of the basis property of cosines and sines in Lp(0, π) (1 < p < ∞) and then show that the solution system above forms a basis in Lp(0, π) if and only if certain cosine system (or sine system, depending on type of initial conditions) forms a basis in Lp(0, π) .