Riesz bases of solutions of Sturm-Liouville equations

He, Xionghui; Volkmer, Hans

doi:10.1007/bf02511815

Cited by 47 publications

(43 citation statements)

References 5 publications

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“…In this appendix we gather some well known facts about Riesz bases of sines and cosines (see, e.g., [12,15] and references therein for a detailed exposition of this topic).…”

Section: Appendix a Riesz Basesmentioning

confidence: 99%

Inverse spectral problems for Sturm Liouville operators with singular potentials

2003

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Abstract. The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space W −1 2 (0, 1). The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

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“…In this appendix we gather some well known facts about Riesz bases of sines and cosines (see, e.g., [12,15] and references therein for a detailed exposition of this topic).…”

Section: Appendix a Riesz Basesmentioning

confidence: 99%

Inverse spectral problems for Sturm Liouville operators with singular potentials

2003

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“…Recently, Theorem 3.3 has been used to construct Riesz bases for weighted L 2 -spaces consisting of solutions to certain Sturm-Liouville problems; cf. [35].…”

Section: Frames and Riesz Bases 281mentioning

confidence: 99%

Frames, Riesz bases, and discrete Gabor/wavelet expansions

Christensen

2001

Bull. Amer. Math. Soc.

106

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Abstract. This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.

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“…Various methods were developed for establishing these properties. For more information, we refer the reader to [1,2,3,4,5,6,7,8,9].…”

Section: Introductionmentioning

confidence: 99%

“…This approach is well studied (see, e.g., the articles [6,7,8,9,14,15,16,17,18,19,28,29,30,31] and the monographs [13,20,21,22,23,24]). On the other hand, it is not difficult to see that the principal part { 1 ; } ∈N is not a standard (in other words, classical) system.…”

Section: Introductionmentioning

confidence: 99%

On basicity of eigenfunctions of second order discontinuous differential operator

Bilalov¹,

Gasymov²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basis for a space by the bases in subspaces. We also consider the cases when the bases for subspaces are isomorphic and the corresponding isomorphisms are not needed. The completeness, minimality and uniform minimality of the corresponding systems are studied. This approach has extensive applications in the spectral theory of discontinuous differential operators.

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Riesz bases of solutions of Sturm-Liouville equations

Cited by 47 publications

References 5 publications

Inverse spectral problems for Sturm Liouville operators with singular potentials

Inverse spectral problems for Sturm Liouville operators with singular potentials

Frames, Riesz bases, and discrete Gabor/wavelet expansions

On basicity of eigenfunctions of second order discontinuous differential operator

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