Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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TL;DR: In this article, the authors introduce a general theory of regular biorthogonal sequences and its physical operators and show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, etc.
Abstract: In this paper, we introduce a general theory of regular biorthogonal sequences and its physical operators. Biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H are said to be regular if Span {ϕn} and Span {ψn} are dense in H. The first purpose is to show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, …, and such an ONB f is unique. The second purpose is to define and study the lowering operators Af and Bf†, the raising operators Bf and Af†, and the number operators Nf and Nf† determined by the non-singular positive self-adjoint operator Tf. These operators connect with quasi-Hermitian quantum mechanics and its relatives. This paper clarifies and simplifies the mathematical structure of this framework and minimizes the required assumptions.
18 citations
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TL;DR: The contents of the paper as discussed by the authors are now covered in two separate papers arXiv:0904.2188v1.4082v2.2602 and ARXiv :0904 v1.262v1, respectively.
Abstract: The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
18 citations
TL;DR: A multidimensional nonuniform Oversampling formula for bandlimited functions with a fairly general frequency domain is derived and a computationally manageable simplification of the main oversampling theorem is given.
18 citations
18 citations
TL;DR: In this paper, an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution, is obtained, using properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.
Abstract: We obtain an approximation of the solution of the nonlinear Volterra integral equation
of the second kind, by means of a new method for its numerical resolution. The main tools used to establish
it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.
18 citations