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: A stabilization of the two-step lifting construction of Sweldens for wavelet bases on irregular meshes on the interval yields biorthogonal bases in which the wavelets on both the primal and the dual side have a chosen number of vanishing moments and have local support.
Abstract: We propose a stabilization of the two-step lifting construction of Sweldens for wavelet bases on irregular meshes on the interval. The method yields biorthogonal bases in which the wavelets on both the primal and the dual side have a chosen number of vanishing moments and have local support. We combine it with a constrained local semiorthogonalization to obtain a stabilized variant.
Numerical results show that the wavelet bases are well-conditioned, having approximately the same condition numbers as wavelet bases obtained by global semiorthogonalization, while the average support is not much larger than in the unstabilized construction.
20 citations
TL;DR: In this paper, a two-dimensional exactly solvable non-Hermitian PT-non-symmetric quantum model with real spectrum, which is not amenable to separation of variables, by supersymmetrical methods is studied.
20 citations
TL;DR: In this article, the authors investigated the topological phase diagrams and the zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger model, based on some analytical results.
Abstract: Intriguing issues in one-dimensional non-reciprocal topological systems include the breakdown of usual bulk-edge correspondence and the occurrence of half-integer topological invariants In order to understand these unusual topological properties, we investigate the topological phase diagrams and the zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger model, based on some analytical results Meanwhile, we provide a concise geometrical interpretation of the bulk topological invariants in terms of two independent winding numbers and also give an alternative interpretation related to the linking properties of curves in three-dimensional space For the system under the open boundary condition, we construct analytically the wavefunctions of zero-mode edge states by properly considering a hidden symmetry of the system and the normalization condition with the use of biorthogonal eigenvectors Our analytical results directly give the phase boundary for the existence of zero-mode edge states and unveil clearly the evolution behavior of edge states In comparison with results via exact diagonalization of finite-size systems, we find our analytical results agree with the numerical results very well
20 citations
TL;DR: In this article, a backward biorthogonalization approach is proposed to generate orthogonal projections onto a reduced subspace, which is relevant to problems amenable to be represented by a general linear model.
Abstract: A backward biorthogonalization approach is proposed, which
modifies biorthogonal functions so as to generate orthogonal
projections onto a reduced subspace. The technique is relevant to
problems amenable to be represented by a general linear model. In
particular, problems of data compression, noise reduction, and
sparse representations may be tackled by the proposed approach.
20 citations
TL;DR: In this article, an asymptotic expansion including error bounds is given for polynomials {Pn, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iα)α−β.
Abstract: An asymptotic expansion including error bounds is given for polynomials {Pn, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iθ)α−β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2F1(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.
20 citations