Biorthogonal Family Research Articles

Uvarov perturbations for matrix orthogonal polynomials

Additive perturbations, specifically matrix Uvarov transformations for matrix orthogonal polynomials, are under consideration. Christoffel–Uvarov formulas are deduced for the perturbed biorthogonal families, along with their matrix norms. These formulations are expressed in terms of the spectral jets of the Christoffel–Darboux kernels.

A search for suitable mother wavelet in discrete wavelet transform based analysis of acoustic emission partial discharge signals

Signal processing helps monitor the condition of power equipment. Partial discharge (PD) signals used in condition-based maintenance give crucial information in the diagnosis of degradation of insulation. The acoustic emission technique (AET) is one of the most widely used techniques in PD signal analysis due to its inherent advantages. Analyzing acoustic emission partial discharge (AEPD) signals in the wavelet-domain provides critical insights into the location and type of the sources of PD. Selection of the most suitable mother wavelet in applying discrete wavelet transform (DWT) on AEPD signals is important as it will directly impact the outcome. For this selection, 36 wavelets belonging to the Daubechies, Symlets, Coiflets, and Bi-orthogonal families are investigated. For this purpose, five experimentally collected AEPD test signals are used. The selection is based on the ?accuracy of wavelet decomposition results? in this work, probably for the first time. One mother wavelet from each family is individually shortlisted for all three performances, namely (a) reconstruction, (b) denoising, and (c) compression, by computing and comparing their commonly used metrics. Further, based on percentage energy criteria, the most suitable mother wavelets are identified as coif3, coif4, and coif5, respectively, for the three performances.

The performance of DST-Wavelet feature extraction for guitar chord recognition

Small systems can be designed to be more energy-efficient compared to larger systems. On small systems, the need for data processing with small data sizes becomes a necessity. In the context of small systems for guitar chord recognition, there are indications that further efforts can be made to reduce the size of feature extraction data. This paper introduces DST (Discrete Sine Transform)-Wavelet feature extraction to achieve this reduction. Basically, this work evaluated the frame blocking length, the number of DST cutting factors, and the type of wavelet filters (Daubechies and biorthogonal families) to obtain the optimal number of feature extraction data. Based on the evaluation, the optimal result obtained was a number of four feature extraction data. This optimal result was obtained by using a frame blocking length of 512 points, a DST cutting factor of 0.5, and a biorthogonal 3.3 wavelet filter. Testing with 140 test chords using these four feature extraction data could give an accuracy of up to 92.86%.

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

BIORTHOGONAL NONSTATIONARY AND DAUBECHIES WAVELETS: A COMPARATIVE STUDY

In this article, we present a constructive technic of a biorthogonal nonstationary wavelets (Bior-NSW) family which are proposed by [12], and we perform a comparative study between the classical Daubechies biorthogonal wavelets family and the constructed nonstationary wavelets, which prooved the dominance of the nonstationary wavelets in approximation of different kinds signals.

Boundary controllability for a coupled system of degenerate/singular parabolic equations

<p style='text-indent:20px;'>In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on a detailed spectral analysis and the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.</p>

Exact controllability to eigensolutions of the bilinear heat equation on compact networks

<p style='text-indent:20px;'>Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.</p><p style='text-indent:20px;'>By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, <i>Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control</i>, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.</p>

Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

In this article, we compute the exact value of the minimal null control time for the Grushin equation controlled on a strip. This result is already known from a recent work by Beauchard et al. (Ann. Inst. Fourier 70(1), 2020); however, we propose a different approach based on the moments method. Our approach requires a careful spectral analysis of a truncated harmonic oscillator. We have also extended known results on biorthogonal families to real exponentials in the absence of a gap condition in order to obtain uniform estimates with respect to the asymptotic behavior of the associated counting function. As a by-product of our approach, we also characterize the minimal null control time for systems of coupled Grushin equations with a distributed control on a strip or with a boundary control.

Coupled Susy, pseudo-bosons and a deformed Lie algebra

In a recent paper a pair of operators a and b satisfying the equations a † a = bb † + γ and aa † = b † b + δ , has been considered, and their nature of ladder operators has been deduced and analyzed. Here, motivated by the spreading interest in non self-adjoint operators in quantum mechanics, we extend this situation to a set of four operators, c, d, r and s, satisfying dc = rs + γ and cd = sr + δ , and we show that they are also ladder operators. We show their connection with biorthogonal families of vectors and with the so-called -pseudo bosons. Some examples are discussed.

The cost of controlling strongly degenerate parabolic equations

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

Tridiagonality, supersymmetry and non self-adjoint Hamiltonians

In this paper we consider some aspects of tridiagonal, non self-adjoint, Hamiltonians and of their supersymmetric counterparts. In particular, the problem of factorization is discussed, and it is shown how the analysis of the eigenstates of these Hamiltonians produce interesting recursion formulas giving rise to biorthogonal families of vectors. Some examples are proposed, and a connection with bi-squeezed states is analyzed.

A new wavelet family for speckle noise reduction in medical ultrasound images

Ultrasonography is highly used as an image tool for medical diagnosis due to its non-invasive nature, low cost, adaptability and the possibility of real-time visualization. However, the noise present in these images makes medical analysis difficult. Therefore, noise reduction is essential for a more precise and accurate diagnosis. This work proposes a new orthonormal wavelet family designed to reduce noise in ultrasonic medical images, focusing on speckle noise, which is of particular concern in this scenario. A set of 110 real images is used to design the filters and to validate the results. The proposed wavelet family leads to better results than the ones obtained with the wavelets from the Daubechies, Symmlet, Coiflet and Biorthogonal families, in terms of both noise reduction and edge preservation. The results also compares favorably with those from the state-of-the-art BM3D denoising technique, when the noise intensity is not very high.

A Comparative Analysis of Wavelet Families for the Classification of Finger Motions

Wavelet transform (WT) has been widely used in biomedical, rehabilitation and engineering applications. Due to the natural characteristic of WT, its performance is mostly depending on the selection of mother wavelet function. A proper mother wavelet ensures the optimum performance; however, the selection of mother wavelet is mostly empirical and varies according to dataset. Hence, this paper aims to investigate the best mother wavelet of discrete wavelet transform (DWT) and wavelet packet transform (WPT) in the classification of different finger motions. In this study, twelve mother wavelets are evaluated for both DWT and WPT. The electromyography (EMG) data of 12 finger motions are acquired from online database. Four useful features are extracted from each recorded EMG signal via DWT and WPT transformation. Afterward, support vector machine (SVM) and linear discriminate analysis (LDA) are employed for performance evaluation. Our experimental results demonstrate Bior3.3 to be the most suitable mother wavelet in DWT. On the other hand, WPT with Bior2.2 overtakes other mother wavelets in the classification of finger motions. The results obtained suggest that Biorthogonal families are more suitable for accurate EMG signals classification.

Biorthogonal vectors, sesquilinear forms, and some physical operators

Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular, we discuss what happens when they forms two D-quasi-bases.

Galaxy halo expansions: a new biorthogonal family of potential-density pairs

Efficient expansions of the gravitational field of (dark) haloes have two main uses in the modelling of galaxies: first, they provide a compact representation of numerically-constructed (or real) cosmological haloes, incorporating the effects of triaxiality, lopsidedness or other distortion. Secondly, they provide the basis functions for self-consistent field expansion algorithms used in the evolution of $N$-body systems. We present a new family of biorthogonal potential-density pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest-order density basis functions are double-power-law profiles cusped like $\rho \sim r^{-2 + 1/\alpha}$ at small radii with asymptotic density fall-off like $\rho \sim r^{-3 -1/(2\alpha)}$. Here, $\alpha$ is a parameter satisfying $\alpha \ge 1/2$. The family therefore spans the range of inner density cusps found in numerical simulations, but has much shallower -- and hence more realistic -- outer slopes than the corresponding members of the only previously-known family deduced by Zhao (1996) and exemplified by Hernquist & Ostriker (1992). When $\alpha =1$, the lowest-order density profile has an inner density cusp of $\rho \sim r^{-1}$ and an outer density slope of $\rho \sim r^{-3.5}$, similar to the famous Navarro, Frenk & White (1997) model. For this reason, we demonstrate that our new expansion provides a more accurate representation of flattened NFW haloes than the competing Hernquist-Ostriker expansion. We utilize our new expansion by analysing a suite of numerically-constructed haloes and providing the distributions of the expansion coefficients.

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

AbstractQuasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel–Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel–Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form , , these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel–Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval, and the cardioid. The unit circle case, given its exceptional properties, is discussed in more detail. In this case, a particularly relevant role is played by the reciprocal polynomial, and the Christoffel formulas provide now with two possible ways of expressing the same perturbed quantities in terms of the original ones, one using only the nonperturbed biorthogonal family of Laurent polynomials, and the other using the Christoffel–Darboux kernels and its mixed versions, as well. Two examples are discussed in detail.

Precise estimates for biorthogonal families under asymptotic gap conditions

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell [ 12 , 13 ], which is based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behavior: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and thus to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results by Fattorini-Russell [ 12 , 13 ] concerning biorthogonal families, using complex analysis techniques that were developed by Seidman [ 36 ], Guichal [ 20 ], Tenenbaum-Tucsnak [ 37 ] and Lissy [ 27 , 28 ].

On global fluctuations for non-colliding processes

We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.

The cost of controlling weakly degenerate parabolic equations by boundary controls

We consider the one-dimensional degenerate parabolic equation \begin{document}$u_t - (x^α u_x)_x =0 \;\;x∈(0,1), t ∈ (0,T) ,$ \end{document} controlled by a boundary force acting at the degeneracy point \begin{document}$ x=0$\end{document} . We study the reachable targets at some given time \begin{document}$T$\end{document} using \begin{document}$ H^1$\end{document} controls, studying the influence of the degeneracy parameter \begin{document}$ α ∈ [0,1)$\end{document} . First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as \begin{document}$ \mathit{\alpha } \to {{\rm{1}}^{\rm{ - }}}$\end{document} and exponentially fast when \begin{document}$ \mathit{T} \to {{\rm{0}}^{\rm{ + }}}$\end{document} . Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states. Our approach is based on the moment method developed by Fattorini and Russell [ 19 , 20 ]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [ 48 ], Guichal [ 26 ], Tenenbaum-Tucsnak [ 49 ] and Lissy [ 35 , 36 ].

Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients

We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a central limit theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai class of an interval. Our results also extend previous results on unitary ensembles in the one-cut case. Finally, we will illustrate our results by deriving central limit theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.