D-Orthogonal Analogs of Classical Orthogonal Polynomials

Connection and linearization coefficients of the Askey–Wilson polynomials

One of the most effective methods in ECG processing is the so-called transformation technique. The most popular systems that are used frequently include the classical trigonometric Fourier, orthogonal polynomial, and spline systems. In recent applications the wavelet methods turned to be especially effective. Yet we propose another system that is built from basic rational functions. The rational orthogonal systems, i.e. the Malmquist-Takenaka systems, we use are determined by parameters called poles and multiplicities. This way we have infinitely many systems at hand. The main advantage of our approach is its flexibility. The parameters that control the system can be adjusted to the particular problem. It means that in ECG processing we may change them even from heartbeat to heartbeat. We have worked out the proper discretization of the original mathematical model [2-3] and designed optimization processes adapted to these systems [4]. The result is a special variable projection method. The algorithms we constructed turned to be very efficient. They outperform the existing ones in many respects [7]. Problems we considered so far include QRS modeling [4], R peak detection, ECG compression, heartbeat classification [1]. The tests were evaluated on the standard public MIT-BIH Arrhythmia Database available on PhysioNet [5]. We note that our algorithms are fast enough for real time applications, and also for big databases. one may use them as a tool for supporting cardiologists among others as a preprocessing long time records (Holter) or as a built in alert function. So far we have concentrated on ECG processing methods that work well in general circumstances. For instance in case of classification we constructed a mixed feature vector that consists of dynamic and morphological features. The morphological features, which can be divided into patient specific and heartbeat specific components, come from rational transforms. Then we used a so called Support Vector Machine classifier for them. The method can be performed for multiple leads independently or even together, and then we can combine the individual results to obtain an improved final result. Recently, we have been focusing on more specific situations where we take various aspects into consideration. In particular we have been working on a version designed for subject-based classification Moreover we have been refining the classification algorithm in order to reduce the rate of false negative results. In connection with it various evaluation and fusion techniques have been investigated. We note that our method can be efficient for medical signals other than ECG. An example for epileptic seizure detection in EEG signals can be found in [6].

The theory of non-selfadjoint differential operators has great importance in several applications. Developing a fruitful method of V. A. II'in [1], in [21 I. Jod described a new method which makes possible to avoid the use of the basic solution in such investigations. Using this method, a general equiconvergence theorem was proved in [9] for the SchrSdinger operator, having generalized the results of the papers [5], [6], [7] and [8]. This theorem concerns the expansions by Riesz bases formed by eigenfunctions of higher order of the SchrSdinger operator (but seems to be new also for the case of orthonormal bases). The existence of such Riesz bases was proved for a wide class of operators (also of higher order) by V. P. Mikhailov [10] and G. M. Keselman [111. Later on, the method of [2] was generalized for the case of differential operators of higher order in [41, [12] and [13]. These generalizations are based on a new formula which extends the well-known mean-value formula of E. C. Titchmarsh for more general operators. Using the method and the results of these papers, we shall prove a general equiconvergence theorem for linear differential operators of higher order. In this theorem the eigenvalues of the Riesz basis may be arbitrary complex numbers. Our result is new also for orthonormal bases.

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question.

In this paper we are concerned with the problem of constructing the functional calculus for an operator with continuous spectrum from a given collection of weak eigenvectors.A functional calculus is a homomorphism/- Tf of an algebra of functions defined over the 'spectrum' V into a subalgebra of bounded linear operators acting in a banach space E; we may regard an 'eigenfunction expansion' for such a functional calculus as an expression for all Tf over a banach space D dense in E of the form Tf= vf(X)Txdm(X), where m is a measure over Fand Tx is a bounded linear operator from D into the space of 'distributions' D'.In general, it is hoped that the range of Tx, lmTx, will consist of rather precisely specified weak eigenvectors (specified, for example, by boundary values).Since a family of weak eigenvectors is often available, a problem that often appears in applications is the following: Given a family of weak eigenvectors at X, when is it true that this family coincides with lmTxl There are many results in the literature for a closed operator with dense domain in a banach space with continuous spectrum which guarantee a rich functional calculus and a representation of the functional calculus in terms of an eigenfunction expansion where, in fact, ImTA consists of weak eigenvectors.F. Browder [3] established such results for a large class of nonsymmetric partial differential operators, and Gel'fand and Silov [5] were the first to consider the problem of the existence of eigenfunction expansions in general operator theoretic terms.However there are few results that tell us which weak eigenvectors to choose for the eigenfunction expansion.Most results of this nature are confined to differential operators with strong regularity theorems, the most classical being Weyl's construction of the Plancherel formula for a singular second order ordinary differential operator [10].The crucial step is the 'Weyl lemma' which asserts that certain weak eigenvectors are actually infinitely differentiable, and so, classical uniqueness theorems may be invoked to determine precisely which weak eigenvectors contribute to the eigenfunction expansion.The results given here, in contrast, are well within the context of functional

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux–Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2 m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2ℓ + 3 recurrence relation where ℓ is the length of the partition λ. Explicit expressions for such recurrence relations are given.

The zeros and asymptotic limits of two new classes of orthogonal polynomials, which are derived by applying two orthogonalization procedures due to Lowdin to a set of monomials, are calculated. It is established that they possess all the properties ofthe zeros of a polynomial. Their asymptotic limits are found. A Unified view of all the Lowdin orthogonal polynomials together with the standard classical orthogonal polynomials are presented in a unique graph.

Fourth-order difference equation for the associated classical discrete orthogonal polynomials

The concept of a well-conditioned BVP, as considered in recent literature, essentially corresponds to the concept of a uniformly stable IVP. But this terminology suggests a shift from absolute to relative quantities, which makes it more difficult to understand the conditioning of numerical discretizations, especially for higher order differential operators. In this note we wish to make it clear that the notion of well-conditioning of BVPs is not always a direct analogue of the u The condition number of a discretization of a well-conditioned BVP generally grows unboundedly when higher accuracy is desired, because it contains a discretization of a differential operator. This condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted. For collocation, an implementation which uses a local (Runge-Kutta) solution representation produces a much better conditioning, because the repeated numerical differentiation is avoided. Instead, the highest derivative is approximated locally by a (discontinuous) piece-wise polynomial, and local integrations yield lower solution derivatives. Upon imposing the collocation equations the highest derivative approximation can be locally eliminated, resulting in a sophisticated discretization of the corresponding first order system.

Orthogonal matrix polynomials and higher-order recurrence relations

In applications of mathematics involving either the Laplace or the Helmholtz equation in spherical coordinates the associated Legendre equation occurs. Its solutions are called associated Legendre functions. They have some relations to classical Legendre orthogonal polynomials and classical Jacobi orthogonal polynomials. The classical Legendre polynomials and the associated Legendre functions have been used by some authors at the approximation of length of curves and cylindrical surfaces and the approximation of Earth shape in geosciences and engineering computations. In the present paper we investigate some relations between these functions pointing out possible applications in geosciences.

Voice signal is an indicator finding a progression of diseases such as nerve disorder and muscle dysfunction. To improve the performance of medical diagnosis system using the voice signal, this paper suggests a new feature extraction method which combines higher-order statistics (HOS) and higher-order differential energy operator (DEO). For the experiment, Saarbruecken Voice Database (SVD) was used, and 687 healthy voice samples and 263 pathological voice samples which consist of Cysts, Paralysis, and Polyp were selected. In addition, the OpenSmile script which provides 6,373 features was used for comparison with our new features. To decide the most effective features, Gradient Boosting was conducted as a feature selector. Finally, 20 features including 15 combinations of HOS and DEO were chosen, and deep neural network(DNN) was trained using the new features. The best accuracy of 87.4% was obtained, which exceeds the best accuracy of 84.5% with the existing features. The finding suggests a possibility that the pathological voice can be efficiently detected with only statistical information without heavy computations such as convolutional neural networks. Due to the simple structure, we expect this approach will be easily applied to a variety of mobile systems.

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.