Biorthogonal Functions Research Articles

About families of orthogonal polynomials satisfying Heun’s differential equation

We consider special families of orthogonal polynomials satisfying differential equations. Besides known hypergeometric cases, we look especially for Heun’s differential equations. We show that such equations are satisfied by orthogonal polynomials related to some classical weight functions modified by Dirac weights or by division of powers of binomials. An appropriate set of biorthogonal rational functions, or 2-point Padé approximations, is also described.

A unified algebraic underpinning for the Hahn polynomials and rational functions

An algebra denoted mH with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on mH. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of mH in U(sl2) is presented. A Padé approximation table for the binomial function is obtained as a by-product.

An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type

The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.

A boundary value problem in the theory of elasticity for a rectangle: exact solutions

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas. The obtained formulas remain the same for other types of homogeneous boundary conditions, for example, when the horizontal sides of the rectangle are firmly clamped, have stiffening ribs that work in tension–compression and/or bending, etc. Obviously, in this case, the Papkovich–Fadle eigenfunctions will change, as well as the corresponding biorthogonal functions and normalizing factors. To solve a specific boundary value problem, it is enough to find the Lagrange coefficients, which are determined from simple formulas, as Fourier integrals of boundary functions specified at the ends of the rectangle, and then substitute them into the necessary formulas. Examples of solving two problems (even-symmetric deformation relative to the central coordinate axes) are given: (a) The normal stresses are known at the rectangle ends, and the tangential ones are zero; and (b) the longitudinal displacements conditioned by the action of some normal stresses are known at the rectangle ends (the tangential stresses are zero). These solutions are compared with the known solutions in trigonometric Fourier series. The basis of the exact solutions obtained is the theory of expansions in Papkovich–Fadle eigenfunctions based on the Borel transform in the class of quasi-entire functions of exponential type (developed by the authors in their previous studies).

A Note on Bi-Orthogonal Polynomials and Functions

The theory of orthogonal polynomials is well established and detailed, covering a wide field of interesting results, as, in particular, for solving certain differential equations. On the other side the concepts and the related formalism of the theory of bi-orthogonal polynomials is less developed and much more limited. By starting from the orthogonality properties satisfied from the ordinary and generalized Hermite polynomials, it is possible to derive a further family (known in literature) of these kind of polynomials, which are bi-orthogonal with their adjoint. This aspect allows us to introduce functions recognized as bi-orthogonal and investigate generalizations of families of orthogonal polynomials

Kompresi Gambar dengan Transformasi Wavelet

Current technological developments have increased quite rapidly, one of which is in digital image processing. The use of digital images sometimes experience problems during the upload process or when users want to insert image files on a platform but are limited by the maximum image file size. The image file to be uploaded must go through a digital image processing so that the image file size can match the requested maximum limit. One technique that can be used in this problem is compression. Compression is one of the main techniques of image processing. Large image size results in reduced data or memory storage. So that compression is also carried out to reduce the use of data storage and hence communication costs. There are many methods used in image compression today. One method used is to use wavelet transforms. In this research, the mother wavelet used is the Biorthogonal wavelet. The purpose of this study is to find out how much the accuracy of the Biorthogonal wavelet to be implemented in color images and to determine the image quality and performance of the Biorthogonal wavelet function in image compression. Decomposition levels used in this study are level 1, level 2, and level 3. The results of the study are displayed in the form of images before compression and image compression results, equipped with PSNR, MSE values, compression ratio, and the size of the compressed images.

On some nonlocal elliptic systems with multiple parameters

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Products of Complex Rectangular and Hermitian Random Matrices

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations

Efficient Legendre dual-Petrov-Galerkin methods for solving odd-order differential equations are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like series. Numerical results indicate that the suggested methods are extremely accurate and efficient, and suitable for the odd-order equations.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

An image classification framework exploring the capabilities of extreme learning machines and artificial bee colony

A hybridized image classification strategy is proposed based on discrete wavelet transform, artificial bee colony (ABC) and extreme learning machine (ELM). The proposed methodology works in three phases: (a) in preprocessing phase, images are decomposed and features are extracted from images using bi-orthogonal wavelet functions; (b) secondly, modified ABC (MABC) optimization algorithm is proposed to determine the optimal parameters such as hidden layer weights and biases to be used by ELM for classification; (c) the ELM in the third phase has been trained and tested with three brain image datasets for different diseases along with normal brain images. The performance recognition of the proposed MABC-ELM in terms of accuracy, rate of per-image classification and speedup has been made with variants of ELM such as ELM, ABC-ELM and MABC-ELM and also with MLPNN, naive Bayesian, linear regression classifiers. Finally, the percentage of accuracy observed by the proposed MABC-ELM, for acute stroke-speech arrest, glioma and multiple sclerosis datasets, is 90%, 90% and 100% with eight hidden nodes in the ELM architecture, and it can be concluded that MABC-ELM gives better generalization performance, more compact network architecture and the hybridization of ELM with modified ABC is worth investigated.

On Approximation by Hyperbolic Splines

The paper considers the minimal hyperbolic splines and their properties. Formulas for constructing quadratic splines and the corresponding biorthogonal (dual) functionals are obtained. Numerical results, demonstrating how approximation quality can be improved by using hyperbolic splines and changing control parameters, are presented.

Research and Modeling of Photovoltaic Array Channel Noise Characteristics

The photovoltaic array can be used as a medium for carrier communication to realize monitoring of photovoltaic components. Photovoltaic array channel noise, especially the pulse-type noise therein, seriously interferes carrier communication, so it is necessary to grasp the characteristics of the photovoltaic array channel noise. Photovoltaic array channel noise modeling is a key process when conducting anti-noise immunity tests of monitoring equipment. Based on the time-domain waveform of photovoltaic series channel noise which is measured in a photovoltaic power station, this paper proposes a photovoltaic array noise modeling method of Wavelet Peak-Type Markov chain, and studies the influence on modeling accuracy when different mother wavelets are adopted for modeling. From the simulation results, root mean square errors of the predicted output for Haar, Biorthogonal and Daubechies wavelet-based function modeling case are 0.9614 V, 1.4915 V and 0.7928 V, respectively, validating that Daubechies wavelet-based function is the best wavelet-based function of modeling. In the case that the peak of original noise reaches 20 V, the predicted mean absolute error of this model is only 0.4926 V, which not only verifies the applicability of the Wavelet Peak-Type Markov chain model to the photovoltaic array channel noise, but also verifies the applicability to the pulse-type noise.

Biorthogonal rational functions of $R_{II}$-type

In this work, a sequence of orthonormal rational functions that is also biorthogonal to another sequence of rational functions arising from recurrence relations of $R_{II}$ type is constructed. The biorthogonality is proved by a procedure which we call Zhedanov method. A particular case of a sequence of orthonormal rational functions having denominators of special form is considered to motivate the general case. The particular case provides a Christoffel type transformation of the generalized eigenvalue problem with a reformulation different from the existing literature.

The Heun operator of Hahn-type

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.

Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

Rosengren and Schlosser introduced notions of RN-theta functions for the seven types of irreducible reduced affine root systems, RN = AN−1, BN, BN∨, CN, CN∨, BCN, DN, N∈N, and gave the Macdonald denominator formulas. We prove that if the variables of the RN-theta functions are properly scaled with N, they construct seven sets of biorthogonal functions, each of which has a continuous parameter t ∈ (0, t*) with given 0 < t* < ∞. Following the standard method in random matrix theory, we introduce seven types of one-parameter (t ∈ (0, t*)) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with N → ∞, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t ∈ (0, t*). We give new expressions for the Macdonald denominators using the Karlin–McGregor–Lindström–Gessel–Viennot determinants for noncolliding Brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration t*, which are specified by the pinned configurations at time t = 0 and t = t*.

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

AbstractA variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated. Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B‐splines of high continuity and Bézier elements with a standard C0 continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.

Phaselet Transform Based Approach for Detecting Voltage Flickers Due to Distributed Generation Units

This paper presents the development and performance evaluation of a new approach for detecting voltage flicker events in power systems hosting distributed generating units. The proposed approach is based on extracting the magnitudes and phases of the different frequency sub-band contents present in the voltage signal(s). Desired magnitudes and phases are extracted by processing the voltage signal(s) using nine phaselet frames that are realized by a modulated filter bank. The modulated filter bank is designed using nine digital high-pass filters (HPFs), each of which implements one phaselet frame. The coefficients of HPFs are determined by bi-orthogonal phaselet basis functions. Extracted high-frequency sub-band contents provide signature information for accurate detection and quantification of voltage flickers. The performance of a phaselet transform based approach is experimentally tested for different conditions, including voltage flicker events that are triggered by an interconnected photovoltaic system. Performance results show accurate, reliable, and fast detection and quantification of voltage flickers, along with a negligible sensitivity to the source producing voltage flicker events.

Mixed boundary value problems in the theory of elasticity in an infinite strip

We propose a method of solving mixed boundary value problems in the theory of elasticity in an infinite horizontal strip that has two points of change in the type of boundary conditions located on the upper and lower sides of the strip and lying at the ends of a vertical segment—the line of the joint between the left and right half-strips. The solutions to the right and the left of the joint line are represented as series in eigenfunctions that correspond to particular boundary conditions on the horizontal sides of the right and left half-strips. Solution continuity conditions or, on the contrary, discontinuities of the displacements or stresses can be specified at the joint between the half-strips. We illustrate the main idea of the method initially using a mixed boundary value problem for the harmonic equation as an example. Here, biorthogonal functions are constructed, their properties and the properties of the coefficients of expansions are investigated, and examples are given to illustrate the correctness of the obtained solutions. Subsequently we solve a mixed boundary value problem in which two trigonometric systems of functions with different eigenvalues are involved at the joint between the half-strips. We consider examples where (1) displacement and stress continuity conditions, (2) a discontinuity of the longitudinal displacements, and (3) a discontinuity of the transverse displacements are specified at the joint between the half-strips.

Higher Order Mixed FEM for the Obstacle Problem of the p-Laplace Equation Using Biorthogonal Systems

Abstract We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ∈ ( 1 , ∞ ) {p\in(1,\infty)} , where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case p = 1.5 {p=1.5} and the degenerated case p = 3 {p=3} . We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.