Orthonormal System Research Articles

Zernike-like systems in polygons and polygonal facets.

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett.32, 74 (2007)10.1364/OL.32.000074OPLEDP0146-9592] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided.

Kernel Multivariate Spectral–Spatial Analysis of Hyperspectral Data

This paper contributes the concept of spectral-spatial kernel-based multivariate analysis (KMVSSA) based on the statistical principle of multivariate statistics. The essence of proposed framework is to expose the inherent structure and meaning revealed within spectral and spatial features through various statistical methods in hyperspectral remotely sensed data. This kernel-based framework is investigated to incorporate the spectral and spatial information simultaneously for dimension reduction and classification of hyperdimensional datasets. The method uses multivariate analysis to choose and apply a transform matrix that the transformed components are as orthogonal as possible. This nonlinear framework is derived by means of the theory of complete orthonormal systems. KMVSSA exhibits great flexibility by the combination of spectral and spatial features. We investigate the possibility of using KMVSSA for the classification of hyperspectral images and dimension reduction. The proposed framework is examined and compared in different merits with several hyperspectral images in different conditions (urban/agricultural area and size of the training set). Experimental results show that the proposed framework can meaningfully enhance the dimensionality reduction and also it greatly improves the overall as well as per class classification accuracies. We demonstrate a comprehensive comparison of some state of the art hyperspectral image classification methods.

Fourier coefficients of characteristic functions of intervals with respect to a complete orthonormal system bounded in L p ([0, 1]), 2 < p < ∞

Fourier coefficients of characteristic functions of intervals with respect to a complete orthonormal system bounded in L p ([0, 1]), 2 < p < ∞

Unconditionality of orthogonal spline systems in H1

We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order $k$ is an unconditional basis in the atomic Hardy space $H^1[0,1]$.

Integrated Pearson family and orthogonality of the Rodrigues polynomials: A review including new results and an alternative classification of the Pearson system

An alternative classification of the Pearson family of probability densities is related to the orthogonality of the corresponding Rodrigues polynomials. This leads to a subset of the ordinary Pearson system, the Integrated Pearson Family. Basic properties of this family are discussed and reviewed, and some new results are presented. A detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented, including an algorithm that enables to decide whether a given Pearson density belongs to the integrated system, or not. Recurrences between the derivatives of the corresponding orthonormal polynomial systems are also given.

Spectral properties of Schrödinger operators on superconducting surfaces

In this work we focus on the characterization of the space L^2 (S; \mathbb C) on Riemannian 2-manifolds S induced by a fixed magnetic vector potential A_0 in the nonlinear Ginzburg-Landau (GL) superconductivity model. The linear differential operator governing the GL model is the surface Schrödinger operator (i \nabla + A_0)^2 on S . We obtain a complete orthonormal system in L^2 (S; \mathbb C) from a collection of nontrivial solutions of the weak-form of the spectral problem associated with (i \nabla + A_0)^2 . Then, after proving that any member of this basis satisfies a higher regularity condition, we conclude that each is also an eigenfunction of the strong-form of the surface Schrödinger operator, and must satisfy a natural Neumann condition over any nonempty component of the manifold boundary \partial S . These results form the theoretical foundations used to develop efficient computational tools for simulating the Langevin version of the surface GL model.

Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

Abstract In the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz, analysis, FEM approximations and computations

In this work we focus on the efficient representation and computation of the eigenvalues and eigenfunctions of the surface Schrödinger operator (i∇+A0)2 that governs a class of nonlinear Ginzburg–Landau (GL) superconductivity models on rotationally symmetric Riemannian 2-manifolds S. We identify and analyze a complete orthonormal system in L2(S;C) of eigenmodes having a variable-separated form. For the unknown functions in this ansatz, our analysis facilitates the identification of approximate spectral problems whose eigenvalues lie arbitrarily near corresponding eigenvalues of the Schrödinger operator. We then develop and implement an arbitrary order finite element method for the efficient numerical approximation of the eigenvalue problem. We also demonstrate our analysis, algorithm and its convergence rate using parallel computations performed on a variety of choices of smooth and non-smooth surfaces S.

SISTEM ORTONORMAL DALAM RUANG HILBERT

Hilbert space is one of the important inventions in mathematics. Historically, the theory of Hilbert space originated from David Hilbert’s work on quadratic form in infinitely many variables with their applications to integral equations. This paper contains some definitions such as vector space, normed space and inner product space (also called pre-Hilbert space), and which is important to construct the Hilbert space. The fundamental ideas and results are discussed with special attention given to finite dimensional pre-Hilbert space and some basic propositions of orthonormal systems in Hilbert space. This research found that each finite dimensional pre- Hilbert space is a Hilbert space. We have provided that a finite orthonormal systems in a Hilbert space X is complete if and only if this orthonormal systems is a basis of X.

A method for characterisation of the static elastic properties of the porous frame of orthotropic open-cell foams

This paper proposes a method to identify the static, fully relaxed elastic Hooke’s matrix of a porous open-cell material. The moduli are estimated through an inverse estimation method, by performing a fit of a numerical model on the measured displacements on the faces of the porous material. These displacements are obtained from a static compression along each of the three coordinate axes. The material is modelled as an orthotropic equivalent solid, of which the principal directions are not necessarily aligned with the orthonormal coordinate system in which the experiments are conducted. The angles of relative orientation accounting for the misalignment are among the properties to be estimated. The focus in this paper is on the methodology itself, and its validity is verified by applying the method to four artificial materials with different levels of anisotropy. In addition, the robustness of the method to perturbations on the input data is investigated.

Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball

The normalized or game-theoretic $p$-Laplacian operator given by $$ -\Delta_p^Nu:=-\frac{1}{p}|\nabla u|^{2-p}\Delta_p(u) $$ for $p\in(1,\infty)$ with $\Delta_pu=\rm{div}(|\nabla u|^{p-2}\nabla u)$ has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [6]. If it is restricted to the class of radial functions, however, the normalized $p$-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted $L^2$-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation $u_t-\Delta_p^Nu=0$.

Kawahara solitons in Boussinesq equations using a robust Christov–Galerkin spectral method

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L2(-∞,∞) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives.As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

Orthonormal systems in linear spans

We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits an orthonormal system such that the $L^2$ norm of the square variation operator $V^2$ is as small as possible. When applied to the span of the trigonometric system, we obtain an orthonormal system of trigonometric polynomials with a $V^2$ operator that is considerably smaller than the associated operator for the trigonometric system itself.

The stochastic properties of [formula omitted]-regularized spherical Gaussian fields

Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that neither Gaussianity nor isotropy are conserved in general under convex regularization based on ℓ1 minimization over a Fourier dictionary, such as the orthonormal system of spherical harmonics.

The Weighted Hermite Polynomials Form a Basis for L2(ℝ)

We present a student-friendly proof that the weighted Hermite polynomials form a complete orthonormal system (a basis) for the collection L2(ℝ) of real-valued functions.

Strengthened Chernoff-type variance bounds

Let $X$ be an absolutely continuous random variable from the integrated Pearson family and assume that $X$ has finite moments of any order. Using some properties of the associated orthonormal polynomial system, we provide a class of strengthened Chernoff-type variance bounds.

Absolute convergence of multiple Fourier–Haar series

Abstract. The class of functions of s variables H α / s ( s ) $H_{\alpha /s}^{(s)}$ is defined. For the functions of this class, Sász's theorem for s-dimensional Fourier–Haar series is proved. When α = 1 $\alpha =1$ , it is shown that the obtained theorem is best possible for general complete orthonormal systems.

Fractal Functions of Discontinuous Approximation

A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give rise to a new one, called in the paper fractal convolution of the originals, whose graph is discontinuous and has a fractal structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous. The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose elements are as close to a classical orthonormal system as desired.

Об операторе дифференцирования на компактных нуль-мерных группах

Saratov State University, 83, Astrakhanskaya str., Saratov, 410012, Russia, KrussUS@gmail.comWe define strong derivative on zero-dimensional compact group and find conditions under which the differential operator does notdepend from an orthonormal system that defines this derivative. For multidimensional case we find conditions under which thedifferential operator does not depend from method of conversion multidimensional group in one-dimensional group. We obtain aclear view of annihilators in a multidimensional compact zero-dimensional group.Key words: zero-dimensional groups, pseudo-differential operator, strong P-adic derivative, strong P-adic integral.

Unconditionality of orthogonal spline systems in Lp

Given any natural number $k$ and any dense point sequence $(t_n)$, we prove that the corresponding orthonormal spline system of order $k$ is an unconditional basis in reflexive $L^p$.