Orthonormal System Research Articles

Absolute convergence of double Walsh–Fourier series and related results

Abstract We consider the double Walsh orthonormal system on the unit square , where {wm(x)} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1<p≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series where {amn} is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0<r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f.

Absolute convergence of Walsh-Fourier series and related results

We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients \( \hat f \) (n), n ∈ ℕ, of functions f ∈ Lp for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σn=1∞an|\( \hat f \)(n)|r, where {an} is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones.

Biorthonormal systems in Freud-type weighted spaces with infinitely many zeros – an interpolation problem

In a Freud-type weighted (w) space, introducing another weight (v) with infinitely many roots, we give a complete and minimal system with respect to vw, by deleting infinitely many elements from the original orthonormal system with respect to w. The construction of the conjugate system implies an interpolation problem at infinitely many nodes. Besides the existence, we give some convergence properties of the solution.

Oscillation and spectral theory for symplectic difference systems with separated boundary conditions

We consider symplectic difference systems involving a spectral parameter together with general separated boundary conditions. We establish the so-called oscillation theorem which relates the number of finite eigenvalues less than or equal to a given number to the number of focal points of a certain conjoined basis of the symplectic system. Then we prove Rayleigh's principle for the variational description of finite eigenvalues and we describe the space of admissible sequences by means of the (orthonormal) system of finite eigenvectors. The principle role in our treatment is played by the construction where the original system with general separated boundary conditions is extended to a system on a larger interval with Dirichlet boundary conditions.

On the basis property of root vectors of a perturbed self-adjoint operator

We study perturbations of a self-adjoint operator T with discrete spectrum such that the number of its points on any unit-length interval of the real axis is uniformly bounded. We prove that if ‖Bϕn‖ ≤ const, where ϕn is an orthonormal system of eigenvectors of the operator T, then the system of root vectors of the perturbed operator T + B forms a basis with parentheses. We also prove that the eigenvalue-counting functions of T and T + B satisfy the relation |n(r, T) − n(r, T + B)| ≤ const.

Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L p and uniform norms.

Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity

Numerical integration of complex linear systems of ODEs depending analytically on an eigenvalue parameter are considered. Complex orthogonalization, which is required to stabilize the numerical integration, results in non-analytic systems. It is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non-analytic characteristic function. The orthonormal systems are constructed using the geometry of Stiefel bundles. Different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one-form on the Stiefel bundle. For the numerical integration, Gauss–Legendre Runge–Kutta algorithms are the principal choice for preserving orthogonality, and performance results are shown for a range of GLRK methods. The theory and methods are tested by application to example boundary value problems including the Orr–Sommerfeld equation in hydrodynamic stability.

A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel

Let {φk} be an orthonormal system on a quasi-metric measure space \({\mathbb{X}}\), {lk} be a nondecreasing sequence of numbers with lim k→∞lk=∞. A diffusion polynomial of degree L is an element of the span of {φk:lk≤L}. The heat kernel is defined formally by \(K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}\). If T is a (differential) operator, and both Kt and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖p≤c1Lc‖P‖p. In particular, we are interested in the case when \({\mathbb{X}}\) is a Riemannian manifold, T is a derivative operator, and \(p\not=2\). In the case when \({\mathbb{X}}\) is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion polynomial valued operators.

Estimates of the Fourier widths of classes of Nikol’skii—Besov and Lizorkin—Triebel types of periodic functions of several variables

where 〈 · , · 〉 is the inner product of the functions and the lower bound is taken over all orthonormal systems {gj}j=1 ⊂ L∞(T). Fourier widths were introduced by Temlyakov [1] in 1982. Estimates of the Fourier widths of different functional classes was studied in many works, among them the monographs [2], [3] and the papers[4]– [7], where a detailed bibliography and the history of the problem are given; also see Remark 2 in this paper.

Multiplication of distributions and Dirac formalism of quantum mechanics

We define multiplication and convolution of distributions and ultradistributions by introducing the notions of evaluation of distributions and integration of ultradistributions. An application is made to an interpretation of the Dirac formalism of quantum mechanics. The role of the Hilbert space of states is played by what is termed a Hermitian orthonormal system, and operators are replaced by the generalized matrices. We describe a simple example of one dimensional free particle and construct explicitly a representation of the Weyl algebra as the generalized matrices.

Estimation of the Effects in the Experimental Design Using Fourier Transforms

We propose that the model in experimental design be expressed in terms of an orthonormal system. Then, we can easily estimate the effects using Fourier transforms. We also provide the theorems with respect to the sum of squares needed in analysis of variance. Using these theorems, it is clear that we can execute the analysis of variance in this model.

A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures

We characterize sequences of numbers (a n ) such that ∑n≥1 a n Φ n converges a.e. for any orthonormal system (Φ n ) in any L 2-space. In our criteria we use the set A={∑m≥n||a m |2;n≥1}, majorizing measures on A, and new descriptions of the complexity of A.

On the possibility of strengthening the Lieb-Thirring inequality

In 1976, Lieb and Thirring obtained an upper bound for the square of the normon L2(ℝ2) of the sum of the squares of functions from finite orthonormal systems via the sum of the squares of the norms of their gradients. Later, a series of Lieb-Thirring inequalities for orthonormal systems was established by many authors. In the present paper, using the standard theory of functions, we prove Lieb-Thirring inequalities, which have applications in the theory of partial differential equations.

Manifestations of the Parseval identity

In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity. In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L2(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás-Mordell. Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions.

Uniqueness of series with respect to general Franklin systems

We give some sufficient condition for the uniqueness of series with respect to general Franklin systems and discuss analogous results for orthonormal spline systems of higher order and some wavelet systems.

Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L 2 ( [ 0 , 1 ] ) and an orthonormal system of L 2 ( [ 0 , + ∞ [ ) . Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.

P-Adic Brownian motion over ℚ p

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚp. First, we construct directly a p-adic white noise over ℚp by using a specific complete orthonormal system of \( \mathbb{L}^2 \)(ℚp). A p-adic Brownian motion over ℚp is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.

Geometrical modelling of scanning probe microscopes and characterization of errors

Scanning probe microscopes (SPMs) allow quantitative evaluation of surface topography with ultra-high resolution, as a result of accurate actuation combined with the sharpness of tips. SPMs measure sequentially, by scanning surfaces in a raster fashion: topography maps commonly consist of data sets ideally reported in an orthonormal rectilinear Cartesian coordinate system. However, due to scanning errors and measurement distortions, the measurement process is far from the ideal Cartesian condition. The paper addresses geometrical modelling of the scanning system dynamics, presenting a mathematical model which describes the surface metric x-, y- and z- coordinates as a function of the measured x′-, y′- and z′-coordinates respectively. The complete mathematical model provides a relevant contribution to characterization and calibration, and ultimately to traceability, of SPMs, when applied for quantitative characterization.

Uniformly bounded Riesz bases and equiconvergence theorems

We review some classical results on the convergence of classical trigonometric and polynomial Fourier series. Then we present a not well-known short proof of the local uniform boundedness of many classical orthonormal systems. Finally, we formulate a strong generalization of Haar’s classical equiconvergence theorem

HARMONIC ANALYSIS OF THE LEBEDEV-STIELTJES INTEGRALS

We expand the Bochner technique on the following Lebedev-Stieltjes integrals [Formula: see text] which are related to the Kontorovich-Lebedev transformation. Mapping and inversion properties are investigated. The Fourier type series with respect to an uncountable orthonormal system of the modified Bessel functions are considered in the Bohr type pre-Hilbert space. The Bessel inequality and Parseval equality are proved.