A property of an orthonormal system of functions

1963 ◽  
pp. 75-89
Author(s):  
O. A. Ziza
Keyword(s):  
Orthonormal System

Modeling the Quasigeoid Heights on Local Areas of the Earth Surface by the Results of Expansion into a Generalized Fourier Series

2020 ◽  
Vol 28 (4) ◽  
pp. 82-94
Author(s):  
V.F. Kanushin ◽  
◽  
I.G. Ganagina ◽  
D.N. Goldobin ◽  
◽  
...  
Keyword(s):  
Fourier Series ◽  
Orthonormal System ◽  
Local Area ◽  
Spherical Functions ◽  
Trigonometric Functions ◽  
Computing Technology ◽  
Generalized Fourier Series ◽  
The Earth ◽  
Local Areas ◽  
Discrete Values

The article presents two methods of modeling discrete heights of a quasigeoid on a local area of the earth’s surface using a gen-eralized Fourier series. The first method is based on modeling the characteristics of the earth’s gravitational field on a plane and involves the use of a two-dimensional Fourier transform by an orthonormal system of trigonometric functions. The second method consists in the expansion of the quasigeoid heights in a Fourier series by an orthonormal system of spherical functions on a local area of the earth’s surface. The errors of approxima-tion of the obtained discrete values of the quasigeoid heights on the local territory are analyzed. It is shown that with the modern computing technology, the most accurate and technologically simple way to model the quasigeoid heights on local areas is to expand them into a Fourier series by an orthonormal system of spherical functions.

scholarly journals A Determinantal Expansion for a Glass of Definite Integral. Part 1

1953 ◽  
Vol 9 (1) ◽  
pp. 44-52 ◽  
Author(s):  
L. R. Shenton
Keyword(s):  
Weight Function ◽  
Fourier Coefficients ◽  
Finite Interval ◽  
Orthonormal System ◽  
Definite Integral ◽  
Negative Weight ◽  
Image Position

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficientsParseval's formula gives

Practical Aspects of Using Orthonormal System Parameterisations in Estimation Problems

2000 ◽  
Vol 33 (15) ◽  
pp. 463-468 ◽  
Author(s):  
Brett Ninness ◽  
Stuart Gibson ◽  
Steve Weller
Keyword(s):  
Orthonormal System ◽  
Estimation Problems

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

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Pengbin Feng ◽  
Erkinjon T. Karimov
Keyword(s):  
Wave Equation ◽  
Mixed Type ◽  
Orthonormal System ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Initial Time ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
Inverse Source Problems ◽  
Well Posed

AbstractIn 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.

On the (C, –1 < α < 0)-Summability of Series with Respect to Block-Orthonormal Systems

2000 ◽  
Vol 7 (3) ◽  
pp. 565-575
Author(s):  
G. Nadibaidze
Keyword(s):  
Orthonormal System ◽  
Almost Everywhere

Abstract Statements connected with the (C, α) (–1 < α < 0)-summability almost everywhere of series with respect to block-orthonormal systems are given. In particular, it is stated that if , then the condition guarantees the (C, α)-summability almost everywhere of the series with respect to an {ϕ n } – (Nk , Nk +1]-orthonormal system.

scholarly journals The bicomplex quantum Coulomb potential problem

2013 ◽  
Vol 91 (12) ◽  
pp. 1093-1100 ◽  
Author(s):  
J. Mathieu ◽  
L. Marchildon ◽  
D. Rochon
Keyword(s):  
Differential Equation ◽  
Coulomb Potential ◽  
Potential Problem ◽  
Orthonormal System ◽  
Mathematical Formulation ◽  
Algebraic Method ◽  
Hamiltonian Operator ◽  
Number System ◽  
Bicomplex Numbers ◽  
Bicomplex Number

Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [Formula: see text], where ξ is a bicomplex number. Following Pauli’s algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrödinger’s time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.

scholarly journals An orthonormal system on the construction of the generalized cantor set

1993 ◽  
Vol 16 (4) ◽  
pp. 737-748 ◽  
Author(s):  
Raafat Riad Rizkalla
Keyword(s):  
Orthonormal System ◽  
Walsh System ◽  
Cantor Set ◽  
Complete Orthonormal System

This paper presents a new complete orthonormal system of functions defined on the interval[0,1]and whose supports shrink to nothing. This system related to the construction of the Cantor ternary set. We defined the canonicalmap ξand proved the equivalence between this system and the Walsh system. The generalized Cantor set with any dissection ratio is established and the constructed system is defined in the general case.

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