The zero-one law for a complete orthonormal system

2004 ◽  
Vol 339 (5) ◽  
pp. 335-337
Author(s):  
Kazaros Kazarian
Keyword(s):  
Orthonormal System ◽  
Complete Orthonormal System

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.

scholarly journals Eigenstructure of the equilateral triangle, Part II: The Neumann problem

2002 ◽  
Vol 8 (6) ◽  
pp. 517-539 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Complete Orthonormal System ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Lines ◽  
Mathematical Techniques ◽  
The Neumann Problem

Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

scholarly journals Investigation of eigenvibrations of a loaded bar

2018 ◽  
Vol 224 ◽  
pp. 04013 ◽  
Author(s):  
Anton A. Samsonov ◽  
Sergey I. Solov’ev
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Orthonormal System ◽  
Model Problem ◽  
Uniform Grid ◽  
Complete Orthonormal System ◽  
Eigenvalues And Eigenfunctions ◽  
Limit Problems ◽  
Plates And Shells ◽  
Theoretical Results

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

scholarly journals Eigenstructure of the equilateral triangle. Part III. The Robin problem

2004 ◽  
Vol 2004 (16) ◽  
pp. 807-825 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Robin Boundary Condition ◽  
Complete Orthonormal System ◽  
Robin Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Robin Boundary

Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored.

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