Topic
Orthogonal functions
About: Orthogonal functions is a research topic. Over the lifetime, 1635 publications have been published within this topic receiving 44812 citations.
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01 Jan 1966
TL;DR: In this article, the authors present a model for vector analysis based on the Calculus of Variations and the Sturm-Liouville theory, which includes the following: Curved Coordinates, Tensors.
Abstract: Vector Analysis. Curved Coordinates, Tensors. Determinants and Matrices. Group Theory. Infinite Series. Functions of a Complex Variable I. Functions of a Complex Variable II. Differential Equations. Sturm-Liouville Theory. Gamma-Factrial Function. Bessel Functions. Legendre Functions. Special Functions. Fourier Series. Integral Transforms. Integral Equations. Calculus of Variations. Nonlinear Methods and Chaos.
7,811 citations
01 Jul 1982
2,574 citations
Book•
01 Jan 1996TL;DR: In this article, the authors introduce spectral methods via orthogonal functions and finite differences, and compare computational cost of spectral methods with FD and PS methods in polar and spherical geometries.
Abstract: 1. Introduction 2. Introduction to spectral methods via orthogonal functions 3. Introduction to PS methods via finite differences 4. Key properties of PS approximations 5. PS variations/enhancements 6. PS methods in polar and spherical geometries 7. Comparisons of computational cost - FD vs. PS methods 8. Some application areas for spectral methods Appendices.
1,447 citations
TL;DR: In this article, the Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm.
Abstract: The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.
1,365 citations
TL;DR: A new set of orthogonal moment functions based on the discrete Tchebichef polynomials is introduced, superior to the conventional Orthogonal moments such as Legendre moments and Zernike moments, in terms of preserving the analytical properties needed to ensure information redundancy in a moment set.
Abstract: This paper introduces a new set of orthogonal moment functions based on the discrete Tchebichef polynomials. The Tchebichef moments can be effectively used as pattern features in the analysis of two-dimensional images. The implementation of the moments proposed in this paper does not involve any numerical approximation, since the basis set is orthogonal in the discrete domain of the image coordinate space. This property makes Tchebichef moments superior to the conventional orthogonal moments such as Legendre moments and Zernike moments, in terms of preserving the analytical properties needed to ensure information redundancy in a moment set. The paper also details the various computational aspects of Tchebichef moments and demonstrates their feature representation capability using the method of image reconstruction.
865 citations