Topic
Series expansion
About: Series expansion is a research topic. Over the lifetime, 6677 publications have been published within this topic receiving 127157 citations.
Papers published on a yearly basis
Papers
More filters
Book•
01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.
7,426 citations
TL;DR: The approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings and the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption.
Abstract: Approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1/n), where n is the number of nodes. The approximated function is assumed to have a bound on the first moment of the magnitude distribution of the Fourier transform. The nonlinear parameters associated with the sigmoidal nodes, as well as the parameters of linear combination, are adjusted in the approximation. In contrast, it is shown that for series expansions with n terms, in which only the parameters of linear combination are adjusted, the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption, where d is the dimension of the input to the function. For the class of functions examined, the approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings. >
2,857 citations
TL;DR: A compilation of nuclear charge density-distribution parameters, obtained from elastic electron scattering, is presented in five separate tables as discussed by the authors, and the positions and amplitudes for the expansion in a sum of gaussians are given.
1,605 citations
CERN1
TL;DR: In this article, the behavior of a renormalized field theory under scale transformations x → λx; p → p/λ can be found in a simple way when the theory is regularized by the continuous dimension method.
1,104 citations
TL;DR: Magnusson expansion as discussed by the authors provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory (TEPT).
1,013 citations