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Linear Differential Operators

Linear Differential Operators

$100.00 (P)

Part of Classics in Applied Mathematics

  • Date Published: January 1987
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • format: Paperback
  • isbn: 9780898713701

$ 100.00 (P)

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About the Authors
  • Don't let the title fool you! If you are interested in numerical analysis, applied mathematics, or the solution procedures for differential equations, you will find this book useful. Because of Lanczos' unique style of describing mathematical facts in nonmathematical language, Linear Differential Operators also will be helpful to nonmathematicians interested in applying the methods and techniques described. Originally published in 1961, this Classics edition continues to be appealing because it describes a large number of techniques still useful today. Although the primary focus is on the analytical theory, concrete cases are cited to forge the link between theory and practice. Considerable manipulative skill in the practice of differential equations is to be developed by solving the 350 problems in the text. The problems are intended as stimulating corollaries linking theory with application and providing the reader with the foundation for tackling more difficult problems.

    Reviews & endorsements

    'This scholarly volume describes Lanczos' life (1893–1974) and presents a clear development of the many fields he opened. … a collection of seven photographs of Lanczos from 1910 to 1972 … is followed by a fascinating twenty-eight page annotated story, 'Cornelius Lanczos: A Biographical Essay', by Barbara Gellai. The fortuitous inclusion of this excellent biography makes the Proceedings a more complete and desirable volume!' Mathematics of Computation

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    Product details

    • Date Published: January 1987
    • format: Paperback
    • isbn: 9780898713701
    • length: 582 pages
    • dimensions: 228 x 153 x 27 mm
    • weight: 0.78kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • Table of Contents

    1. Interpolation. Introduction
    The Taylor expansion
    The finite Taylor series with the remainder term
    Interpolation by polynomials
    The remainder of Lagrangian interpolation formula
    Equidistant interpolation
    Local and global interpolation
    Interpolation by central differences
    Interpolation around the midpoint of the range
    The Laguerre polynomials
    Binomial expansions
    The decisive integral transform
    Binomial expansions of the hypergeometric type
    Recurrence relations
    The Laplace transform
    The Stirling expansion
    Operations with the Stirling functions
    An integral transform of the Fourier type
    Recurrence relations associated with the Stirling series
    Interpolation of the Fourier transform
    The general integral transform associated with the Stirling series Interpolation of the Bessel functions
    2. Harmonic Analysis. Introduction
    The Fourier series for differentiable functions
    The remainder of the finite Fourier expansion
    Functions of higher differentiability
    An alternative method of estimation
    The Gibbs oscillations of the finite Fourier series
    The method of the Green's function
    Non-differentiable functions
    Dirac's delta function
    Smoothing of the Gibbs oscillations by Fejér's method
    The remainder of the arithmetic mean method
    Differentiation of the Fourier series
    The method of the sigma factors
    Local smoothing by integration
    Smoothing of the Gibbs oscillations by the sigma method
    Expansion of the delta function
    The triangular pulse
    Extension of the class of expandable functions
    Asymptotic relations for the sigma factors
    The method of trigonometric interpolation
    Error bounds for the trigonometric interpolation method
    Relation between equidistant trigonometric and polynomial interpolations
    The Fourier series in the curve fitting
    3. Matrix Calculus. Introduction
    Rectangular matrices
    The basic rules of matrix calculus
    Principal axis transformation of a symmetric matrix
    Decomposition of a symmetric matrix
    Self-adjoint systems
    Arbitrary n x m systems
    Solvability of the general n x m system
    The fundamental decomposition theorem
    The natural inverse of a matrix
    General analysis of linear systems
    Error analysis of linear systems
    Classification of linear systems
    Solution of incomplete systems
    Over-determined systems
    The method of orthogonalisation
    The use of over-determined systems
    The method of successive orthogonalisation
    The bilinear identity
    Minimum property of the smallest eigenvalue
    4. The Function Space. Introduction
    The viewpoint of pure and applied mathematics
    The language of geometry
    Metrical spaces of infinitely many dimensions
    The function as a vector
    The differential operator as a matrix
    The length of a vector
    The scalar product of two vectors
    The closeness of the algebraic approximation
    The adjoint operator
    The bilinear identity
    The extended Green's identity
    The adjoint boundary conditions
    Incomplete systems
    Over-determined systems
    Compatibility under inhomogeneous boundary conditions
    Green's identity in the realm of partial differential operators
    The fundamental field operations of vector analysis
    Solution of incomplete systems
    5. The Green's Function. Introduction
    The role of the adjoint equation
    The role of Green's identity
    The delta function --
    The existence of the Green's function
    Inhomogeneous boundary conditions
    The Green's vector
    Self-adjoint systems
    The calculus of variations
    The canonical equations of Hamilton
    The Hamiltonisation of partial operators
    The reciprocity theorem
    Self-adjoint problems
    Symmetry of the Green's function
    Reciprocity of the Green's vector
    The superposition principle of linear operators
    The Green's function in the realm of ordinary differential operators
    The change of boundary conditions
    The remainder of the Taylor series
    The remainder of the Lagrangian interpolation formula

  • Author

    Cornelius Lanczos

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