Linear Differential Operators

Table of Contents

• Preface
• Bibliography
• 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
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