Self-adjoint Systems Research Articles

The stability of linear, real, periodic self-adjoint systems of differential equations

The stability of linear, real, periodic self-adjoint systems of differential equations

On optimal control of self-adjoint systems

On optimal control of self-adjoint systems

Linear Differential Operators

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 Fejer'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

Oscillation criteria for self-adjoint differential systems

Oscillation criteria for self-adjoint differential systems

Principal solutions of non-oscillatory self-adjoint linear differential systems

Principal solutions of non-oscillatory self-adjoint linear differential systems

The Expansion Theorem For Singular Self-Adjoint Linear Differential Operators

1. There has recently appeared three papers giving independently and almost simultaneously a much simplified approach to the expansion theorem for selfadjoint singular ordinary differential equations. Levitan [7] treats by means of a simple lemma the differential operator of arbitrary order but does not deal with the problem of uniqueness of the expansion. Yosida [11] and Levinson [5, 6] considered only the second order case but with uniqueness proved in the limit point case. All three accounts make important use of the Helly selection theorem for a set of functions of bounded variation. Here it will be shown that using the lemma of Levitan [7] and a treatment rather similar to that used by Levinson [6] for the inverse transform theorem, it is possible to obtain a simple self-contained proof of uniqueness under broad conditions. In particular none of the apparatus of Hilbert space or the theory of singular integral equations will be required. One of the results that follow, Theorem II, has been demonstrated by Coddington [2] with the use of Levitan's lemma and theorems from Hilbert space theory. Moreover Theorem III is implicit in Coddington's results because of the remarks which follow his Theorem 3 [2, p. 735]. The methods of the present paper can easily be carried over to singular selfadjoint systems using the formulations given by Bliss [1] for the non-singular

On Separation, Comparison and Oscillation Theorems for Self-Adjoint Systems of Linear Second Order Differential Equations

On Separation, Comparison and Oscillation Theorems for Self-Adjoint Systems of Linear Second Order Differential Equations

Algebraic Properties of Self-Adjoint Systems

Algebraic Properties of Self-Adjoint Systems

Algebraic properties of self-adjoint systems

The general definition of adjoint systems of boundary conditions associated with ordinary linear differential equations was given by Birkhoff.t In a paper of Bocher, I in which the idea is further developed, there is obtained a condition that a system of the second order be self-adjoint. It is proposed here to extend the discussion of this problem to the case of systems of any order.? A condition for self-adjointness of the boundary conditions is simply expressed in matrix form, without any requirement that a corresponding propertv be possessed by the differential equations; the condition gains in symmetry if it is assumed that the differential expression which forms the lefthand member of the given differential equation is itself identical with its adjoint or with the negative of its adjoint. As a preliminary, it will be well to recall a well-known rule for the combination of matrices, which will be found particuilarly convenient. Let a,, cx2, a3, and a4 be n-rowed square matrices. The square matrix of order 2n which is made up of the elements of these four, with the elements of al, arranged in order, in its upper left-hand corner, and the elements of the other matrices correspondingly disposed, may be indicated by the notation