### Biorthogonal Systems of Functions

Transactions of the American Mathematical Society, Vol. 12, No. 2 (April 1911), 135-164

Presented to the Society (Chicago) April 10, 1909

Introduction

In boundary value problems of differential equations which are not self-adjoint, biorthogonal systems of functions play the same role as the orthogonal systems do in the self-adjoint case. Liouville has considered special non-self-adjoint differential equations with real characteristic values of the parameter; Birkhoff has proved the existence of the characteristic values (in general complex) for the differential equation of the nth order and obtained the related expressions.

If the integral equation
\[u(s) = \lambda \int_a^b L(s,t)u(t)\,dt\]
with the unsymmetric kernel \(L(s,t)\) has solutions \(u(s)\), and therefore the integral equation
\[v(s)=\lambda \int_a^b L(t,s)v(t)\,dt\]
solutions \(v(s)\), it has been shown by Plemelj and Goursat that the solutions or functions closely related to them form a biorthogonal system. But expansions in terms of these solutions have not been obtained, and no criteria have been given for the existence of real characteristic numbers of an unsymmetric kernel.

The object of this paper is the development of a theory of biorthogonal systems of functions independent of their connection with integral or differential equations. In the theory frequent use is made of the theorems by Riesz, Fischer, and Toeplitz.

Necessary and sufficient conditions for the existence of the adjoint system {v_{i}} of any system of linearly independent functions {u_{i}} are deduced. Theorems for biorthogonal systems analogous to those of Riesz and Fischer for orthogonal systems are [given].

The equivalence of two biorthogonal systems is defined and a classification into types is made.

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Applications of Biorthogonal Systems of Functions to the Theory of Integral Equations

Transactions of the American Mathematical Society, Vol. 12, No. 2 (April 1911), 165-180

Presented to the Society, September, 1909

Introduction

In this paper we give a sufficient condition that the characteristic numbers of an unsymmetric kernel exist and be real, and prove the expansibility of arbitrary functions in terms of the corresponding characteristic function. This sufficient condition is stated in terms of a functional transformation *T(f)* defined by certain general properties, and for the special case *T(f) = f* we obtain the known theory of the orthogonal integral equations. The method employed is that of infinitely many variables and is based to some extent on an earlier paper.