Linear transformations and functions of positive type

In a communication received by the Society on April 12 last I gave in outline a proof that a symmetrisable function k ( s, t ), which belongs to one of certain classes, admits an expansion in terms of its bi-orthogonal fundamental functions. Two classes were explicitly mentioned, viz., (i) the functions defined by k ( s, t ) = α( s ) ϒ ( s, t ), and (ii) those defined by k ( s, t ) = ∫ a b α( s, x ) ϒ ( x, t ) dx . It was assumed in each case that ϒ( s, t ) was of positive type in the square a ≤ s ≤ b , a ≤ t ≤ b ; in the first case, that α( s ) was a function defined in the interval ( a, b ); and, in the second case, that α ( s, t ) was a symmetric function defined in the square a ≤ s ≤ b , a ≤ t ≤ b . When k ( s, t ) belongs to either of the classes mentioned and certain conditions of continuity are satisfied it was shown that k ( s, t ) = Σ n=1 ⌽ n ( s ) Ψ n ( t )/λ n + Σ n=1 ξ n ( s ) η n ( t )/ v n , where (i) ⌽ 1 ( s ), ⌽ 2 ( s ), ... ⌽ n ( s ), ... Ψ 1 ( s ), Ψ 2 ( s ), ... Ψ n ( s ), ... is a bi-orthogonal system of fundamental functions for the interval ( a, b ), and ⌽ n ( s ), Ψ n ( t ) satisfy the homogeneous equations ⌽ n ( s ) = λ n ∫ a b k ( s, t ) ⌽ n ( t ) dt , Ψ n ( t ) = λ n ∫ a b Ψ n ( s ) k ( s, t ) ds ;}