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
;}
The Jones Polynomial and Functions of Positive Type on the Oriented Jones–Thompson Groups
$$\vec {F}$$
F
→
and
$$\vec {T}$$
T
→
