The analytic properties of these forms and the associated operators have been studied extensively, with much attention given to the relationship between the properties of B and B, and those of b. The idea of bilinear forms given by a representation such as (1-1), and thus only depending on the product of the arguments, can certainly be extended to other function spaces. One investigation of those more general forms is in [Janson et al. 1987]. In the more general contexts operators based on expressions such as (1-1) are sometimes called small Hankel operators. There is another generalization, the large Hankel operators] the two types agree for the Hardy space. Recently there has also been consideration of more general classes, the Hankel forms of higher type or order. For each nonnegative integer n there is a class, iJn, of Hankel forms of type n. The elements of H1 are the traditional Hankel forms, and Hn ⊂ Hn+1 for each n.

The characteristic property of such a form is that for any polynomial, p, the new bilinear form Cp(f,g) = E(pf,g) — E(f,pg) is a Hankel form.

Thus such forms are obtained by perturbing Hankel forms in a controlled way. Higher-order forms were introduced in [Janson and Peetre 1987].