Some Extreme Forms Defined in Terms of Abelian Groups

On the other hand, no one has yet constructed a sequence of forms for which yn{f) is unbounded, let alone of order n, as n -> oo. We have therefore thought it worthwhile to describe, in some detail, a new class of extreme forms yielding values yn{f) of order n% for suitable large n. More specifically, corresponding to each N = 2 (n = 2, 3, . . .) and to each sequence of integers Ao, Xv . . ., Xn satisfying Ao = 0, Xr 1 ^ Ar_i ^ <*r ( l ^ r ^ »). we construct a positive iV-variable form /(A), which we show to be extreme in most cases. We prove also that, for each N, there is an /U ) satisfying YNU) = ( W * . whence (1.4) 7^ ^ (JiV)* for iV = 2", » ^ 2.