On the extensions of positive definite functions

I n a beaut i fu l p a p e r [13], 1~I. Kre in has given a pene t r a t i ng analys is of a general p rob lem of momen t s (see also [17]). This p a p e r is the cu lmina t ion of a series of notes and papers b y K r e i n s t re tch ing over a lmos t a decade (see [13] for a bibl iog raphy) . His resul ts a p p e a r in the form of a t h e o r y a b o u t a special class of symmet r i c opera to r s on a Hi ]ber t space (see also [16]). The p r o t o t y p e of an opera to r in th is class m a y be found in the t h e o r y of the classical H a m b u r g e r m o m e n t problem. The genera l p rob lem of momen t s which can be t r e a t e d b y K re in ' s me thods is concerned wi th condi t ions on a p re -Hi ]be r t space E of ana ly t i c funct ions of a single real va r iab le for which there exis ts a measure d/~ (t)~>0 so t h a t if /, g E C, then