On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one 4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in 6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .