Let (mn)n=1∞ be an increasing
divergent sequence of positive numbers. Then we are interested in characterising
those sequences (αn)n=1∞ for which αn=∫01xmnf(x)dx for n = 1,2,⋯ and some
f ∈ L2([0,1)). It is shown that if (mn)n=1∞ diverges sufficiently rapidly, then
∑n−1∞|αn|2< ∞ if and only if an=∫01xmnf(x)dx for n = 1,2,⋯ and some
f ∈ L2([0,1)). It is also shown that if (mn)n=1∞ is a lacunary sequence of integers
then the Hilbert subspace of L2([0,1)) generated by the functions xmn(n = 1,2,⋯)
has a reproducing kernel.
