Is there a recently published book on the Classical Moment Problems and related theory?
I have seen a couple of old books by Tamarkin and a few other books by Russian authors. Want to know what else can be a good reference.
The classic reference and still one of the best places to learn about this stuff is Akhiezer's book The classical moment problem and some related problems in analysis, (English translation 1965). This isn't a recently published book, but it is in my view the gold standard.
A much more recent book Unbounded selfadjoint operators on Hilbert spaces by K Schmudgen (Springer 2012) has a whole chapter on the classical moment problem. However, this chapter refers quite often to Akhiezer and to Barry Simon's article mention by Alexandre Eremenko.
Another beautiful source, though much older, is Widder's classic The Laplace Transform. The book has a chapter on the moment problem, but the connection with selfadjoint operators is not touched. It contains however beautiful morsels of classical real analysis I could not find anywhere else.
There is a book-size paper: MR1627806 Simon, Barry The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137 (1998), no. 1, 82–203.