This paper solves the point mass problem on the real line when the recurrence
coefficients are asymptotically periodic. First, we give formulae for the
perturbed orthogonal polynomials and the perturbed recurrence coefficients when
a point mass is added to any non-trivial measure supported on the real line.
Then we analyze the recurrence relation and prove new asymptotic formulae for
the orthogonal polynomials associated to a measure with asymptotically periodic
coefficients. Finally, we apply those formula to solve the point mass problem
on the real line when the recurrence coefficients are asymptotically periodic.
The method introduced allows us to consider the point mass problem with a
finite number point masses using an inductive argument.