I understand that for any sequence of reals there is an analytic interpolation, and for any positive sequence there is an analytic interpolation that is positive. I am wondering if there is a technique to construct analytic interpolations that preserve monotonicity, perhaps by mitigating spurious oscillations in more obvious interpolations by adding appropriate analytic correction terms.
A weaker question is whether for any convergent sequence, there is an analytic interpolation whose derivatives all have existing limits that vanish for arbitrarily large arguments. I am having trouble finding material on this kind of subject, I have learned about monotone cubic splines but these are not analytic. Any advice, knowledge, or direction would be very much appreciated!