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- the classical moment problem and some related questions in analysis
$$ x p_n(x) = a_n p_n+1(x) + b_n p_n(x) + a_n-1 p_n-1(x). $$
This analytic function has a power series expansion $S(z) = -\sum_n=0^\infty \fracm_nz^n+1$ for large $|z|$. The moment problem asks: Given the asymptotic expansion, can we recover the analytic function $S(z)$ uniquely? This becomes a problem of analytic continuation and branch cuts. Indeterminacy means there are multiple analytic functions with the same asymptotic series—these differ by a "Nevanlinna" function.
$$ H_n = \beginpmatrix s_0 & s_1 & \dots & s_n \ s_1 & s_2 & \dots & s_n+1 \ \vdots & \vdots & \ddots & \vdots \ s_n & s_n+1 & \dots & s_2n \endpmatrix \geq 0 $$
The implications of the moment problem permeate various corners of analysis, creating a web of related questions that validate its status as a central theme.
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