The use of the word “moment” in mathematics is related to its use in physics, as in moment arm or moment of inertia. For a non-negative integer n, the nth moment of a function f is the integral of xn f(x) over the function’s domain.
Uniqueness
If two continuous functions f and g have all the same moments, are they the same function? The answer is yes for functions over a finite interval, but no for functions over an unbounded interval.
Existence
Now let’s consider starting with a set of moments rather than starting with a function. Given a set of moments m0, m1, m2, … is there a function that has these moments? Typically no.
A better question is what conditions on the moments are necessary for there to exist a function with these moments. This question breaks into three questions
- The Hausdorff moment problem
- The Stieltjes moment problem
- The Hamburger moment problem
depending on whether the function domain is a finite interval, a half-bounded interval, or the real line. For each problem there are known conditions that are necessary and sufficient, but the conditions are different for each problem.
Interestingly, each of the three names Hausdorff, Stieltjes, and Hamburger are well known. Felix Hausdorff is best known for his work in topology: Hausdorff spaces, etc. Thomas Stieltjes is best known for the Riemann-Stieltjes integral, and for his work on continued fractions. Hans Ludwig Hamburger is not as well known, though his last name is certainly familiar.
Finite moments
A practical question in probability is how well a finite number of moments determine a probability distribution. They cannot uniquely determine the distribution, but the do establish bounds for how different the two distributions can be. See this post.
I think the Hausdorff, Stieltjes, and Hamburger moment problems are a little bit different from what you describe.
(1) They’re about _measures_ rather than _functions_; e.g., suppose our moment sequence is (1,1,1,…); then the usually-given conditions for all three problems are satisfied and “a solution exists”, but it’s “f(x) = delta(x-1)” — i.e., the measure that gives measure 1 to {1} and 0 to everything else — which isn’t actually a function.
(2) They’re about _measures_ rather than _signed measures_ — i.e., if we pretend they’re functions then the functions are required to be nonnegative. E.g., if we took “f(x) = delta(x-1) – delta(x+1)”, an infinite spike of area +1 at x=1 and one of area -1 at x=-1, then we would get the moment sequence (2,0,2,0,2,0,…), which doesn’t satisfy the conditions for any of those standard moment problems. Or, if you want an actual function, suppose we take f(x) = -1 on (0,1), f(x) = +1 on (1,2), and f(x) = 0 everywhere else; then the n’th moment is (2^(n+1)-2)/(n+1) so the sequence is (0, 1, 2, 3.5, 6, …) which doesn’t satisfy the usually-given conditions for any of the problems.
I’m not sure what happens if we restrict ourselves only to functions, or allow signed measures, or do both at once so that we allow only functions but let them have negative values. I vaguely think that for the Hamburger case _every_ sequence of real numbers is the measure-sequence for some signed measure. Or something.