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I have been reading about moment problem and I have been curious about the following question.

What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?

I have been asking myself out of curiosity, what is gained by people who considered this case? or let me say how can I explain my reasons to someone who has little knowledge about the subject the reasons for for studying hamburger moment problem.

I will be more than happy if useful literatures can be suggested or provided with some explanations.

Thanks for your useful suggestions and explanations.

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It has many applications, especially in probability: e.g. you want to know if there is a probability distribution satisfying some condition on its moments. Or you know the moments of a random variable match those of a certain distribution, and you want to know if that implies this is the distribution of the random variable.

I'm not claiming this as a major motivation, but one of my papers happens to contain an application of the Hamburger moment problem to mathematical physics: a characterization of reflection positivity for pair interactions in one-dimensional lattice systems (see Prop. 3.2).

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