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On a variational approach to truncated problems of moments. (English) Zbl 1274.44010

The author considers a problem of finding an a.e. non-negative function \(f\) on a closed set \(T\subset \mathbb R^n\) such that \(\int _T | t^i| f(t) dt < \infty \) and \(\int _T t^i f(t) dt = g_i\), \(i\in I\), where \((g_i)_{i\in I}\) is a prescribed set of numbers, \(g_0 = 1\) and \(I\subset \mathbb Z^n_+\). Under some conditions on \(T\) and \(I\), the existence of \(f\) is shown to be equivalent to the condition that a functional \(L\) (Lagrangian) is bounded from above and \(\sup L\) is attained at a (unique) point.

MSC:

44A60 Moment problems
30E05 Moment problems and interpolation problems in the complex plane
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