On the multilinear Hausdorff problem of moments

Given a multi-index sequence mu k , k=(k 1 ,...,k n )∈N n 0 , necessary and sufficient conditions are given for the existence of a regular Borel polymeasure gamma on the unit interval I=[0,1] such that mu k =∫ I n t k 1 1 ⊗⋯⊗t k n n gamma . This problem will be called the weak multilinear Hausdorff problemof moments for mu k . Comparison with classical results will allow us to relate the weak multilinear Hausdorff problem with the multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff problem of moments will be provided by exhibiting necessary and sufficient conditions for the existence of a Radon measure mu on [0,1] such that L mu (f 1 ,...,f n )=∫ I f 1 (t)⋯f n (t)mu(dt) where L mu is the n -linear moment functional on the space of continuous functions on the unit interval defined by the sequence mu k . Finally the previous results will be used to provide a characterization of a class of weakly harmonizable stochastic processes with bimeasures supported on compact sets.

On the multilinear Hausdorff problem of moments Articles