Let be a positive Radon measure on the real line having moments of all orders. We prove that the set of polynomials is note dense in for any , if is indeterminate. If is determinate, then is dense in for , but not necessarily for . The compact convex set of positive Radon measures with same moments as is studied in some details.
Given a finite set X⊆ ℝ we characterize the diagonals of self-adjoint operators with spectrum X. Our result extends the Schur-Horn theorem from a finite-dimensional setting to an infinite-dimensional Hilbert space analogous to Kadison's theorem for orthogonal projections (2002) and the second author's result for operators with three-point spectrum (2013).
We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group that are invariant for changes on null-sets (e.g. measurable...
We show that the maximal operator associated to the family of rectangles in one of whose sides is parallel to for some j,k is bounded on , . We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.