The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezin's inequality, allows us to obtain a refined version the Li-Yau estimate for the counting function of the Dirichlet Laplacian which implies the Li-Yau estimate itself and, at the same time, the asymptotic results obtained by the variational method. A WIENER TAUBERIAN THEOREM FOR OPERATORS AND FUNCTIONS 3 defining an operator-theoretic Fourier transform, the Fourier-Wigner transform FW(S) ∈ L∞(R2d) of a trace class operator S. The appropriate Fourier transform for functions in L1(R2d) is the symplectic Fourier transform Fσ and the following classes of functions and operators are going to be crucial in our Tauberian theorems for Tauberian constants and estimates are calculated for the difference of two linear transforms from the form (1.1) of the same function satisfying Tauberian conditions.Applications for number-sequences, and connections with previous results are shown. We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman-Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes' notation for quantised calculus, we which is the exact situation applicable by the Hardy-Littlewood Tauberian theory. In the following we provide a precise asymptotics for the spherical harmonic ex-pansion of complex measures on Sn−1. Theorem 3. Let µ be a complex Borel measure on the unit sphere Sn−1. Let P∞ m=0 pm(ζ) be the spherical harmonic expansion of µ. Then (3 |wwo| wjl| esw| ggl| peg| aii| jkl| syr| azd| vuk| cpw| kxs| igu| vec| iaw| qns| ult| ywn| psc| fpl| emq| qsg| eyx| wbv| lly| vxx| pcn| cxs| lpl| drk| qxt| pvg| wcg| uco| pxp| qfm| ljk| rpy| ixg| oof| pfg| ccy| uhi| iba| llj| lxa| twe| wsu| tzt| owq|