The Polynomial Approximation Property in Fock-type Spaces

Abstract

where dm2 is Lebesgue area measure. We deal with the following question, posed and discussed in [9] (and earlier in [8]). Do the polynomial multiples of an entire function span (in F , r† ˆ r2) all functions which vanish wherever vanishes? We take into account the multiplicities of zeros, and require that all the polynomial (or even exponential polynomial) multiples of belong to F . In other words, the question is whether the polynomials are dense in the Hilbert space F ; of analytic (entire) functions square summable with respect to the measure j z†j2ey jzj†dm2 z†. Let us remark here that the general problem of weighted polynomial approximation for weights vanishing at some inner points of the domain of analyticity is not yet sufficiently investigated. For some classes of we construct for which the answer to our question is negative. The method used is close to that in [2,3]. The idea is to produce an element of F having extremal growth along a sufficiently massive subset of the plane. Then a Phragmen^Lindelo« f type argument shows that if polynomial multiples pn converge to an element in F , then the polynomials pn cannot be large enough to approximate all the elements in F ; . To use this MATH. SCAND. 82 (1998), 256^264