A class of 𝑃^{𝑡}(𝑑𝜇) spaces whose point evaluations vary with 𝑡

Akeroyd, John; Saleeby, Elias

doi:10.1090/S0002-9939-99-04617-1

A class of $P^t(d\mu )$ spaces whose point evaluations vary with $t$
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by John Akeroyd and Elias G. Saleeby PDF

Proc. Amer. Math. Soc. 127 (1999), 537-542 Request permission

Abstract:

Extending an example given by T. Kriete, we develop a class of measures each of which consists of a measure on $\{z: |z|{}=1\}$ along with a series of weighted point masses in $\mathbf {D:=}\{z: |z|{}<1\}$. This class provides relatively simple examples of measures $\mu$ which have the property that the collection of analytic bounded point evaluations for $P^{t}(d\mu )$ varies with $t$. The first known measures with this property were recently constructed by J. Thomson.

References

John Akeroyd, An extension of Szegő’s theorem. II, Indiana Univ. Math. J. 45 (1996), no. 1, 241–252. MR 1406692, DOI 10.1512/iumj.1996.45.1195
Thomas L. Kriete III, Cosubnormal dilation semigroups on Bergman spaces, J. Operator Theory 17 (1987), no. 2, 191–200. MR 887217
Thomas L. Kriete III and Henry Crawford Rhaly Jr., Translation semigroups on reproducing kernel Hilbert spaces, J. Operator Theory 17 (1987), no. 1, 33–83. MR 873462
James E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477–507. MR 1109351, DOI 10.2307/2944317
James E. Thomson, Bounded point evaluations and polynomial approximation, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1757–1761. MR 1242106, DOI 10.1090/S0002-9939-1995-1242106-1