We present a characterization of sets for which Cartwrights theorem holds true. The connection is discussed between these sets and sampling sets for entire functions of exponential type.
In this paper, we prove a $Tb$ theorem on product spaces $Bbb R^ntimes Bbb R^m$, where $b(x_1,x_2)=b_1(x_1)b_2(x_2)$, $b_1$ and $b_2$ are para-accretive functions on $Bbb R^n$ and $Bbb R^m$, respectively.
The classical Szeg{o}--Kolmogorov Prediction Theorem gives necessary and sufficient condition on a weight $w$ on the unite cirlce $T$ so that the exponentials with positive integer frequences span the weighted space $L^2(T,w)$. We consider the proble
m how many of these exponentials can be removed while still keeping the completeness property.
We characterize even measures $mu=wdx+mu_s$ on the real line with finite entropy integral $int_{R} frac{log w(t)}{1+t^2}dt>-infty$ in terms of $2times 2$ Hamiltonian generated by $mu$ in the sense of inverse spectral theory. As a corollary, we obtain
criterion for spectral measure of Krein string to have converging logarithmic integral.
In this paper, we provide a non-homogeneous $T(1)$ theorem on product spaces $(X_1 times X_2, rho_1 times rho_2, mu_1 times mu_2)$ equipped with a quasimetric $rho_1 times rho_2$ and a Borel measure $mu_1 times mu_2$, which, need not be doubling but
satisfies an upper control on the size of quasiballs.
In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification result related to normal numbers.