Abstract. This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in H ∞ , BMOA and the Bloch space are discussed. A counterpart of the Hardy-Stein-Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.
Abstract. Sufficient conditions for solutions ofand their derivatives to be in H ∞ ω (D) are given by limiting the growth of coefficientsconsists of those analytic functions f in a domain D for which |f (z)|ω(z) is uniformly bounded. In particular, the case where D is the unit disc is considered. The theorems obtained generalize and improve certain results in the literature. Moreover, by using one of the main results, one can give a straightforward proof of a classical result regarding the situation where the coefficients are polynomials.
We consider the spectral spatial coherence characteristics of scalar light fields in second-harmonic generation in an optically non-linear medium. Specifically, we take the fundamental-frequency (incident) field to be a Gaussian Schell-model (GSM) beam with variable peak spectral density and different coherence properties. We show that with increasing intensity the overall degree of coherence of both the fundamental and the second-harmonic field in general decreases on passage through the non-linear medium. In addition, the spectral density distributions and the two-point degree of coherence may, for both beams, deviate significantly from those of the GSM, especially at high intensities. Propagation in the non-linear medium is numerically analyzed with the Runge–Kutta and the beam-propagation methods, of which the latter is found to be considerably faster. The results of this work provide means to synthesize, via non-linear material interaction, random optical beams with desired coherence characteristics.
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