On a space of smooth functions on a convex unbounded set in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> admitting holomorphic extension in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ? ⁿ . Due to the conditions on M each function of this space admits a holomorphic extension in ? ⁿ . In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L ²-bounds in a domain of holomorphy in ? ⁿ are obtained with the aid of L. Hörmander’s method of L ²-bounds for the \(\bar \partial\) operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.     

Hypersensitive transitions of Tm3+, Ho3+ and Dy3+ rare-earth ions in garnet crystals

In this work, spectroscopic characteristics (oscillator strengths, intensity parameters) of intermanifold f–f transitions of Tm^{3 +}, Ho³⁺and Dy³⁺ions in garnet crystals are investigated. It has been found that the intensity change of the hypersensitive intermanifold f–f transitions of rare-earth ions in garnet crystals is determined by the characteristics of their local environment.