Gel’fond-Leont’ev generalized differentiation operators in spaces of type S

Еволюційні Псевдодиференціальні Рівняння З Аналітичними Символами В Просторах Типу $S$

“…For example, if b n = n nβ , 0 < β < 1, then ρ(y) ∼ exp{|y| 1/β }. In addition, as proved in [7], ln ρ is a convex function on (0, +∞) in the sense…”

“…In the monograph [4], the case a k = k kα , α > 0, b n = n nβ , β > 0, was studied in detail; the corresponding spaces are called S spaces and are denoted by S β α . In [7], topological structure of the spaces S b n a k , properties of functions, basic operations in such spaces were studied. Known spaces of the W type, introduced by B.L.…”

“…In [7], it was established that a function ϕ ∈ C ∞ (R) is an element of the space S b n a k with a k = k!d k , b n = n!ρ n if and only if it analytically extends into the complex plane to the entire function ϕ(z), z ∈ C, which satisfies the condition…”

“…Note that ρ is a continuously differentiable even function on R that is monotonically increasing on the interval [1, +∞). It follows from property d) (see [7]) that…”

“…From the results given in [7] it follows that the sequence {ϕ ν : ν ≥ 1} ⊂ S b n a k converges to zero in this space if the functions ϕ ν and their derivatives of arbitrary order uniformly converge to zero on each segment [a, b] ⊂ R and the following inequalities are satisfied…”