In the mathematical field of functional analysis, a Gelfand–Shilov space S α β {\displaystyle S_{\alpha }^{\beta }} is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov (1968, Chapter IV). The space S α β {\displaystyle S_{\alpha }^{\beta }} is characterized as the space of smooth functions f : R n → C {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {C} } such that there exist constants A , B , C {\displaystyle A,B,C} such that, for every pair of multi-indices i , j {\displaystyle i,j} and every x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , the inequality
| x i ∂ j f ( x ) | ≤ C A | i | B | j | ( i ! ) α ( j ! ) β . {\displaystyle |x^{i}\partial ^{j}f(x)|\leq CA^{|i|}B^{|j|}(i!)^{\alpha }(j!)^{\beta }.}
The Fourier transform sends S α β {\displaystyle S_{\alpha }^{\beta }} to S β α {\displaystyle S_{\beta }^{\alpha }} .
References
Chung, Jaeyoung; Chung, Soon-Yeong; Kim, Dohan (1996), "Characterizations of the Gel'fand–Shilov spaces via Fourier transforms", Proceedings of the American Mathematical Society, 124 (7): 2101–2108, doi:10.1090/S0002-9939-96-03291-1, ISSN 0002-9939, MR 1322917 Gelfand, I. M.; Shilov, G. E. (1968) [1958], Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, vol. 2, Boston, MA: Academic Press, MR 0230128