Gelfand–Shilov space

In the mathematical field of functional analysis, a Gelfand–Shilov space $S_{\alpha }^{\beta }$ is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov (1968, Chapter IV).

The space $S_{\alpha }^{\beta }$ is characterized as the space of smooth functions $f:\mathbb {R} ^{n}\to \mathbb {C}$ such that there exist constants $A,B,C$ such that, for every pair of multi-indices $i,j$ and every $x\in \mathbb {R} ^{n}$, the inequality $$|x^{i}\partial ^{j}f(x)|\leq CA^{|i|}B^{|j|}(i!)^{\alpha }(j!)^{\beta }.$$ The Fourier transform sends $S_{\alpha }^{\beta }$ to $S_{\beta }^{\alpha }$.

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