Characterizations of the Gelfand-Shilov spaces via Fourier transforms

Abstract

Abstract. We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type S and type W . These results explain more clearly the invariance of these spaces under the Fourier transformations.

Ornstein-Uhlenbeck Semigroup on the dual space of Gelfand-Shilov Spaces of Beurling type

“…In [1], J. Chung et al proved symmetric characterizations for Gelfand-Shilov spaces via the Fourier transform in terms of the growth of the function and its Fourier transform which imposes no conditions on the derivative.…”

“…In this paper, we use the characterization of Gelfand-Shilov spaces of Beurling type of test functions of tempered ultradistribution in terms of their Fourier transform obtained in [1] to prove structure theorem for functionals in dual space (Σ β α ) ′ . Using the structure theorem of the dual space(Σ β α ) ′ equipped with the weak topology, we study the action of Ornstein-Uhlenbeck semigroup on the dual space (Σ β α ) ′ .…”

Wavelet transforms on Gelfand-Shilov spaces and concrete examples

“…For the proof refer to [ 6 , 12 ], etc. Taking Lemma 2.1 into account, we denote another Banach Gelfand-Shilov space combining with the infinite vanishing moment condition , We remark that corresponds to with , i.e., …”

A note on holomorphic functions and the Fourier-Laplace transform

“…We mention that the construction of M is reminiscent of the definition of the (larger) space of temperate ultra-distributions of J. Sebastião e Silva [18], as well as the distributions of exponential growth, see Hasumi [7] or Yoshinaga [22]. In contrast, Gelfand-Shilov spaces [6] are (in analogy with S ) characterized by symmetric conditions with respect to the Fourier transform, see Chung, Chung and Kim [2]. For an alternative treatment of Fourier transforms of arbitrary distributions, see Ehrenpreis [4].…”