• Title/Summary/Keyword: Schwartz space

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BOCHNER-SCHWARTZ THEOREM ON LOCALLY COMPACT ABELIAN GROUPS

  • Kim, Jin-Man;Cho, Jong-Gyu
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.7-16
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    • 2001
  • We study the Fourier transformation on the Gelfand-Bruhat space of type S and characterize this space by means of Fourier transform on a locally compact abelian group G. Also, we extend Bochner-Schwartz theorem to the dual space of the Gelfand-Bruhat space and the space of Fourier hyperfunctions on G. respectively.

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REAL VERSION OF PALEY-WIENER-SCHWARTZ THEOREM FOR ULTRADISTRIBUTIONS WITH ULTRADIFFERENTIABLE SINGULAR SUPPORT

  • Cho, Jong-Gyu;Kim, Kwang-Whoi
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.3
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    • pp.483-493
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    • 1999
  • We extend the Paley-Wiener-Schwartz theorem to the space of ultradistributions with respect to ultradifferentiable singular support and obtain its real version. That is, we obtain the growth condition in some tubular neighborhood of n of the Fourier transform of ultradistributions of Roumieu (or Beurling) type with ultradifferentiable singular support contained in a ball centered at the origin, and its real version.

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HYERS-ULAM-RASSIAS STABILITY OF QUADRATIC FUNCTIONAL EQUATION IN THE SPACE OF SCHWARTZ TEMPERED DISTRIBUTIONS

  • CHUNG JAEYOUNG
    • The Pure and Applied Mathematics
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    • v.12 no.2 s.28
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    • pp.133-142
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    • 2005
  • Generalizing the Cauchy-Rassias inequality in [Th. M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.] we consider a stability problem of quadratic functional equation in the spaces of generalized functions such as the Schwartz tempered distributions and Sato hyperfunctions.

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WHITE NOISE HYPERFUNCTIONS

  • Chung, Soon-Yeong;Lee, Eun-Gu
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.329-336
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    • 1999
  • We construct the Gelfand triple based on the space \ulcorner, introduced by Sato and di Silva, of analytic and exponentially decreasing function. This space denoted by(\ulcorner) of white noise test functionals are defined by the operator cosh \ulcorner, A=-(\ulcorner)\ulcorner+x\ulcorner+1. We also note that many properties like generalizations of the Paley-Wiener theorem and the Bochner-Schwartz theorem hold in this space as in the space of Hida distributions.

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BOEHMIANS ON THE TORUS

  • Nemzer, Dennis
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.831-839
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    • 2006
  • By relaxing the requirements for a sequence of functions to be a delta sequence, a space of Boehmians on the torus ${\beta}(T^d)$ is constructed and studied. The space ${\beta}(T^d)$ contains the space of distributions as well as the space of hyperfunctions on the torus. The Fourier transform is a continuous mapping from ${\beta}(T^d)$ onto a subspace of Schwartz distributions. The range of the Fourier transform is characterized. A necessary and sufficient condition for a sequence of Boehmians to converge is that the corresponding sequence of Fourier transforms converges in $D'({\mathbb{R}}^d)$.

Suboptimal Homing Guidance Law by Synthesis of the Aided Loop for Impact Angle Constraint (충돌각 구속조건을 위한 보조루프 합성을 통한 준최적 호밍 유도법칙)

  • Lee, Jin-Ik
    • Journal of the Korean Society for Aeronautical & Space Sciences
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    • v.35 no.11
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    • pp.1006-1012
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    • 2007
  • In this paper, a suboptimal homing guidance law for the homing missiles with an impact angle constraint is presented. Unlike general LQ optimal control, the aided loop ensuring some degrees of freedom for the constraint is introduced. Then an optimal feedback loop in consideration of the aided loop is designed by using Schwartz inequality. The aided loop is synthesized with the optimal control to produce the guidance command. Furthermore, to investigate the characteristics of the guidance law we carry out the comparative studies with other guidance laws. The results of the various computer simulations show the good performance of the proposed law.

WICK DERIVATIONS ON WHITE NOISE FUNCTIONALS

  • Chung, Dong-Myung;Chung, Tae-Su
    • Journal of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.993-1008
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    • 1996
  • The white noise analysis, initiated by Hida [3] in 1975, has been developed to an infinite dimensional distribution theory on Gaussian space $(E^*, \mu)$ as an infinite dimensional analogue of Schwartz distribution theory on Euclidean space with Legesgue measure. The mathematical framework of white noise analysis is the Gel'fand triple $(E) \subset (L^2) \subset (E)^*$ over $(E^*, \mu)$ where $\mu$ is the standard Gaussian measure associated with a Gel'fand triple $E \subset H \subset E^*$.

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THE PRODUCT OF ANALYTIC FUNCTIONALS IN Z'

  • Li, Chenkuan;Zhang, Yang;Aguirre, Manuel;Tang, Ricky
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.455-466
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    • 2008
  • Current studies on products of analytic functionals have been based on applying convolution products in D' and the Fourier exchange formula. There are very few results directly computed from the ultradistribution space Z'. The goal of this paper is to introduce a definition for the product of analytic functionals and construct a new multiplier space $F(N_m)$ for $\delta^{(m)}(s)$ in a one or multiple dimension space, where Nm may contain functions without compact support. Several examples of the products are presented using the Cauchy integral formula and the multiplier space, including the fractional derivative of the delta function $\delta^{(\alpha)}(s)$ for $\alpha>0$.