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On the non-triviality of certain spaces of analytic functions : hyperfunctions and ultrahyperfunctions of fast growth

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Abstract
We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type and Roumieu type are non-trivial if and only if sup p >= 2 (log p)(p)/h(p)M(p) < infinity, for all h > 0 and some h > 0, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.
Keywords
:Spaces of analytic functions, Hyperfunctions, Ultrahyperfunctions, Ultradistributions, Boundary values, Analytic representations, Non-triviality, Laplace transform, Gelfand-Shilov spaces, ULTRA-HYPERFUNCTIONS, ULTRADISTRIBUTIONS, OPERATORS, THEOREM

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Chicago
Debrouwere, Andreas, and Jasson Vindas Diaz. 2018. “On the Non-triviality of Certain Spaces of Analytic Functions : Hyperfunctions and Ultrahyperfunctions of Fast Growth.” Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas 112 (2): 473–508.
APA
Debrouwere, A., & Vindas Diaz, J. (2018). On the non-triviality of certain spaces of analytic functions : hyperfunctions and ultrahyperfunctions of fast growth. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, 112(2), 473–508.
Vancouver
1.
Debrouwere A, Vindas Diaz J. On the non-triviality of certain spaces of analytic functions : hyperfunctions and ultrahyperfunctions of fast growth. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS. 2018;112(2):473–508.
MLA
Debrouwere, Andreas, and Jasson Vindas Diaz. “On the Non-triviality of Certain Spaces of Analytic Functions : Hyperfunctions and Ultrahyperfunctions of Fast Growth.” REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 112.2 (2018): 473–508. Print.
@article{8558880,
  abstract     = {We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type and Roumieu type are non-trivial if and only if 

sup p {\textrangle}= 2 (log p)(p)/h(p)M(p) {\textlangle} infinity, 

for all h {\textrangle} 0 and some h {\textrangle} 0, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.},
  author       = {Debrouwere, Andreas and Vindas Diaz, Jasson},
  issn         = {1578-7303},
  journal      = {REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS},
  language     = {eng},
  number       = {2},
  pages        = {473--508},
  title        = {On the non-triviality of certain spaces of analytic functions : hyperfunctions and ultrahyperfunctions of fast growth},
  url          = {http://dx.doi.org/10.1007/s13398-017-0392-9},
  volume       = {112},
  year         = {2018},
}

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