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On Borel summability and analytic functionals

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Abstract
We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function $f$ is of exponential type if and only if the formal power series $\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$ is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon.
Keywords
Analytic functionals, Entire functions of exponential type, Borel summability, Borel polygon, Silva tempered ultradistributions

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Citation

Please use this url to cite or link to this publication:

Chicago
Estrada, Ricardo, and Jasson Vindas Diaz. 2013. “On Borel Summability and Analytic Functionals.” Rocky Mountain Journal of Mathematics 43 (3): 895–903.
APA
Estrada, R., & Vindas Diaz, J. (2013). On Borel summability and analytic functionals. ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 43(3), 895–903.
Vancouver
1.
Estrada R, Vindas Diaz J. On Borel summability and analytic functionals. ROCKY MOUNTAIN JOURNAL OF MATHEMATICS. 2013;43(3):895–903.
MLA
Estrada, Ricardo, and Jasson Vindas Diaz. “On Borel Summability and Analytic Functionals.” ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 43.3 (2013): 895–903. Print.
@article{4115876,
  abstract     = {We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function \$f\$ is of exponential type if and only if the formal power series \${\textbackslash}sum\_\{n=0\}\^{ }\{{\textbackslash}infty\}f\^{ }\{(n)\}(0)z\^{ }\{n\}\$ is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon.},
  author       = {Estrada, Ricardo and Vindas Diaz, Jasson},
  issn         = {0035-7596},
  journal      = {ROCKY MOUNTAIN JOURNAL OF MATHEMATICS},
  keyword      = {Analytic functionals,Entire functions of exponential type,Borel summability,Borel polygon,Silva tempered ultradistributions},
  language     = {eng},
  number       = {3},
  pages        = {895--903},
  title        = {On Borel summability and analytic functionals},
  url          = {http://dx.doi.org/10.1216/RMJ-2013-43-3-895},
  volume       = {43},
  year         = {2013},
}

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