The Annals of Probability

Multivariable approximate Carleman-type theorems for complex measures

Isabelle Chalendar and Jonathan R. Partington

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Abstract

We prove a multivariable approximate Carleman theorem on the determination of complex measures on ℝn and ℝn+ by their moments. This is achieved by means of a multivariable Denjoy–Carleman maximum principle for quasi-analytic functions of several variables. As an application, we obtain a discrete Phragmén–Lindelöf-type theorem for analytic functions on ℂ+n.

Article information

Source
Ann. Probab., Volume 35, Number 1 (2007), 384-396.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1174324134

Digital Object Identifier
doi:10.1214/009117906000000377

Mathematical Reviews number (MathSciNet)
MR2303955

Zentralblatt MATH identifier
1120.26027

Subjects
Primary: 26E10: $C^\infty$-functions, quasi-analytic functions [See also 58C25] 44A60: Moment problems
Secondary: 32A22: Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30} 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords
Moments of measures on ℝn functions of exponential type Denjoy–Carleman maximum principle Phragmen–Lindelof theorems

Citation

Chalendar, Isabelle; Partington, Jonathan R. Multivariable approximate Carleman-type theorems for complex measures. Ann. Probab. 35 (2007), no. 1, 384--396. doi:10.1214/009117906000000377. https://projecteuclid.org/euclid.aop/1174324134


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References

  • Boas, R. P. (1954). Entire Functions. Academic Press, New York.
  • Chalendar, I., Habsieger, L., Partington, J. R. and Ransford, T. J. (2004). Approximate Carleman theorems and a Denjoy--Carleman maximum principle. Arch. Math. (Basel) 83 88--96.
  • de Jeu, M. (2003). Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Probab. 31 1205--1227.
  • de Jeu, M. (2004). Subspaces with equal closure. Constr. Approx. 20 93--157.
  • Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Springer, Berlin.
  • Hryptun, V. G. (1976). An addition to a theorem of S. Mandelbrojt. Ukrain. Mat. Ž. 28 849--853, 864.
  • Koosis, P. (1998). The Logarithmic Integral. I. Cambridge Univ. Press.
  • Korenblum, B., Mascuilli, A. and Panariello, J. (1998). A generalization of Carleman's uniqueness theorem and a discrete Phragmén--Lindelöf theorem. Proc. Amer. Math. Soc. 126 2025--2032.
  • Mikusiński, J. G. (1951). Remarks on the moment problem and a theorem of Picone. Colloquium Math. 2 138--141.
  • Musin, I. Kh. (1994). On the Fourier--Laplace representation of analytic functions in tube domains. Collect. Math. 45 301--308.
  • Ronkin, L. I. (1974). Introduction to the Theory of Entire Functions of Several Variables. Amer. Math. Soc., Providence, RI.