Multivariable approximate Carleman-type theorems for complex measures. (English) Zbl 1120.26027

A multivariable Denjoy-Carleman maximum principle is obtained by means of a multivariable version of Bernstein’s inequality for functions of exponential type. On the base of these ideas a multivariable approximate Carleman theorem on \({\mathbb R}^n\) is proved. Such theorem holds even in the case of complex measures. At the proof a variant of the Paley-Wiener-Schwartz theorem for Fourier transforms of distributions is used. An analogous result is obtained for the complex measures supported on the half-space \({\mathbb R}^n_{+}\).


26E10 \(C^\infty\)-functions, quasi-analytic functions
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A60 Moment problems
Full Text: DOI arXiv


[1] Boas, R. P. (1954). Entire Functions . Academic Press, New York. · Zbl 0058.30201
[2] 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. · Zbl 1056.26017
[3] de Jeu, M. (2003). Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Probab. 31 1205–1227. · Zbl 1050.44003
[4] de Jeu, M. (2004). Subspaces with equal closure. Constr. Approx. 20 93–157. · Zbl 1134.41337
[5] Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis . Springer, Berlin. · Zbl 0521.35001
[6] Hryptun, V. G. (1976). An addition to a theorem of S. Mandelbrojt. Ukrain. Mat. Ž. 28 849–853, 864.
[7] Koosis, P. (1998). The Logarithmic Integral. I. Cambridge Univ. Press. · Zbl 0931.30001
[8] 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. JSTOR: · Zbl 0895.30025
[9] Mikusiński, J. G. (1951). Remarks on the moment problem and a theorem of Picone. Colloquium Math. 2 138–141. · Zbl 0044.32302
[10] Musin, I. Kh. (1994). On the Fourier–Laplace representation of analytic functions in tube domains. Collect. Math. 45 301–308. · Zbl 0823.32001
[11] Ronkin, L. I. (1974). Introduction to the Theory of Entire Functions of Several Variables . Amer. Math. Soc., Providence, RI. · Zbl 0286.32004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.