On Small Complete Sets of Functions

Aizenberg, Lev ; Vidras, Alekos

Canadian journal of mathematics, Tome 52 (2000) no. 1, pp. 3-30

Résumé

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T. Carleman and A. F. Leontiev is proven for the space of holomorphic functions defined on a suitable open strip ${{T}_{\alpha }}\,\subset \,{{\mathbf{C}}^{2}}$ . The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in ${{\mathbf{C}}^{n}}$ .

Détail

DOI : 10.4153/CJM-2000-001-4

Mots-clés : 32A10, 42C30

Bibliographie

[1] [1] Aizenberg, L., The general formof the linear continuous functional in spaces of functions holomorphic in convex domains in n. Soviet Math. Dokl. 7 (1966), 198–202. Google Scholar

[2] [2] Aizenberg, L. and Yuzhakov, A. Integral representations and residues in multidimensional complex analysis. Amer. Math. Soc. (1983). Google Scholar

[3] [3] Berndtsson, B., Zeros of analytic functions in several complex variables. Ark. Mat.16 (1978), 251–262. Google Scholar

[4] [4] Boas, R., Entire functions. Academic Press, 1954. Google Scholar

[5] [5] Carleman, T., Ark. Mat. Astronom. Fys. 17 (1922). Google Scholar

[6] [6] Leontiev, A. F., Completeness of the system ﹛eƛnz﹜ in a closed strip. Soviet Math. Dokl. 4 (1963), 1286–1289. Google Scholar

[7] [7] Leontiev, A. F., On the completness of the system of exponential functions in curvilinear strips. Mat. B. (N. S.) (78) 36 (1955), 555–568. in Russian. Google Scholar

[8] [8] Levin, B. Ja., Distribution of zeros of entire functions. Trans. Math. Monographs 5 , Amer. Math. Soc., Providence, 1964. Google Scholar

[9] [9] Martineau, A., Sur la topologie des espaces des fonctiones holomorphes. Math. Ann. (1963), 62–88. Google Scholar

[10] [10] Ronkin, L., On the real sets of uniqueness for entire functions of several variables and on the completness of the system of functions eiƛ,x. Siberian J. Math. (3) 13 (1972), 638–644. Google Scholar

[11] [11] Rudin, W., Real and Complex Analysis. Mc Graw-Hill, 1974. Google Scholar

[12] [12] Tillman, H. G., Randverteilungen analytischer funktionen und distributionen. Math. Z. 59 (1953), 61–83. Google Scholar

[13] [13] Sekerin, A. B., On the representation of analytic functions of several variables by exponential series. Russian Acad. Sci. Izv. Math. (3) 40 (1993), 503–527. Google Scholar

[14] [14] Tsikh, A., Use of Residues to Compute the Sum of the Squares of the Taylor Coefficients of a Rational Function of Two Variables. Translation of Multidimensional Complex Analysis (eds., Aizenberg, L. et al.), Krasnoyarsk, 1985, 198–209. Google Scholar

[15] [15] Tsikh, A., Multidimensional Residues and Their Applications. Transl. Amer. Math. Soc. 103 (1992). Google Scholar

[16] [16] Vidras, A., Local Residues and discrete sets of uniqueness. Complex Var., to appear. Google Scholar