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The divisor of a generalized analytic function. (English. Russian original) Zbl 0914.30038
Math. Notes 61, No. 5, 547-552 (1997); translation from Mat. Zametki 61, No. 5, 655-661 (1997).
Let \({\mathcal O}(\Delta_G)\) be the space of generalized analytic functions in the open big disc \(\Delta_G\), \(\widehat G\subset {\mathbb R},\) in the sense of Arens and Singer. The spectrum Sp \(f\) of \(f\) is the set of all \(a\in \widehat G\cap [0,\infty)\) with non-zero Fourier coefficients \(c_a=\int_G\frac{f(rg)}{\chi^a(rg)} d\sigma(g)\), where \((rg)=s\in\Delta_G\), \(g\in G\), \(0\leq r<1\), \(\chi^a\in\widehat G\), and \(\sigma\) is the normalized Haar measure on \(G\). It is well known that for any given \(s\in\Delta_G\) there is an embedding of the upper half plane into \(\Delta_G\) through \(s\) and the restrictions \(\widetilde f\) of generalized analytic functions \(f\) on its image are analytic (and almost periodic) on the upper half-plane. The author introduces the notion of divisor of a generalized analytic function. Namely, \(\text{div,} f(s)= \text{ord } \widetilde f(s)\) for \(s\in\Delta_G\setminus \{O\}\), and, \(\text{div }f(O) = \inf\{\text{Sp } f\}\). A function \(g \in{\mathcal O}(\Delta_G)\) is said to divide \(f\in {\mathcal O}(\Delta_G)\) if \(f=gh\) for some \(h\in{\mathcal O}(\Delta_G)\).
Main theorem: If \(f,g\in {\mathcal O}(\Delta_G)\), then \(g\) divides \(f\) if and only if \(\text{div }g\leq \text{div } f\). As a consequence, given \(\text{div } f = \text{div } g\) one can find an \(h\in{\mathcal O}(\Delta_G)\) such that \(f=g\exp h\).
MSC:
30G35 Functions of hypercomplex variables and generalized variables
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[1] S. A. Grigoryan, ”Generalized analytic functions,”Uspekhi Mat. Nauk [Russian Math. Surveys],49, No. 2, 3–43 (1994). · Zbl 1108.30319
[2] S. A. Grigoryan, ”Maximal algebras of generalized analytic functions,”Izv. Akad. Armyan. SSR Ser. Mat. [Soviet J. Contemporary Math. Anal.],15, No. 5, 358–365 (1981). · Zbl 0491.46037
[3] S. A. Grigoryan, ”Generalized meromorphic functions,”Izv. Ross. Akad. Nauk Ser. Mat. [Math. Izv.],57, No. 1, 147–166 (1993). · Zbl 0801.30039
[4] R. Arens and I. Singer, ”Generalized analytic functions,”Trans. Amer. Math. Soc.,81, 379–393 (1956). · Zbl 0078.10902 · doi:10.1090/S0002-9947-1956-0078657-5
[5] B. M. Levitan,Almost Periodic Functions, [in Russian], Gostekhizdat, Moscow (1953). · Zbl 1222.42002
[6] B. Jessen, ”Über die Nullstellen einer analytischen fastperodischen Function,”Math. Ann.,108, 485–516 (1933). · JFM 59.1027.04 · doi:10.1007/BF01452849
[7] R. Arens, ”The boundary integral of log |\(\Phi\)| for generalized analytic functions,”Trans. Amer. Math. Soc.,86, 57–69 (1957). · Zbl 0202.40803
[8] K. Hoffman,Banach Spaces of Analytic Functions Prentice-Hall, Englewood Cliffs (1962). · Zbl 0117.34001
[9] H. Bohr,Fastperiodische Functionen, Springer-Verlag, Berlin (1932).
[10] Th. Gamelin,Uniform Algebras, Prentice-Hall, Englewood Cliffs (1969). · Zbl 0213.40401
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