Generalized analytic functions.

*(English. Russian original)*Zbl 1108.30319
Russ. Math. Surv. 49, No. 2, 1-40 (1994); translation from Usp. Mat. Nauk 49, No. 2(296), 3-42 (1994).

Summary: This article presents a broad survey on generalized analytic and generalized meromorphic functions on subsets of a big plane generated by a compact abelian group with ordered dual. It is based on several of the author’s papers [see, in particular, Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 24, No. 3, 20–40 (1989); translation from Izv. Akad. Nauk Arm. SSR, Mat. 24, No. 3, 226–247 (1989; Zbl 0702.46038); Russ. Acad. Sci., Izv., Math. 42, No. 1, 133–147 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 1, 147–166 (1993; Zbl 0801.30039)]. Let \(G\) be a compact abelian group with ordered dual group \(\Gamma\) that is a subgroup of \(\mathbb R\). The big plane over \(G\) is the infinite cone \([0,\infty)\cdot G\), and the unit big disc over \(G\) is the set \(\Delta_G\) of points \(rg\) in the big plane whose “modulus” \(r\) is \(\leq1\). A continuous function \(f\) in a domain \(D\subset \mathbb C_G\) is generalized analytic in \(D\) if \(f\) can be approximated locally by polynomials on \(\mathbb C_G\) (i.e., linear combinations \(\sum \alpha(a)\phi^a\) , where \(\alpha(a)\in\mathbb C, r\in [0,\infty), g\in G\), \(a\in\Gamma\cap[0,\infty)\) and \(\phi^a\) is a natural extension of the character \(\chi^a\) on the big plane) in \(D\). The results on generalized analytic functions in the present paper are from the author’s 1989 paper [loc. cit.]. (A more detailed exposition on the development of the theory of generalized analytic functions on subsets of the big plane, including the related results of the reviewer and D. Stankov, is presented in the reviewer’s book [Big-planes, boundaries and function algebras, North-Holland, Amsterdam (1992; Zbl 0755.46020)].) In addition, the following version of the Bohr-van Kampen theorem is proved: If a generalized analytic function \(f\) on the big disc \(\Delta_G\) does not vanish outside the origin, then there are a character \(\chi^a, a\geq 0\), on \(G\) and a function \(g\) generalized analytic on \(\Delta_G\) such that \(f=\phi^a\exp g\). The author introduces also the concept of differentiability for generalized analytic functions. The quantity \(n(r,f)=r\Phi'(r+0)\), where \(\Phi(r)= \int_G\log|f(rg)|d\sigma(g)\), \(\sigma\) being the Haar measure on \(G\), is used to measure the zero set of a generalized analytic function \(f\) on \(\Delta_G\). Let the spectrum of \(f\), defined by \(S(f)=\{b\in[0,\infty)\colon c_b(f)=\int f\overline{\chi}{}^b\delta \sigma\neq 0,\;\chi^b\in \widehat G\}\), contain its infimum \(a\). Then the following formula is shown to hold for the first Fourier-Stieltjes coefficient \(c_a(f)\) of \(f\):
\[
\log|c_a(f)|=\int_{\{r\}\times G}\log|f|d\sigma-\int_0^r\frac{n(t,f)-a}{t}dt-a\log r.
\]

The basics of the author’s pioneering work on generalized meromorphic functions presented in this paper are from his 1993 paper [loc. cit.]. Various descriptions of zero sets, peak sets, interpolation sets and measures orthogonal to the big disc algebra are given. The author’s factorization theorems for generalized meromorphic functions on the big disc and on the big annulus are presented as well. Let \(H^\infty(d\sigma)\) be the weak\(^*\) closure of the big disc algebra in \(L^\infty(d\sigma)\). A well-known theorem due to W. Rudin says that the set \(H^\infty(d\sigma)+C(G)\) is not an algebra unless \(G\) is the unit circle. If \(H^\infty(\Delta_G)\) is the algebra of bounded generalized analytic functions on the open big disc, the author shows that the space \(H^\infty(\Delta_G)|_G+C(G)\) is a closed algebra. Versions of this result have been proved previously by R. E. Curto, P. S. Muhly and J. Xia [Integral Equ. Oper. Theory 8, No. 5, 650–673 (1985; Zbl 0583.47030)], and by D. Stankov [e.g., T. Tonev, loc. cit., Zbl 0755.46020]. Most of the results in the present paper imply nice and useful applications to almost periodic functions.

The basics of the author’s pioneering work on generalized meromorphic functions presented in this paper are from his 1993 paper [loc. cit.]. Various descriptions of zero sets, peak sets, interpolation sets and measures orthogonal to the big disc algebra are given. The author’s factorization theorems for generalized meromorphic functions on the big disc and on the big annulus are presented as well. Let \(H^\infty(d\sigma)\) be the weak\(^*\) closure of the big disc algebra in \(L^\infty(d\sigma)\). A well-known theorem due to W. Rudin says that the set \(H^\infty(d\sigma)+C(G)\) is not an algebra unless \(G\) is the unit circle. If \(H^\infty(\Delta_G)\) is the algebra of bounded generalized analytic functions on the open big disc, the author shows that the space \(H^\infty(\Delta_G)|_G+C(G)\) is a closed algebra. Versions of this result have been proved previously by R. E. Curto, P. S. Muhly and J. Xia [Integral Equ. Oper. Theory 8, No. 5, 650–673 (1985; Zbl 0583.47030)], and by D. Stankov [e.g., T. Tonev, loc. cit., Zbl 0755.46020]. Most of the results in the present paper imply nice and useful applications to almost periodic functions.

Reviewer: T. Tonev (Missoula)

##### MSC:

30G35 | Functions of hypercomplex variables and generalized variables |

43A17 | Analysis on ordered groups, \(H^p\)-theory |

30D30 | Meromorphic functions of one complex variable, general theory |

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |

46J99 | Commutative Banach algebras and commutative topological algebras |