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- W1982552702 abstract "LetK be the class of trigonometric series of power type, i.e. Taylor series (sumnolimits_{n = 0}^infty {c_n z^n } ) forz=eix, whose partial sums for allx inE, whereE is a nondenumerable subset of [0, 2π), lie on afinite number of circles (a priori depending onx) in the complex plane. The main result of this paper is that for every member of the classK, there exist a complex number ω, |ω|=1, and two positive integers (nu , kappa , nu < kappa ), such that for the coefficientscn we have: $$c_{mu + lambda left( {kappa - nu } right)} = c_mu omega ^lambda , mu = nu ,nu + 1, ldots , kappa - 1, lambda = 1,2,3, ldots $$. Thus, every member of the classK has (with minor modifications) a representation of the form: $$P(x)sumnolimits_{n = 0}^infty {e^{iknx} ,} $$where P(x) is a suitable trigonometric polynomial andk a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of (C, 1) summable Taylor series." @default.
- W1982552702 created "2016-06-24" @default.
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- W1982552702 date "1989-12-01" @default.
- W1982552702 modified "2023-09-25" @default.
- W1982552702 title "On a theorem of Marcinkiewicz and Zygmund for Taylor series" @default.
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- W1982552702 doi "https://doi.org/10.1007/bf02386363" @default.
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