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Representation of the solution of a homogeneous convolution equation in the form of a series. (English. Russian original) Zbl 0741.30021
Math. Notes 49, No. 3, 259-262 (1991); translation from Mat. Zametki 49, No. 3, 46-51 (1991).
Let $$f_ j (j=0,1,2,\ldots)$$ be entire functions of exponential type and let $$f_ 0(z)=\prod^ \infty_{j=1}f_ j(z).$$ The main result of the paper contains the non-improvable conditions which provide the representability of any entire solution of the equation $$f_ 0(d/dz)y_ 0(z)=0$$ in the form $$y_ 0(z)=\sum^ \infty_{j=1}y_ j(z),$$ where $$y_ j(z)$$ are entire solutions of the equations $$f_ j(d/dz)y_ j(z)=0 (j=1,2,\ldots)$$.
##### MSC:
 30D10 Representations of entire functions of one complex variable by series and integrals 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 47A50 Equations and inequalities involving linear operators, with vector unknowns
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##### References:
 [1] V. V. Napalkov, ?Factorization of an operator of convolution type,? Mat. Zametki,15, No. 1, 165-171 (1974). · Zbl 0335.42014 [2] V. V. Napalkov, Convolution Equations in Multidimensional Spaces [in Russian], Nauka, Moscow (1982). · Zbl 0582.47041 [3] A. F. Leont’ev, ?On an application of an interpolation method,? Mat. Zametki,18, No. 5, 735-752 (1975). [4] A. F. Leont’ev, Exponential Series [in Russian], Nauka, Moscow (1976). [5] T. T. Kuzbekov, ?Ideals in certain rings of entire functions,? Otd. Fiz. Mat. Bashkir. Filiala Akad. Nauk SSSR, Ufa (1988). Manuscript deposited at VINITI, March 23, 1988, No. 2237-B88. Dep.
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