Borel transformation, entire function, quasi entire function.

The W class of quasi-entire functions is introduced and the properties of the Borel transformation in this class are investigated. It has to be noted that quasi- entire functions of exponential type were for the first time considered apparently by A. Pfluger in 1935 [2], and we have not heard of any other works in this field since then. In essence, the Pfluger’s work amounted to the construction of an analog of Borel transform for integer function of exponential type. The extraction of sub-class W – square summable functions on real axis — from the class of integer, exponential type functions allowed Paley and Winer to obtain fundamental results for the theory of Fourier integral, which have proven to be a very powerful tool for the theory of function’s basis [3]. Subclass W for quasi- entire, exponential type functions is introduced in the same way as it is introduced for entire functions, i.e. it includes quasi- entire functions which are square summable on real axis. The results thus obtained were used as essential input data in the studies of basic properties of Fadle-Papkovich functions of the elasticity theory. In the theory of entire, exponential type functions, an entire function is presented as a Borel transform of a function which is defined on a complex plane and is analytical outside a circle over which a corresponding integral is taken [3], [4]. If an entire function belongs to the W class, the circle over which integration is made can be fixed to the segment of imaginary axis which coincides with the circle’s diameter. In this case, Borel transform turns into an ordinary Fourier transform on a segment. Following this way, Paley and Winer managed to link Fourier transform on a segment with the powerful apparatus of the theory of analytical functions. For quasi- entire, exponential type functions the situation is basically similar, but in order to ensure single-valuedness of the function, associated by Borel with the considered quasi-entire function, it is necessary to consider it on the Riman logarithmic surface where it is analytical; this time, analyticity is observed outside a spiral with a certain radius. Now, if we require this quasi- entire function to be square summable on real axis, then (like in the theory of Paley and Winer), it will be possible to affix it (which is easy to guess) — this time not to a single segment, but to an infinite system of segments located on imaginary axis on the sheets of the Riman logarithmic surface, one over another. A next step in our theory - building is transition from the Riman logarithmic surface to an ordinary complex plane with a single cross-section on the imaginary axis (like in the Paley and Winer theory), outside which the function, associated by Borel with the quasi- entire function, is analytical. Then, in order to ensure that this function is single-valued over the considered plane, a corresponding cross-section needs to be made in it. Cross-section is usually chosen to coincide either with the negative or with the positive segment of real axis. The results obtained by Paley and Winer represent a specific case within the framework of this theory. 

1. Kovalenko, Mikhail Denisovich Dr. Sci. Phys. & Math., Professor, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow

2. Menchova, Irina Vladimirovna Candidate of Phys.&Math., Assoc. Professor, Senior Research Officer, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow