الفهرس 
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Abstract
It is known that the famous functional Hilbert space LiR).
1t”(R) (Sobolev space) contain elements that are not entire (even not
smooth in the space L2(R)). The aim of the present thesis is to
introduce and study some Hilbert spaces consisting of entire
functions. The second aim of the thesis is to study the Fourier
transformation as an operator by which it is possible to define entire
jUnctions. For satisfying these aims it was necessary to present some
elementary ideas and concepts on analytic jUnctions of a complex
variable. generalized function and some jUndamental theorems from
the theory of real analysis.
The thesis consists of five sections.
The first section. Smooth and Analytic Functions of a complex
or Real Variable. deals with analytic jUnctions of a complex variable.
analytic jUnctions of a real variable. and the test space and test
jUnctions in one dimension.
The second section. Generalized Functions. deals with: The
space of generalized functions in one dimension. and derivatives of
generalized function.
The third section. Hilbert Spaces and Fourier Transformations.
deals with : Abstract Hilbert spaces. Sobolev spaces. and the Fourier
transformation in LiR).
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The jourth section. Hilbert Spaces oj Entire Functions. deals
with: Some theorems oj Paley and Wiener, Paley-Wiener spaces. a
mod(fied Paley-Wiener theorem, and mod(fied Paley-Wiener spaces.
The fifth section. Characterization oj Fowter Transjormations.
deals with : A characterization oj Fourier transjormation in LiRJ. and
a characterization oj Fourier transjormation in LiRnJ.