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Abstract:
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This doctoral dissertation comprises two parts. Trigonometric polynomial rings
are the central topic of the first part of the dissertation. It is presented that the
ring of complex trigonometric polynomials,
C
[cos
x,
sin
x
], is a unique factorization
domain, and that the ring of real trigonometric polynomials,
R
[cos
x,
sin
x
], is not a
unique factorization domain. Necessary and sufficient conditions for the case when
in the ring
C
[cos
x,
sin
x
], unlike the ring
R
[cos
x,
sin
x
], the degree of the product of
two trigonometric polynomials is not equal to the sum of degrees of its factors, are
given.
The theory of trigonometric polynomials is extended to hyperbolic-trigonometric
polynomials, or HT-polynomials for short, which are defined similarly to trigonome-
tric polynomials. Real or complex HT-polynomials form a ring and even an integral
domain
R
[cosh
x,
sinh
x
], or
C
[cosh
x,
sinh
x
]. Factorization in these domains is con-
sidered, and it is shown that these are unique factorization domains. The irreducible
elements, as well as the form of the maximal ideals of both these domains are deter-
mined. The algorithms for dividing, factoring, computing greatest common divisors,
as well as the algorithms for simplifying ratios of two HT-polynomials are considered
over the field of rational numbers.
In the second part of the dissertation, related to applications, two methods of
proving inequalities of the form
f
(
x
)
>
0 are described over the given finite in-
terval (
a,b
)
⊂
R
,
a
≤
0
≤
b
, which by using the finite Maclaurin series expan-
sion generate polynomial approximations, when the function
f
(
x
) is element of the
ring extension of
R
[cos
x,
sin
x
], or
R
[cosh
x,
sinh
x
], denoted by
R
[
x,
cos
x,
sin
x
], or
R
[
x,
cosh
x,
sinh
x
]. The completeness of the presented methods is proved and the
concrete results of these methods are illustrated through examples of proving actual
inequalities. |