Browsing Mathematics by Author "Makragić, Milica"
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Makragić, Milica (Beograd , 2018)[more][less]
Abstract: 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 hyperbolictrigonometric polynomials, or HTpolynomials for short, which are defined similarly to trigonome tric polynomials. Real or complex HTpolynomials 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 HTpolynomials 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. URI: http://hdl.handle.net/123456789/4745 Files in this item: 1
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