Polynomials, Rational Functions and Trigonometric Polynomials

Mariano Giaquinta and Giuseppe Modica
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Mariano Giaquinta: Scuola Normale Superiore, Dipartimento di Matematica
Giuseppe Modica: Università degli Studi di Firenze, Dipartimento di Matematica Applicata

Chapter 5 in Mathematical Analysis, 2004, pp 145-186 from Springer

Abstract: Abstract In this chapter we want to illustrate the relevance of complex numbers in some elementary situations. After a brief discussion of the algebra of polynomials in Section 5.1, we prove the fundamental theorem of algebra and discuss solutions by radicals of algebraic equations in Section 5.2. In Section 5.3 we present Hermite’s decomposition formulas for rational functions, which are useful for the integration of rational functions, see Chapter 4 of [GM1]. Finally, in Section 5.4, we discuss some basic facts about trigonometric polynomials and, more generally, sums of sinusoidal signals. In particular we shall see that the spectrum of a signal completely identifies the signal itself, we shall prove the energy identity and present a sampling formula.

Keywords: Rational Function; Positive Root; Fundamental Theorem; Trigonometric Polynomial; Sinusoidal Signal (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-8176-4414-7_5

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