By Michael W. Frazier

This article was once initially written for a "Capstone" direction at Michigan nation collage. A Capstone direction is meant for undergraduate arithmetic majors, as one of many ultimate classes taken of their undergraduate curriculum. Its function is to assemble assorted themes coated within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their purposes. easy wavelet conception appears to be like an ideal subject for one of these path. As a subject matter, it dates again purely to 1985. because then there was an explosion of wavelet examine, either natural and utilized. Wavelet concept is at the boundary among arithmetic and engineering. specifically it's a stable subject for demonstrating to scholars that arithmetic examine is prospering within the modern-day: scholars can see non-trivial arithmetic rules resulting in average and demanding purposes, similar to video compression and the numerical answer of differential equations. the single necessities assumed are a easy linear algebra history and a little bit research historical past. this article is meant to be as straightforward an advent to wavelet idea as attainable. it isn't meant as an intensive or authoritative reference on wavelet conception.

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N ∈ F. Note that our deﬁnition requires a linear combination to be a ﬁnite sum. 35 Let V be a vector space over a ﬁeld F, and suppose U ⊆ V. The span of U (denoted span U) is the set of all linear combinations of elements of U. In particular, if U is a ﬁnite set, say U {u1 , u2 , . . un }, then span U n j 1 αj uj : αj ∈ F for all j 1, 2, . . , n . 3. Vector Spaces and Bases 33 To visualize the span, note that the span of a single nonzero vector u in Rn consists of all vectors lying on the line through the origin that contains u.

5) is a useful formula that we apply for other purposes in chapter 2. We now consider power series. 2. Complex Series, Euler’s Formula, and the Roots of Unity 19 Fix a point z0 ∈ C. 20 series of the form ∞ an (z − z0 )n , n 0 where an ∈ C for each integer n ≥ 0. 7. A function f deﬁned on an open set O ⊆ C (a set having the property that any point in it has a ball of positive radius around it that is contained in the set) is said to be analytic if at every point z ∈ O, f is represented by a power series about z with a positive radius of convergence.

2. 12) implies that cos θ eiθ eiθ − e−iθ . 14) 1. 15) and |eiθ | Our main interest in the next result is in the special case where z and w are purely imaginary, which yields eiθ eiϕ ei(θ+ϕ) for all θ, ϕ ∈ R. 9(i)). 23 Suppose z, w ∈ C. Then e z +w ez ew . 9(ii). 16), note that for θ ∈ R and n ∈ N, (eiθ )n einθ . 17) This equation can be used to obtain elementary trigonometric identities easily. For example, it is clear that by iterating the addition formulae for sine and cosine, we can write sin nθ and cos nθ in terms of sin θ and cos θ (in fact as polynomials in sin θ and cos θ).