# An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier

By Michael W. Frazier

This article used to be initially written for a "Capstone" path at Michigan country 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 compile assorted themes lined within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their functions. easy wavelet conception looks an ideal subject for this kind of direction. As an issue, it dates again in basic terms to 1985. on account that then there was an explosion of wavelet examine, either natural and utilized. Wavelet thought is at the boundary among arithmetic and engineering. specifically it's a stable subject for demonstrating to scholars that arithmetic examine is flourishing within the modern-day: scholars can see non-trivial arithmetic rules resulting in average and critical purposes, similar to video compression and the numerical answer of differential equations. the one necessities assumed are a uncomplicated linear algebra heritage and just a little research heritage. this article is meant to be as user-friendly an creation to wavelet concept as attainable. it isn't meant as a radical or authoritative reference on wavelet concept.

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**Extra info for An Introduction to Wavelets Through Linear Algebra (Undergraduate Texts in Mathematics)**

**Sample text**

6. (Ratio test) Let ∞ n M zn be a series and suppose that lim n→∞ zn + 1 zn ρ exists. i. Prove that if ρ < 1, then ∞ n M zn converges absolutely, whereas if ρ > 1, ∞ n M zn diverges. 16. Assume the fact that the sequence {r n }∞ n 0 diverges if r > 1. ii. Give an example of a convergent series for which ρ 1 and a divergent series for which ρ 1. Thus the ratio test gives no conclusion when ρ 1. You can assume the fact that 1/np converges if and only if p > 1. 7. (Radius of convergence) Consider a power series ∞ an (z − z0 )n .

Argue that this is a contradiction, keeping in mind that < is not necessarily the usual ordering when restricted to R. 5. 7. 6. 8. Suggestion: Do not write it out in terms of the real and imaginary parts. 7. 7. For z, w ∈ C, deﬁne d(z, w) |z − w|. 2 for the deﬁnition). 2 is called the triangle inequality. 8. 2) is a ﬁeld by checking properties A1–A5, M1–M5, and D. 9. 9. 10. Let {zn }n∞ M be a sequence of complex numbers. For each n, let zn xn + iyn , where xn , yn ∈ R. 16 1. Background: Complex Numbers and Linear Algebra i.

24 1. Background: Complex Numbers and Linear Algebra If √we multiply out the left side of the equation (a + ib)5 2 + 2 3i and set the real and imaginary parts equal, we obtain two 5th degree polynomial equations in two variables a and b to solve simultaneously. Even one 5th degree polynomial equation in one variable cannot be solved by a general algebraic formula (by a deep theorem in Galois theory, a topic in abstract algebra), so another approach must be found. Consider a nonzero complex number z.