Download PDF by Michael W. Frazier: An Introduction to Wavelets Through Linear Algebra

By Michael W. Frazier

ISBN-10: 0387986391

ISBN-13: 9780387986395

This article used to be initially written for a "Capstone" path at Michigan country collage. A Capstone path is meant for undergraduate arithmetic majors, as one of many ultimate classes taken of their undergraduate curriculum. Its goal is to collect diversified issues coated within the undergraduate curriculum and introduce scholars to present advancements in arithmetic and their functions. simple wavelet idea looks an ideal subject for this kind of direction. As a subject matter, it dates again basically to 1985. seeing that then there was an explosion of wavelet study, either natural and utilized. Wavelet concept is at the boundary among arithmetic and engineering. particularly it's a strong subject for demonstrating to scholars that arithmetic examine is prospering within the modern-day: scholars can see non-trivial arithmetic principles resulting in traditional and demanding purposes, akin to video compression and the numerical answer of differential equations. the single necessities assumed are a uncomplicated linear algebra heritage and a little bit research heritage. this article is meant to be as effortless an creation to wavelet thought as attainable. it's not meant as a radical or authoritative reference on wavelet conception.

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Factor theorem) Let P be a polynomial and let a ∈ C. We say z − a divides P, or z − a is a factor of P (written (z − a)|P), if there exists a polynomial Q such that P(z) (z − a)Q (z). Prove that (z − a)|P if and only if P(a) 0; that is, z − a is a factor of P if and only if a is a root of P. Hint: The “only if” direction is immediate. For the “if” direction, the result follows easily from the division algorithm for polynomials, if you are familiar with that. If not, a more elementary proof can be obtained by first showing that (z − a)|(z k − ak ) for any integer k ≥ 1.

In other words, v1 , v2 , . . vn are linearly independent if the only linear combination α1 v1 + α2 v2 + · · · + αn vn that is 0 is the trivial one (αj 0 for all j 1, 2, . . , n). If there is a nontrivial linear combination of v1 , v2 , . . vn that is 0, then v1 , v2 , . . vn are linearly dependent. 7. 8). 37 Let V be a vector space over a field F. A subset U of V is a basis for V if U is a linearly independent set such that span U V. Bases are also characterized in the following way. 38 Let V be a vector space over a field F, and let U be a nonempty subset of V.

Zn αzn zn wn the components of           z1 + w1 z2 + w2 ..    .  z n + wn It is easy to check that Cn , with these operations, is a vector space over C. 32 1. Background: Complex Numbers and Linear Algebra In these examples, the vector space elements, or vectors, are of the type encountered in calculus. 1(i) and the following. 33 Let [0, 1] {x ∈ R : 0 ≤ x ≤ 1}. The field is C in this example. Let V be the set of all complex-valued functions on [0, 1], that is, {f : [0, 1] → C}.

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An Introduction to Wavelets Through Linear Algebra (Undergraduate Texts in Mathematics) by Michael W. Frazier


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