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

By Michael W. Frazier

Mathematics majors at Michigan nation collage take a "Capstone" path close to the top in their undergraduate careers. The content material of this path varies with every one supplying. Its objective is to assemble assorted themes from the undergraduate curriculum and introduce scholars to a constructing sector in arithmetic. this article used to be initially written for a Capstone direction. simple wavelet conception is a typical subject for this kind of path. via identify, wavelets date again in simple terms to the Eighties. at the boundary among arithmetic and engineering, wavelet conception indicates scholars that arithmetic learn continues to be thriving, with very important purposes in components comparable to photograph compression and the numerical answer of differential equations. the writer believes that the necessities of wavelet conception are sufficiently straightforward to study effectively to complicated undergraduates. this article is meant for undergraduates, so just a simple heritage in linear algebra and research is thought. we don't require familiarity with advanced numbers and the roots of unity.

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Vn are linearly dependent. 7. 8). 37 Let V be a vector space over a field lF. 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 lF, and let U be a nonempty subset of V. i. Suppose U is finite, say U {u 1 , u2 , •.. , Un}, for some n E N, with ui =f:. uk for j =f:. k. Then U is a basis for V if and only if = 34 1. •• , an e lF such that LJ=l v = ajUj. ii. If U is infinite, then U is a basis for V if and only if for each non-zero v e V, there exist unique m e N, u1 , u 2, ...

V. So there exists vk+I e V \ span{vi. v2, ... , vk}. 9(ii), the set {v1, ... , vk, Vk+d is linearly independent. Continue in this fashion. 11. 38. 12. Suppose n + 1 vectors belong to the span of n vectors, say Wj E span{v 1,v 2 , ... ,vn} for j = 1,2, ... ,n + 1. Prove that w 1 , w 2 , ... , Wn, Wn+l are linearly dependent. Hint: Prove this by induction on n. Let Pn be the statement given. The case n = 1 is not difficult. Suppose Pn- 1 holds. To prove Pn. 6). Otherwise, at least one of its coefficients is nonzero.

Hint: Since L;! 1 ak converges, limk~+oo ak = 0. Since L;! 1 ak does not converge absolutely, the series of positive terms and the series of negative terms in {ak} must each diverge. List the positive terms in decreasing order (call these {bk}) and the negative terms in order of decreasing magnitude (say {ck}). Form the rearranged series by taking enough bks, in order starting at b1 , until their sum is above a. Then add enough cks, starting at c1 and proceeding in order, until the sum is below a.

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