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By Michael W. Frazier

Mathematics majors at Michigan kingdom college take a "Capstone" direction close to the tip in their undergraduate careers. The content material of this direction varies with every one supplying. Its objective is to compile various subject matters from the undergraduate curriculum and introduce scholars to a constructing region in arithmetic. this article used to be initially written for a Capstone direction. uncomplicated wavelet concept is a typical subject for the sort of path. by way of identify, wavelets date again simply to the Eighties. at the boundary among arithmetic and engineering, wavelet conception exhibits scholars that arithmetic study continues to be thriving, with vital purposes in components comparable to snapshot compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet conception are sufficiently basic to learn effectively to complicated undergraduates. this article is meant for undergraduates, so just a simple history in linear algebra and research is thought. we don't require familiarity with complicated numbers and the roots of unity.

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Extra resources for An Introduction to Wavelets Through Linear Algebra

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Iz + 3! + 5! + .. ·) z4 z2 = (1 - 2 + 4! z6 - 6! = cosz + i sinz. + .. ·) + i(z- z3 3! zs + 5! - . ") • This remarkable formula includes curious facts such as -eirr = 1. 8) gives the alternate formula e-iz = cosz- isinz. 11) Although these formulae hold for all complex numbers z, our main interest in them is in the case when z is real. 2. 9(i)). 23 Suppose z, we C. Then tf+w = ezew. 2. 9(ii). 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 nO and cos nO in terms of sin 0 and cos 0 (in fact as polynomials in sin 0 and cos 0).

Suppose T : U --+ V is a linear transformation. Let R be a basis for U and S a basis for V. 49. Then Tis an invertible linear transformation if and only if Ar is an invertible matrix. Proof First suppose T is invertible. z2, ... ,zn. Suppose R = {u1, u2, ... , Un}. Let u = LJ=I ziui; in other words, [u]R = z. Let Ar-l be the matrix that represents r- 1 with respect to S and R. Then z = [u]R = [T- 1 (T(u))]R = Ar-I[T(u)]s = Ar-IAr[u]R = Ar-IAyz. 6(i). A symmetric argument shows that also ArAr-l = I.

6. (Ratio test) Let ,E:,M Zn be a series and suppose that . 'Zn+ll 11m - - = p exists. n~oo Zn i. Prove that if p < 1, then ,E:,Mzn converges absolutely, whereas if p > 1, ,E:,M Zn diverges. 16. Assume the fact that the sequence {rn} ~ 0 diverges ifr > 1. ii. Give an example of a convergent series for which p = 1 and a divergent series for which p = 1. Thus the ratio test gives no conclusion when p = 1. E 1/nP converges if and only ifp > 1. 7. (Radius of convergence) Consider a power series 00 Lan(z -zot.

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