By Michael W. Frazier

Mathematics majors at Michigan nation college take a "Capstone" direction close to the top in their undergraduate careers. The content material of this path varies with each one delivering. Its goal is to collect assorted subject matters from the undergraduate curriculum and introduce scholars to a constructing quarter in arithmetic. this article was once initially written for a Capstone direction. simple wavelet thought is a ordinary subject for the sort of direction. via identify, wavelets date again in basic terms to the Eighties. at the boundary among arithmetic and engineering, wavelet idea indicates scholars that arithmetic study remains to be thriving, with vital functions in components equivalent to picture compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet conception are sufficiently straightforward to learn effectively to complicated undergraduates. this article is meant for undergraduates, so just a easy heritage in linear algebra and research is thought. we don't require familiarity with complicated numbers and the roots of unity.

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**Example text**

3. 4. 14. 15. 16. A series of complex numbers is conditionally convergent if it is convergent but not absolutely convergent. Let 'L~ 1 ak be a conditionally convergent series of real numbers. Let a be an arbitrary real number. , a 1-1, onto map from N to N) such that the reindexed series L~ 1 arr(k) converges to a. 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.

28 1. ). In addition, we used the fact that if I::o Zn converges and a E C, then I::o azn converges and 00 00 LaZn =a LZn. n=O n=O Prove this also. i. 16). Hint: recall the addition formulae: cos(O +

Vn are linearly independent (or {vi, Vz, ... Vn} is a linearly independent set) if holds only when ai = 0 for every j = 1, 2, 3, ... , n. For a possibly infinite subset U ofV, we say U is linearly independent ifevery finite subset of U is linearly independent, and we say U is linearly dependent if U has a finite subset that is linearly dependent. •• Vn are linearly independent if the only linear combination a 1VI + a 2 v2 + · · · + anVn that is 0 is the trivial one (aj = 0 for all j = 1, 2, ...