By Michael W. Frazier
Mathematics majors at Michigan country collage take a "Capstone" direction close to the top in their undergraduate careers. The content material of this path varies with each one supplying. Its function is to assemble assorted themes from the undergraduate curriculum and introduce scholars to a constructing region in arithmetic. this article used to be initially written for a Capstone path. easy wavelet thought is a traditional subject for the sort of direction. via identify, wavelets date again merely to the Nineteen Eighties. at the boundary among arithmetic and engineering, wavelet thought indicates scholars that arithmetic examine remains to be thriving, with vital functions in parts corresponding to photo compression and the numerical resolution of differential equations. the writer believes that the necessities of wavelet thought are sufficiently straightforward to learn effectively to complex undergraduates. this article is meant for undergraduates, so just a uncomplicated history in linear algebra and research is believed. 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
A3. (Associativity for addition) u + (v + w) = (u + v) + w, for all u, V, WE V. A4. (Existence of additive identity) There exists an element in V, denoted 0, such that u + 0 = u for all u e V. AS. (Existence of additive inverse) For each u E V, there exists an element in V, denoted -u, such that u + (-u) = 0. Ml. (Closure for scalar multiplication) For all a e JF and u e V, a · u is defined and is an element of V. M2. (Behavior of the scalar multiplicative identity) 1 · u = u, for all u e V, where 1 is the multiplicative identity in JF.
57 Suppose A is ann x n matrix over 16 (Comparison test) Let L:,M Zn be a complex series and L:,M an a series of nonnegative real numbers. Suppose that there exists an integer N such that lzn I ~ an for all n 2: N, and that L:,M an converges. Then L:,M Zn converges. 3. • If the elements of the series are real numbers, they can be regarded as complex and the definitions of convergence for real and complex series are consistent. So from now on we use the term "series" without specifying whether the terms are real or complex. 17 converges.
16 (Comparison test) Let L:,M Zn be a complex series and L:,M an a series of nonnegative real numbers. Suppose that there exists an integer N such that lzn I ~ an for all n 2: N, and that L:,M an converges. Then L:,M Zn converges. 3. • If the elements of the series are real numbers, they can be regarded as complex and the definitions of convergence for real and complex series are consistent. So from now on we use the term "series" without specifying whether the terms are real or complex. 17 converges.