An Introduction to Wavelets Through Linear Algebra

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Springer Science & Business Media, Aug 17, 2001 - Mathematics - 503 pages
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Mathematics majors at Michigan State University take a “Capstone” course near the end of their undergraduate careers. The content of this course varies with each offering. Its purpose is to bring together different topics from the undergraduate curriculum and introduce students to a developing area in mathematics. This text was originally written for a Capstone course. Basicwavelettheoryisanaturaltopicforsuchacourse. Byname, wavelets date back only to the 1980s. On the boundary between mathematics and engineering, wavelet theory shows students that mathematics research is still thriving, with important applications in areas such as image compression and the numerical solution of differential equations. The author believes that the essentials of wavelet theory are suf?ciently elementary to be taught successfully to advanced undergraduates. This text is intended for undergraduates, so only a basic background in linear algebra and analysis is assumed. We do not require familiarity with complex numbers and the roots of unity. These are introduced in the ?rst two sections of chapter 1. In the remainder of chapter 1 we review linear algebra. Students should be familiar with the basic de?nitions in sections 1. 3 and 1. 4. From our viewpoint, linear transformations are the primary object of study; v Preface vi a matrix arises as a realization of a linear transformation. Many students may have been exposed to the material on change of basis in section 1. 4, but may bene?t from seeing it again. In section 1.
 

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Contents

IV
7
VI
16
VII
29
VIII
40
IX
56
X
79
XI
101
XIII
128
XXI
298
XXII
309
XXIII
321
XXIV
330
XXV
349
XXVI
362
XXVII
380
XXVIII
398

XIV
151
XV
165
XVI
196
XVII
225
XVIII
265
XIX
271
XX
279
XXIX
429
XXX
451
XXXII
459
XXXIII
470
XXXIV
484
XXXV
491
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