An Introduction to Wavelets Through Linear Algebra
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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Algebra apply assumption basis for 2(ZN Cauchy sequence change of basis complete orthonormal set complex inner product complex multiplications complex numbers compression compute condition number consider Corollary corresponding D6 wavelet Daubechies's define Definition 5.30 diagonal matrix diagonalizable dimensional vector space eigenvalues eigenvectors element equivalent example exists fceZ Figure filter bank first-stage wavelet basis follows formula Fourier basis Fourier inversion Fourier transform frequency graph Haar Hence Hilbert space Hint identity implies induction infinite inner product space integral invertible matrix Lebesgue Lebesgue point Lemma linear transformation linearly independent multiresolution analysis n x n matrix norm notation Note obtain orthogonal orthonormal basis Parseval's relation polynomial Proof Exercise Prove equation real numbers real Shannon relation Relative error result satisfies Shannon wavelets shows signal solution subspace Suppose f system matrix Theorem 3.8 translation invariant triangle inequality unitary values variable wavelet coefficients wavelet system