School of Mathematics
School of Mathematics
Course 415 - Topics in Analysis (Fourier Analysis and Wavelets)
2000-01 (JS & SS Mathematics
)
Lecturer: Prof. R. M. Aron & Dr. R. M. Timoney
Requirements/prerequisites: 221
Duration: 21 weeks
Number of lectures per week: 3 including tutorials
Assessment: No continuous assessment.
End-of-year Examination: 3-hour end of year examination.
Description:
- Basics of Functional Analysis, including
- Baire category theorem and consequences: Uniform Boundedness
principle, closed graph theorem, open mapping theorem, et. al.
- Schauder basis,
Haar system for L_{2}[0,1], Schauder system for C[0,1] (with
proofs). Comparison with Hamel basis.
- Hilbert Spaces
- revision of basis properties of inner product spaces
- L_{p}, conjugate indices, Hölder's inequality, completeness of
L_{p}
- Convexity and Hilbert spaces
- Every subspace of a Hilbert space is complemented
- Orthonormal bases and examples.
- Haar system for L_{2}(\mathbbR)
- Fourier Analysis
- Fourier transform, basic definitions of f*g and [^f],
basic properties, Fourier inversion theorem, Plancherel theorem
(without proof).
- Riemann-Lebesgue lemma, Paley-Wiener theorem,
uncertainty principle, Nyquist sampling rate.
- Discrete Fourier transform
- Continuous Wavelet Transform on \mathbbR.
- Orthonormal Wavelet bases, dilation equations,
multiresolution analysis.
- Discrete Wavelet transform
Textbooks: The following may be used as references, but no one
book is being followed closely.
- Berberian, S. K., Introduction to Hilbert Space, Oxford (1961).
- Burke Hubbard, B., The world according to Wavelets:
The story of a Mathematical Technique in the Making (2nd ed.),
A. K. Peters, Natick, Massachusetts (1998).
- Chui, C., Wavelets: a mathematical tool for signal processing,
SIAM, Philadelphia, PA, 1997.
- Daubechies, Ingrid, Ten Lectures on Wavelets,
CBMS-NSF Regional Conference Series in Applied Mathematics volume 61,
Society for Industrial and Applied Mathematics (1992).
- Firth, Jean M., Discrete Transforms, Chapman & Hall (1992).
- Rudin, W., Real and Complex Analysis, McGraw-Hill (3rd ed., 1987).
- G. Strang, Wavelet transforms versus Fourier
transforms, Bull. Amer. Math. Soc. 28 (1993) 288-305.
- Wojtaszczyk, P.,
A mathematical introduction to wavelets.
London Mathematical Society Student Texts, 37, Cambridge University
Press, Cambridge, 1997.
Jan 17, 2001
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On 17 Jan 2001, 11:35.