Syllabus
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Course Code: MMATH21-414 Course Name: Elective VI) Fourier and Wavelet Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Fourier Transform: The finite Fourier transform, the circle group T, convolution to T, (L (T),+,*) as a Banach algebra, convolutions to products, convolution on T, the exponential form of Lebesgue’s theorem, Fourier transform : trigonometric approach, exponential form, Basics/examples. Fourier transform and residues, residue theorem for the upper and lower half planes, the Abel kernel, the Fourier map, convolution on R, inversion, exponential form, inversion, trigonometric form, criterion for convergence, continuous analogue of Dini’s theorem, continuous analogue of Lipschitz’s test, analogue of Jordan’s theorem. (Scope as in relevant parts of Chapter 5 of the book “Fourier and Wavelet Analysis” by Bachman, Narici and Beckenstein ) |
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| 2 | (C,1) summability for integrals, the Fejer-Lebesgue inversion theorem,the continuous Fejer Kernel, the Fourier map is not onto, a dominated inversion theorem, criterion for integrability of Approximate identity for L (R), Fourier Sine and Cosine transforms, Parseval’s identities, the L theory, Parseval’s identities for L , inversion theorem for L functions, the Plancherel theorem, A sampling theorem, the Mellin transform, variations. (Scope as in relevant parts of Chapter 5 of the book “Fourier and Wavelet Analysis” by Bachman, Narici and Beckenstein ) |
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| 3 | Discrete Fourier transform, the DFT in matrix form, inversion theorem for the DFT, DFT map as a linear bijection, Parseval’s identities, cyclic convolution, Fast Fourier transform for N=2 , Buneman’s Algorithm, FFT for N=RC, FFT factor form. ( Scope as in relevant parts of Chapter 6 of the book “Fourier and Wavelet Analysis” by Bachman, Narici and Beckenstein) | |
| 4 | Wavelets : orthonormal basis from one function , Multiresolution Analysis, Mother wavelets yield Wavelet bases, Haar wavelets, from MRA to Mother wavelet, Mother wavelet theorem, construction of scaling function with compact support, Shannon wavelets, Riesz basis and MRAs, Franklin wavelets, frames, splines, the continuous wavelet transform. (Scope as in relevant parts of Chapter 7 of the book “Fourier and Wavelet Analysis” by Bachman, Narici and Beckenstein) | |
