Syllabus
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Course Code: Elective 6 MMATH21 -414 Course Name: 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. |
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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 f Approximate identity for L1 (R), Fourier Sine and Cosine transforms, Parseval’s identities, the L 2 theory, Parseval’s identities for L 2 , inversion theorem for L 2 functions, the Plancherel theorem, A sampling theorem, the Mellin transform, variations. | |
| 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. | |
| 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. | |
