MA4201 : Fourier Analysis (Spring 2017) | Academic Portal of IISER Kolkata

Details of MA4201 (Spring 2017)

Level: 4

Type: Theory

Credits: 3.0

Course Code	Course Name	Instructor(s)
MA4201	Fourier Analysis	Rajib Dutta

Syllabus
Convolution: Convolution, elementary properties of convolutions. The Hardy-Littlewood Maximal Function : Approximations of the identity, weak-type inequality, Marcinkiewicz Interpolation Theorem, Hardy-Littlewood maximal function and its properties, dyadic maximal function, Calderon-Zygmund theorem. Riesz-Thorin Interpolation Theorem. Fourier Series: Fourier coefficients and series, summability in norm, summability at a point, Weiner Algebra, pointwise convergence of Fourier series, convergence of Fourier series in norm. The Hilbert Transform: Conjugate Poisson kernel, principal value distribution, Riesz and Kolmogorov theorems, multipliers. Fourier Transform: Fourier transform of $L^1$ functions, Schwartz class, Fourier transform for $ L^2$ functions, Plancherel theorem. Fourier integrals: Summability in norm, pointwise summability and convergence in norm.

Syllabus

Convolution: Convolution, elementary properties of convolutions.
The Hardy-Littlewood Maximal Function : Approximations of the identity, weak-type inequality, Marcinkiewicz Interpolation Theorem, Hardy-Littlewood maximal function and its properties, dyadic maximal function, Calderon-Zygmund theorem. Riesz-Thorin Interpolation Theorem.
Fourier Series: Fourier coefficients and series, summability in norm, summability at a point, Weiner Algebra, pointwise convergence of Fourier series, convergence of Fourier series in norm.
The Hilbert Transform: Conjugate Poisson kernel, principal value distribution, Riesz and Kolmogorov theorems, multipliers.
Fourier Transform: Fourier transform of $L^1$ functions, Schwartz class, Fourier transform for $ L^2$ functions, Plancherel theorem.
Fourier integrals: Summability in norm, pointwise summability and convergence in norm.

References
J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, American Mathematical Society, 2001. L. Grafakos, Classical Fourier Analysis (2nd Edition), Springer-Verlag, 2008. Y. Katznelson, An Introduction to Harmonic Analysis (3rd Edition), Cambridge University Press, 2004. T. W. Krner, Fourier Analysis, Cambridge University Press, 1989. E. M. Stein and R. Sakarachi, Fourier Analysis: An Introduction, Princeton University Press, 2003. 6. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. 7. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971. University Press, 1971.

References

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, American Mathematical Society, 2001.
L. Grafakos, Classical Fourier Analysis (2nd Edition), Springer-Verlag, 2008.
Y. Katznelson, An Introduction to Harmonic Analysis (3rd Edition), Cambridge University Press, 2004.
T. W. Krner, Fourier Analysis, Cambridge University Press, 1989.
E. M. Stein and R. Sakarachi, Fourier Analysis: An Introduction, Princeton University Press, 2003.

6. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton
University Press, 1971.

7. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton
University Press, 1971.

University Press, 1971.

Sl. No.	Programme	Semester No	Course Choice
1	IP	2	Not Allowed
2	IP	4	Core
3	IP	6	Not Allowed
4	MR	2	Not Allowed
5	MR	4	Not Allowed
6	MS	8	Core
7	RS	1	Elective
8	RS	2	Not Allowed