MATH 750 Fourier Analysis  - Fall 2002 Professor Robert Sharpley Meets: MWF  11:15-12:05 in LeConte College 316 Instructor Information  Office:   LeConte 313 D  Office Hours: (TBD)

Course Topics
The course is the study of the basic principles of Fourier analysis and the necessary prerequisites for the analysis of wavelets. Lectures will drawn from several references (listed below) and will include the following topics:

Fourier series of periodic functions and the Fourier transform on the line: representation of functions, i.e. convergence and divergence (point-wise sense, in the norms of various function spaces, and almost everywhere), convergence of Fejer means and summability; Parseval's relation and the square summable theory; conjugate Fourier series, the conjugate function and the Hilbert transform, the Hardy-Littlewood maximal operator, the Riesz-Thorin and Marcinkiewicz interpolation theorems, function spaces, Riesz' theorem.

Applications will include topics in the theory of partial differential equations and signal processing, in particular the FFT.

Prerequisites
  Real Analysis (Math 703-704)

Lectures:
    Link to Weekly Outline

Primary References

• "Introduction to Fourier Analysis and Wavelets" by Mark A. Pinsky, Brooks/Cole, Oct 2001. (432 pp.)
• "An Introduction to Harmonic Analysis"(2-nd ed.) by Yitzhak Katznelson, Paperback - (1976) Dover Pubns, 1976. (264 pp.) [QA403 .K3 1976].

Additional References

• "Interpolation of Operators," by C. Bennett and R. Sharpley, Academic Press, New York, 1988. (469 pp.) [QA3 .P8 vol. 129]
• "Fourier Series" by G. H. Hardy and Werner Rogosinski, - Paperback - Dover Pubns, 1999. (112 pp.) [QA 404 .H3 1962].

Course Grading
   Assigned Homework (50%), Mid-term exam (25%) and Final (25%).

sharpley@math.sc.edu

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