Applied Analysis Qualifier Syllabus – May 2007

Course descriptions – Math 641 and Math 642

641. Analysis for Applications I. (3-0). Credit 3. Review of preliminary concepts; sequence and function spaces; normed linear spaces, inner product spaces; spectral theory for compact operators; fixed point theorems; applications to integral equations and the calculus of variations. Prerequisites: MATH 447 and 640 or approval of instructor.

642. Analysis for Applications II. (3-0). Credit 3. Distributions and differential operators; transform theory; spectral theory for unbounded self-adjoint operators; applications to partial differential equations; asymptotics and perturbation theory. Prerequisite: MATH 641.

Sequence and function spaces

Normed linear spaces and inner product spaces

Hilbert space

Distributions

Calculus of variations

Special functions

Unbounded linear operators and spectral theory

Asymptotics

References

  1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. J. J. Benedetto, Harmonic Analysis and Applications, CRC Press, Inc., Boca Raton, FL, 1997.
  3. N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, New York, 1981.
  4. A. Erdélyi, Asymptotic Expansions, Dover Publications, New York, 1956.
  5. G. B. Folland, Fourier Analysis and Its Applications, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1992.
  6. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood cliffs, NJ, 1963.
  7. J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Perseus books, Reading, MA, 1995.
  8. F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar Publishing, New York, 1955.
  9. E. M. Stein, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.
  10. G. P. Tolstov, Fourier Series, Dover Publications, New York, 1976.
  11. H. J. Wilcox and D. L. Meyers, An Introduction to Lebesgue Integration and Fourier Series, Dover Publications, New York, 1994.

Updated 5/16/07 (fjn).