[ TM1WAUEPSDO ] UE Pseudodifferential operators and Fourier integral operators

Workload Education level Study areas Responsible person Hours per week Coordinating university
1,5 ECTS B3 - Bachelor's programme 3. year Mathematics Markus Passenbrunner 1 hpw Johannes Kepler University Linz
Detailed information
Original study plan Bachelor's programme Technical Mathematics 2012W
Objectives Practice and Consolidation of important concepts and methods in the theory of Pseudo differential operators
Subject 1 Introduction

  1. Convolution integrals
  2. Fourier transformation
  3. Tempered distributions

2 Differential operators with constant coefficients

3 Pseudo differential operators

  1. Motivation, Definitions
  2. Simple propertis
  3. Asymptotic expansion of symbols
  4. Partition of Unity, Frequency localization
  5. Product of Pseudo differential operators
  6. Adjoints of Pseudo differential operators
  7. Parametrice of Pseudo differential operators
  8. Lp -boundedness of Pseudo differential operators
  9. Sobolev spaces and Pseudo differential operators

4 Oscillating Integrals

  1. Motivation: High frequency asymptotics of the wave equation
  2. Method of the stationary phase
    1. Linear phase
    2. Quadratic phase
    3. Morse Lemma
  3. Application to the wave equation

5 Supplements

  1. Connection between wave equation and Eikonal equation
  2. Wave equation with variable coefficients
  3. Homogenization of elliptic equations

A Applications of Fourier multiplier theorems

  1. Hilbert transform
    1. Properties of the Hilbert transform
  2. Riesz-Transformation
    1. An application of the Riesztransform

B Hermite functions

  1. Definition and simple properties
  2. Expansion in Hermite series
  3. Hermite functions als Eigenfunctions of the Fourier transform
  4. Hermite-Polynomials and recursions

C Inequality of Babenko-Beckner

  1. Formulation of the result
  2. Central limit theorem
  3. Proof of the Babenko-Beckner inequality
Criteria for evaluation “Tick exercise” + Blackboard performance
Language German
Study material Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

G. B. Folland. Lectures on partial differential equations, volume 70 of Tata Institute of Fun- damental Research Lectures on Mathematics and Physics. Published for the Tata Institute of Fundamental Research, Bombay, 1983.

M. W. Wong. An introduction to pseudo-differential operators. World Scientific Publishing Co. Inc., River Edge, NJ, second edition, 1999.

Gerald B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley- Interscience Publication.

Xavier Saint Raymond. Elementary introduction to the theory of pseudodifferential operators. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1991.

Robert S. Strichartz. A guide to distribution theory and Fourier transforms. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1994.

William Beckner. Inequalities in Fourier analysis. Ann. of Math. (2), 102(1):159–182, 1975.

Michael Beals, Charles Fefferma, and Robert Grossman. Strictly pseudoconvex domains in Cn. Bull. Amer. Math. Soc. (N.S.), 8(2):125–322, 1983.

Doina Cioranescu and Patrizia Donato. An introduction to homogenization, volume 17 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1999.

Lawrence C. Evans. Weak convergence methods for nonlinear partial differential equations, vo- lume 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990.

[Gut05] Allan Gut. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005.

Elias M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathe- matical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.

Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.

Elias M. Stein and Guido Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32.

François Trèves. Introduction to pseudodifferential and Fourier integral operators. Vol. 1. Plenum Press, New York, 1980. Pseudodifferential operators, The University Series in Mathematics.

François Trèves. Introduction to pseudodifferential and Fourier integral operators. Vol. 2. Plenum Press, New York, 1980. Fourier integral operators, The University Series in Mathematics.
Changing subject? No
On-site course
Maximum number of participants 25
Assignment procedure Direct assignment