Mathematics BSc
Course description
2013.

Fourier integral
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
term grade
pure math. mm1c1fi6m
mm1c2fi6m
6 recommended
Strong Weak Prerequisites
Practice class
Strong:
Analysis4L-p (mm1c1an4m)
Strong:
Complex analysisP-p (mm1c2kf5m)
Lecture
Weak:
Practice class
Syllabus
  • L1 theory. The Fourier integral of L1 functions. The Fourier integral of a convolution. The inverse formula.
  • The Wiener approximation theorem. Application for Wiener's Tauber-type theorem.
  • The Fourier integral of a complex measure. The Fourier integral of a measure with bounded variation. Inversion formula. Wiener's Parseval-type formula. Characterization of continuous measures.
  • L2 theory. The Parseval formula.
  • Convolution in Lp spaces. The Fourier transform of the derivative. The Young-Hausdorff inequality. The Riesz-Thorin theorem. The Marczinkiewicz interpolation theorem.
  • The Fejér kernel. The estimation of the discrepancy due to Erdős and Turán. The estimation of the discrepancy due to Beck.
  • The Selberg sieve. The Poisson formula. Application in number theory.
  • The complex Fourier integral. The Paley-Wiener theorem.