Harmonic Analysis

Published

18 October 2026

Passport

  • Where and when. Semester IV in the FPMI curricula for the 2025 intake for programs 01.03.02 and 09.03.01.
  • Workload. For 01.03.02: 135 hours, 3 credits: 60 contact hours (30 lectures, 30 seminars), 45 hours of independent work and 30 hours for the exam. For 09.03.01: 180 hours, 4 credits: 60 contact hours, 90 hours of independent work and 30 hours for the exam.
  • Department. Department of Higher Mathematics.
  • Lecturers in the published 2025/26 programs. N. A. Gusev and V. V. Redkozubov, for different FPMI streams.
  • Final assessment. Exam in semester IV.

Contents of the published program

Below is the complete list of sections from N. A. Gusev’s version for 2025/26. It reproduces all 24 items of the program in condensed wording.

  1. Convolution of integrable functions and the Fourier transform of a convolution.
  2. Continuity and differentiability of a proper Lebesgue integral depending on a parameter.
  3. Uniform convergence of an improper Lebesgue integral with a parameter: criterion and the Weierstrass test.
  4. The Cauchy criterion and the Dirichlet test for an improper integral with a parameter.
  5. Passage to the limit in an improper integral with a parameter.
  6. Differentiation and integration of an improper integral with a parameter.
  7. The Riemann–Lebesgue lemma and the decay of Fourier coefficients to zero.
  8. Partial sum of a Fourier series, the Dirichlet kernel and the localization principle.
  9. Sufficient conditions for pointwise convergence of a Fourier series.
  10. Differentiation of a Fourier series and conditions for uniform convergence.
  11. Integration of a Fourier series.
  12. Estimate of the rate of decay of Fourier coefficients.
  13. CesΓ ro summation and FejΓ©r kernels.
  14. FejΓ©r’s theorem and the Weierstrass theorems on approximation by trigonometric and algebraic polynomials.
  15. The extremal property of Fourier coefficients and Bessel’s inequality.
  16. Completeness of an orthogonal system, orthonormal bases and Parseval’s identity.
  17. The Riesz–Fischer theorem.
  18. The relation between completeness of an orthogonal system and closedness of its linear span.
  19. Pointwise convergence of the Fourier integral and the inversion formula.
  20. The Fourier transform of a derivative and the derivative of the Fourier transform.
  21. The du Bois-Reymond theorem.
  22. The spaces \(\mathcal S\), \(\mathcal S_0\), \(\mathcal D\), \(\mathcal D_0\); regular and singular generalized functions.
  23. The Fourier transform on Schwartz spaces and spaces of generalized functions, the delta function and Plancherel’s theorem.
  24. The Kotelnikov theorem.

Grading scheme (spring 2026)

The course’s grading scheme is 100 points: two tests, 0–9 each; two assignments with a notebook and a defense, 0–3 each; theory check, 0–3; seminar work, 0–3; and oral exam, 0–70 (an integer multiple of 7). Scale: 93–100 β€” 10, 86–92 β€” 9, 79–85 β€” 8, 72–78 β€” 7, 65–71 β€” 6, 58–64 β€” 5, 51–57 β€” 4, 44–50 β€” 3, 30–43 β€” 2, 0–29 β€” 1. If the oral answer is 14 or less, special rules apply for grades 1 and 2.

Sources and currency

Gusev’s program, 2025/26, Redkozubov’s program, grading scheme 2026, curriculum 01.03.02, curriculum 09.03.01.