1 semester

1

Name of course

Introduction to harmonic analysis on Euclidean spaces

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Year

1, speciality Mathematics

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Semester

1

4

Credits

4

5

Lecturer’s scientific and academic degree, First Name, Patronic Name, Surname

Doctor of Sciences (habilitated), Professor,

Veniyamin Ruhoravič Krotaū

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Goals of course

Progress in a professional competence of students, forming of a concept about technical potential of the    discipline and role of the Fourier transform in scientific problems.

At the end of the course a student will be able:

  • to use properties of maximal functions and approximative identities for an estimation of harmonic operators;
  • to use principle properties of the Fourier transform;
  • to use basic techniques of harmonic analysis in mathematics.

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Background

Mathematical analysis, functional analysis, equations of mathematical physics.

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Contents

Density classes in . Linear functional representation theorem in  space. Weak compactness theorem for . 

spaces on Euclidean sets. Polar coordinates. Density of continuous functions. -continuity. Density of locally constant functions.

Steklov averages. Hardy-Littlewood maximal functions. Coverings lemma. Weak-type inequalities. -inequalities.

Approximative identities of a general form. Convergence in spaces of summable functions. Estimations for approximative identities. Almost everywhere convergence. Principal lemma of variational calculus.

 Examples of approximative identities: Steklov averages, Poisson’s kernel for half-space, Gauss-Weierstrass kernel.

Linear operators in spaces of summable functions. Operator’s norm linearization. Hadamard’s lemma. Riss-Thorin theorem of convexity operator’s norm. Convolutions on Euclidean spaces and their existence. Basic inequalities for convolutions. Young’s inequality.

Multi-indexes and differential operators. Schwartz functions. Fourier transform and its basic properties. Fourier transform and analysis operations. Inverse formula for smooth functions. Plancherel formula for smooth functions.

Elementary estimates. Inverse formula for . Fourier transform on . Plancherel theorem. Fourier transform for . Properties of Fourier transform on .

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Bibliography

  1. E.Stein, Singular integrals and differentiability properties of functions, Princeton, New Jersey, Princeton University Press, 1970.
  2. E.Stein, G.Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton, New Jersey, Princeton University Press, 1971.
  3. L.Grafakos, Classical Fourier Analysis, Springer, New-York, 2008.
  4. L.Grafakos, Modern Fourier Analysis, Springer, New-York, 2009.

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Methods of teaching

Comparative method, problematic method, dialog and heuristic method, method of forming knowledge of a personal significance.

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Language

English

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Requirements, monitoring

  • evaluation of an exercise.

Final grade on the discipline is a cumulative characteristic which is formed on a base of an undergraduate student’s knowledge during the total period of monitoring and on the semester grade(s).

Monitoring grade component can attain 50% of a final grade.

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Form of assessment

Exam