1 
Name of course 
Introduction to harmonic analysis on Euclidean spaces 
2 
Year 
1, speciality Mathematics 
3 
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 
6 
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:

7 
Background 
Mathematical analysis, functional analysis, equations of mathematical physics. 
8 
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. HardyLittlewood maximal functions. Coverings lemma. Weaktype 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 halfspace, GaussWeierstrass kernel. Linear operators in spaces of summable functions. Operator’s norm linearization. Hadamard’s lemma. RissThorin theorem of convexity operator’s norm. Convolutions on Euclidean spaces and their existence. Basic inequalities for convolutions. Young’s inequality. Multiindexes 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 . 
9 
Bibliography 

10 
Methods of teaching 
Comparative method, problematic method, dialog and heuristic method, method of forming knowledge of a personal significance. 
11 
Language 
English 
12 
Requirements, monitoring 
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. 
13 
Form of assessment 
Exam 