Frontmatter -- Contents -- Preface to the English Edition -- Preface to the Russian Edition -- Introduction -- Chapter 1. SIMPLEST EXTREMAL PROBLEMS -- Chapter 2. APPROXIMATION OF FUNCTIONS OF ONE VARIABLE BY FOURIER SUMS -- Chapter 3. APPROXIMATION OF FUNCTIONS OF MANY VARIABLES BY FOURIER SUMS -- Chapter 4. FEJER SUMS -- Chapter 5. SPHERICAL RIESZ SUMS -- Chapter 6. ROGOSINSKI SUMS -- Chapter 7. FAVARD SUMS -- REFERENCES -- SUBJECT INDEX