A. S. Belov

Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series

AbstractFairly general conditions on the coefficients

On the convergence in mean of trigonometric Fourier series

\left\{ {a_n } \right\}_{n = 1}^\infty

An estimate of the constant term of a nonnegative trigonometric polynomial with integer coefficients

of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation

Norms of lacunary polynomials in functional spaces

\sum\nolimits_{k = \left[ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right]}^{2n} {{{\left| {a_k } \right|} \mathord{\left/ {\vphantom {{\left| {a_k } \right|} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \left( { = O\left( 1 \right),{\text{ respectively}}} \right)

Estimate of the remainder in the asymptotic solution of an extremal problem involving nonnegative trigonometric polynomials

are given.

Use of complex analysis for deriving lower bounds for trigonometric polynomials

It is proved that a trigonometric cosine series of the form ΣnEmphasis>=0/∞an cos(nx) with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.