## On the properties of sums of trigonometric series with monotone coefficients.(English. Russian original)Zbl 0881.42004

Mosc. Univ. Math. Bull. 50, No. 3, 19-27 (1995); translation from Vestn. Mosk. Univ., Ser. I 1995, No. 3, 22-32 (1995).
The authors, among others, extend a well-known theorem of Hardy-Littlewood to the case $$0<p\leq 1$$. Let $$f(x)$$ and $$g(x)$$ denote the sums of the series $${a_0\over 2}+ \sum^\infty_{n=1} a_n\cos nx$$ and $$\sum^\infty_{n=1} a_n\sin nx$$, respectively, where $$a_n\to 0$$ and $$\Delta_ka_n\geq 0$$ for some $$k\geq 1$$ and any $$n$$. The new theorem reads as follows:
a) If $$\Delta_2a_n\geq 0$$, then for any $$p\in(0,\infty)$$, $C_1\left(\sum^\infty_{n=0}(\Delta_1a_n)^p(n+ 1)^{2p-2}\right)^{1/p}\leq|f|_p\leq C_2\Biggl(\sum^\infty_{n=0}(\Delta_1 a_n)^p(n+ 1)^{2p-2}\Biggr)^{1/p}.$ b) If $$\Delta_1a_n\geq 0$$, then for any $$p\in(0,\infty)$$, $C_3\Biggl(\sum^\infty_{n=1} a^p_n n^{p-2}\Biggr)^{1/p}\leq|g|_p\leq C_4\Biggl(\sum^\infty_{n=1} a^p_n n^{p-2}\Biggr)^{1/p},$ where the positive constants $$C_i$$ do not depend on $$\{a_n\}$$. Unfortunately in the English translation there are several upsetting misprints.

### MSC:

 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 42A05 Trigonometric polynomials, inequalities, extremal problems