DIRECT AND CONVERSE THEOREMS OF JACKSON TYPE IN $ L^p$ SPACES, $ 0<p<1$

Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in L 0

“…Zygmund [3] extended inequalities (0.2) and (0.3) to the norms ⋅ p , 1 ≤ p < ∞. The method of Zygmund for 0 < p < 1 (trigonometric interpolation) was used by Ivanov [4] and Storozhenko, Krotov, and Oswald [5], but an additional constant depending on p appeared in inequalities of the type (0.2). In 1981, Arestov [6] proposed a new method for the proof of inequality (0.2) for any 0 ≤ p < 1 and its analog in classes ϕ ( ) L .…”

Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes

“…Multiplying both sides of inequality (14) by the function 2 k ψ τ ( ) and integrating the obtained relation with respect to τ from 0 to h, we get…”

“…Note that the characteristic Ω 1 ( , ) f t p had earlier been used by Storozhenko, Krotov, and Oswald in [14] for the investigation of the behavior of the best approximation of functions by polynomials in the Haar system.…”

The Hadamard product of polynomials inH0

“…Obtaining Bernstein's inequality in Hardy spaces H p : P H p ≤ n P H p , for p < 1 had been a difficult problem (see, e.g. [14,29]). In an important paper [12], N. G. de Bruijn and T. A. Springer proved that if deg(P ) ≤ n, then…”