1.Introduction. Let , and

, (1.1)

(1.2)

be the set of a linear means of Fourier series and conjugate Fourier series. In various questions of the analysis there is a problem of behaviour of (1.1) and (1.2) when . Here are complex Fourier coefficients,

(1.3)

is infinite sequence defined by the values of parameter . In a case of discrete h the similar problems for (1.1) have been studied by L.I.Bausov ([1]).

We consider the semi-continuous methods of summation corresponding, basically, to a case of , =, where

, (1.4)

and function is continuous on [and twice differentiated on (. We are extending, in particular, a case of Poisson-Abel means ().

Let be a norm in Lebesque space (and

be a conjugate function ([2], v.1, p.402). Define ; .

2. Results. Estimates of -norms. The sequence (1.3) is called as convex (concave), if The sequence (1.3) is piecewise -convex, if changes the sign a finite number of times,

Theorem 2.1. If the sequence (1.3) is convex (concave) and

, (2.1)

for each then the estimates

; (2.2)

; (2.3)

(2.4)

hold.

Here will represent a constant, though not necessarily one such constant.

The estimates (2.2) - (2.4) remain valid, if a piecewise -convex sequence (1.3) satisfies to the condition (2.1) and there is constant , such, that

| (2.5)

for all

Proofs of both statements are based on the Abel transform of sums (1.1), (1.2) and on the estimates of Fejér means ([2], v.1, p. 148) by maximal operators

and .

3. Convex and piecewise -convex еxponentially summarising sequences. Consider a case (1.4). Let restrict oneself, basically, to consideration of functions

Theorem 3.1. Let function be twice continuously differentiated on , , ,

=exp (3.1)

and exp for everyone . Then the estimates (2.2)–(2.4)

are valid and the relations

= , (3.2)

= (3.3)

hold almost everywhere (a.e.) for everyone and in metrics ,

The results follow from theorem 2.1, if we note the convexity of (3.1) for and . Тhе convergence a.e. and in metrics follows from (2.4) and (2.2) by standard arguments ([2], v.2, p.464-465).

4. Examples.

4.1. Let then

, (4.1)

The sequence (4.1) satisfies to conditions of theorem 3.1 when and it is piecewise-convex for and satisfies to condition (2.5). Hence, the relations (2.2) – (2.4) and (3.2), (3.3) are valid for all

4.2. If then

. (4.2)

The sequence (4.2) is convex when and it is piecewise-convex for . By this reason the relations (2.2) – (2.4) and (3.2), (3.3) are valid for all

[2] Zygmund A. Trigonometric series. Vol. 1, 2. Moscow: “Mir”, 1965. V.1 –615 p., V.2 – 537 p.