Scientific journal
European Journal of Natural History
ISSN 2073-4972
ИФ РИНЦ = 0,283


Nakhman A.D. 1
1 Tambov State Technical University
1019 KB
We consider the linear means of the conjugate Fourier series of integrable 2π-periodic function f(y), generated by summing infinite sequence λ(h). For the positive values of h, and any x, the behavior of λ-means is investigated, when a point (y, h) tends to (x, 0) within the fixed “corner” area Γ(x). In the case of the summation sequences, decreasing quickly enough, the estimates of strong and weak type of corresponding maximal operators are obtained. We establish the convergence of λ-means to the conjugate function, when (y, h) tends to (x, 0) along the paths within Γ(x). An important special case of received statements is a non-tangential convergence of λ-means for summation methods of exponential type. Results include classical case of Poisson-Abel means.
conjugate series; estimates of the weak and strong type; non-tangential summability

Formulation of the problem

Denote Q = [–π, π]; let Lp = Lp(Q) be Lebesgue class of 2π-periodic functions of a real variable, for which

Nakhman01.wmf p ≥ 1,

set L = L(Q) = L1(Q). Let

λ = {λk(h), k = 0, 1, ...; λ0(h) = 1} (1)

be an arbitrary sequence infinite, generally speaking, determined by values of parameter h > 0. In this paper we study the behavior of λ-means

Nakhman02.wmf (2)

of conjugate Fourier series

Nakhman03.wmf Nakhman04.wmf k = 0, ±1, ±2, ...; (3)

when (y, h) > (x, +0) along the paths within

Nakhman05.wmf d = const, d > 0

(tending along non-tangential paths). We generalize and strengthen some of the results of [3, 4, 5].

Maximal operators


Nakhman06.wmf Nakhman07.wmf (3)

f* and Nakhman08.wmf are defined ([1], vol. 1, p. 60, 401–402, 442, 443) for every f ∈ L; moreover, in this case there is almost everywhere a conjugate function


In accordance with λ-means (2), introduced above, we define maximal operator

Nakhman10.wmf (4)

For each h > 0 denote Nakhman11.wmf. The basis of the results of the behavior of means (2) is the following statement.

Theorem 1. Let the sequence (1) decreases so rapidly that

Nakhman12.wmf N → ∞, (5)


Nakhman13.wmf (6)

Then, for all f ∈ L(Q) the estimate

Nakhman14.wmf (7)


Here and below C will represent constants, which depend only on clearly specified indexes.

Auxiliary assertion

Consider ([2], vol. 1, pp. 86, 153) the conjugate Dirichlet kernel


and the conjugate Fejer kernel

Nakhman16.wmf (8)

where Nakhman17.wmf k = 0, 1, ...; Nakhman18.wmf

Lemma. For all k = 0, 1, ... and (y, h) ∈ Γd(x) the estimate

Nakhman19.wmf (9)


Proof. Let’s start with a few comments. At k = 0 the left side of (9) vanishes, so consider k = 1, 2, ...

If (y, h) ∈ Γd(x), then, obviously, Nakhman20.wmf. Hence, for x and t, such, that Nakhman21.wmf the estimate

Nakhman22.wmf (10)

is valid. Indeed, (10) follows from inequality Nakhman23.wmf for all (y, h) ∈ Γd(x). Then, by definitions (8), the estimates

Nakhman24.wmf Nakhman25.wmf

Nakhman26.wmf Nakhman27.wmf (11)


Assume firstly k ≤ m and obtain the relation (9). By (11) we have

Nakhman28.wmf (12)

It is obvious that

J1(x, k) ≤f*(x). (13)



Taking into account (10), we have




if a positive integer S chosen from the condition



Nakhman33.wmf (14)

Finally, in view of (10) and (11)

J3(x, k) ≤ Cf*(x). (15)

Now, according to (12)–(15), the estimate (9) is valid at all k ≤ m.

Consider now the case of k > m. By (11) we have

Nakhman34.wmf (16)

According to (13) and (14) we obtain

J1(x, m) ≤ f*(x); Nakhman35.wmf

Further, in view of (11) and (10)


It follows now from (16) that


for all k > m.

Thus, the estimate (9) is valid for all k = 1,2, ..., and lemma is proved.

Proof of Theorem 1

Applying (3), Abel transform twice ([2], vol. 1, p. 15), the obvious estimate Nakhman38.wmf N = 1, 2, …, and (11), we obtain for (2)


According to (5) and (9) we have


and, because of the condition (19), we obtain the assertion (7).

Estimates of the weak and strong type

Theorem 2. Under the conditions of Theorem 1 the estimates of weak type

Nakhman41.wmf p ≥ 1

and strong type

Nakhman42.wmf p > 1;


Nakhman44.wmf 0 < p < 1

are valid.

The assertion follows from Theorem 1 and the corresponding estimates of weak and strong type for (3); see ([2], vol.1, pp. 58–59, 404).

Non-tangential summability

Theorem 3. If the sequence (1) satisfies to the conditions (5), (6) and

Nakhman45.wmf k = 0, 1, ..., (17)

then the relation

Nakhman46.wmf (18)

holds almost everywhere for each f ∈ L(Q).

The relations (18) follows from the weak type estimates (theorem 2) and condition (17) by the standard method ([2], vol. 2, pp. 464–465).

Piecewise convex summation methods

It noted in [3–5] (cf. [2], p. 476–478) that under the condition (5) every piecewise-convex sequence (1) satisfies the condition


By virtue of piecewise convexity of sequence (1), the second finite differences Δ2λk(h) retain the sign; suppose for definiteness, it will be a plus sign at all sufficiently large k (depending, generally speaking, from h), namely k ≥ τ(m), where τ = τ(m) – some positive integer,

 τ = τ(m) = τ(m, λ) ≤ m. (20)

The sum (6) does not exceed

Nakhman48.wmf (21)

In the second sum of (21) all Δ2λk(h) are positive by (20); applying twice Abel transform, we have


Thus, under conditions (5) and

Nakhman50.wmf (22)

the assertions of Theorems 2 and 3 are valid for each piecewise-convex sequence (1).

Exponential summation methods

Summation methods

λ0(h) = 1, Nakhman51.wmf k = 1, 2, ..., where λ(x, h) = exp(–hφ(x))

were studied in [3–4] in the case of “radial” convergence; in particular, it was given the condition of piecewise convexity of sequence { λk(h)} In this paper we consider

λ(x, h) = exp(–hxα), α ≥ 1

It is easy to show that this function is a piecewise-convex; verify now the satisfialibity of condition (22). We have

Nakhman52.wmf (23)

where Nakhman53.wmf is Euler gamma function. For α ≥ 1 the right side of (23) does not exceed a constant that depends only on α. Thus, Theorems 2 and 3 are valid for exponential summation methods λk(h) = exp(–hkα), α ≥ 1; for α = 1 we have classical Poisson-Abel means.