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European Journal of Natural History

ISSN 2073-4972

ИФ РИНЦ = 0,301

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Nakhman A.D. 1

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1 Tambov State Technical University

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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

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

(2)

of conjugate Fourier series

k = 0, ±1, ±2, ...; (3)

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

d = const, d > 0

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

Maximal operators

Denote

(3)

f* and 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

(4)

For each h > 0 denote . 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

N → ∞, (5)

and

(6)

Then, for all f ∈ L(Q) the estimate

(7)

holds.

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

(8)

where k = 0, 1, ...;

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

(9)

holds.

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, . Hence, for x and t, such, that the estimate

(10)

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

(11)

hold.

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

(12)

It is obvious that

J1(x, k) ≤f*(x). (13)

Further,

Taking into account (10), we have

Here

if a positive integer S chosen from the condition

Hence

(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

(16)

According to (13) and (14) we obtain

J1(x, m) ≤ f*(x);

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 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

p ≥ 1

and strong type

p > 1;

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

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

then the relation

(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 Δ²λ_k(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

(21)

In the second sum of (21) all Δ²λ_k(h) are positive by (20); applying twice Abel transform, we have

Thus, under conditions (5) and

(22)

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

Exponential summation methods

Summation methods

λ₀(h) = 1, 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 { λ_k(h)} In this paper we consider

λ(x, h) = exp(–hx^α), α ≥ 1

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

(23)

where 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 λ_k(h) = exp(–hk^α), α ≥ 1; for α = 1 we have classical Poisson-Abel means.

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