ON THE SUMMABILITY OF THE DISCRETE HILBERT TRANSFORM

Abstract: In this paper, we study the asymptotic behavior of the distribution function of the discrete Hilbert transform of sequences from the class l1 and find a necessary condition and a sufficient condition for the summability of the discrete Hilbert transform of a sequence from the class l1.

In this paper, we study the asymptotic behavior of the distribution function (Hb) (λ) of the Hilbert transform of a sequence b ∈ l 1 as λ → 0 and find a necessary condition and a sufficient condition for the summability of the discrete Hilbert transform of a sequence from the class l 1 .

Asymptotic behavior of the distribution function of the discrete Hilbert transform
Theorem 1.Let b ∈ l 1 .Then the following equation holds: We first prove an auxiliary lemma.
Lemma 1.Let b ∈ l 1 and n∈Z b n = 0. Then the following equation holds: P r o o f.Assume first that the sequence b ∈ l 1 is concentrated on some finite interval [−m, m], i. e., b n = 0 for |n| > m.In this case, from the equality for large values of n, whence the asymptotic equation (1.2) follows.
Let us now consider the general case.From the condition n∈Z b n = 0, it follows that, for all ε > 0 there exist sequences b where the sequence b ′ ∈ l 1 is concentrated on some finite interval [−m, m] and n∈Z b ′ n = 0, and the sequence b ′′ ∈ l 1 satisfies the inequality 2) is satisfied for the sequence b ′ ∈ l 1 , and, therefore, there exists λ (ε) > 0 such that the inequality for all λ > 0, where (Hb ′′ ) (λ) = {n∈Z: |(Hb ′′ ) n |>λ} 1.From inequalities (1.3) and (1.4) and the inclusion Since (Hb ′′ t) n = α/n for n = 0 and (Hb ′′ ) 0 = 0, we have For all 0 < ε < 1, by the inclusions This implies equation (1.1) and completes the proof of Theorem 1. (2.1) P r o o f.We first we prove that, if h = {h n } n∈Z ∈ l 1 , then the distribution function h (λ) = {n∈Z: |hn|>λ} 1 of the sequence h satisfies the condition

A necessary condition and a sufficient condition for the summability of the discrete Hilbert transform
Note that the condition h = {h n } n∈Z ∈ l 1 implies that the set of {n ∈ Z : |h n | > λ} is finite for all λ > 0.Then, the inequality Hence, taking into account that the function h(λ) is decreasing, we obtain (2.2).It follows from (2.1) that, if Hb ∈ l 1 , then (2.4) From condition (i) for n = 0, we obtain that It follows from inequalities (2.4) and (2.5) that Let us estimate the summands J k , k = 1, 2, 3, 4. From condition (ii) and f equalities of the form n<0 for m > 0, and for m < 0, we obtain that From this and (2.6), we obtain (2.3).The proof of Theorem 3 is complete.
It follows from condition (ii) that From this and (2.8), we obtain (2.7).The proof of Theorem 4 is complete.
Denote by l p , p ≥ 1, the class of numeric sequences b = {b n } n∈Z satisfying the condition b lp = n∈Z |b n | p 1/p < ∞, where Z is the set of integers.Let b = {b n } n∈Z ∈ l 1 .The sequence H(b) = {(Hb) n } n∈Z is called the Hilbert transform of the sequence b = {b n } n∈Z , where (Hb) n = m =n b m n − m , n ∈ Z.
This shows that equality (1.2) holds for all b ∈ l 1 satisfying the condition n∈Z b n = 0.This completes the proof of Lemma 1. P r o o f of Theorem 1.In the case n∈Z b n = 0, the statement of the theorem follows from Lemma 1.Consider the case n∈Z b n = α = 0. We use the following notation: b ′ n = b n for n = 0, b ′ 0 = b 0 − α, b ′′ n = 0 for n = 0, and b ′′ 0 = α.Then b = b ′ + b ′′ , where b

Theorem 2 .
Let b ∈ l 1 .If Hb ∈ l 1 , then it is necessary that the following equation holds: n∈Z b n = 0.

Theorem 3 .
and, therefore, by Theorem 1, we obtain that the equation (2.2) holds.The proof of Theorem 2 is complete.If asequence b ∈ l 1 satisfies the conditions (i) n∈Z b n = 0; (ii) m∈Z |b m | ln (e + |m|) < ∞, then Hb ∈ l 1 and the following inequality holds: Hb l 1 ≤ 6 m∈Z |b m | ln (e + |m|) .(2.3)P r o o f.It follows from the definition of the discrete Hilbert transform that |(Hb) 0 | = m =0 b m m ≤ b l 1 .

Theorem 4 .
The following equation holds under the conditions of Theorem 3: n∈Z (Hb) n = 0.(2.7)P r o o f.By the conditions of Theorem 3,