Determination of jumps in terms of linear operators

A theorem of Luk´acs [4] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. M´oricz [5] proved a similar theorem for the rectangular partial sums of double variab...

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Datum:	2015
1. Verfasser:	Zviadadze, Sh.
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spelling	irk-123456789-1659222020-02-18T01:28:22Z Determination of jumps in terms of linear operators Zviadadze, Sh. Статті A theorem of Luk´acs [4] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. M´oricz [5] proved a similar theorem for the rectangular partial sums of double variable functions. We consider analogs of the M´oricz theorem for generalized Ces´aro means and for positive linear means. We consider a similar theorem in terms of linear operators satisfying certain conditions. Теорема Лукаша стверджує, що частиннi суми спряжених рядiв Фур’є перiодичної функцiї f, iнтегровної по Лебегу, розбiгаються з логарифмiчною швидкiстю в точках розриву першого роду функцiї f. Морiч довiв подiбну теорему для прямокутних частинних сум (функцiй двох змiнних). Розглянуто теореми, що аналогiчнi теоремi Морiча для узагальнених середнiх Чезаро та для позитивних лiнiйних середнiх. Аналогiчну теорему також розглянуто в термiнах лiнiйних операторiв, що задовольняють певнi умови. 2015 Article Determination of jumps in terms of linear operators / Sh. Zviadadze // Український математичний журнал. — 2015. — Т. 67, № 12. — С. 1649–1657. — Бібліогр.: 21 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165922 517.5 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
topic	Статті Статті
spellingShingle	Статті Статті Zviadadze, Sh. Determination of jumps in terms of linear operators Український математичний журнал
description	A theorem of Luk´acs [4] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. M´oricz [5] proved a similar theorem for the rectangular partial sums of double variable functions. We consider analogs of the M´oricz theorem for generalized Ces´aro means and for positive linear means. We consider a similar theorem in terms of linear operators satisfying certain conditions.
format	Article
author	Zviadadze, Sh.
author_facet	Zviadadze, Sh.
author_sort	Zviadadze, Sh.
title	Determination of jumps in terms of linear operators
title_short	Determination of jumps in terms of linear operators
title_full	Determination of jumps in terms of linear operators
title_fullStr	Determination of jumps in terms of linear operators
title_full_unstemmed	Determination of jumps in terms of linear operators
title_sort	determination of jumps in terms of linear operators
publisher	Інститут математики НАН України
publishDate	2015
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/165922
citation_txt	Determination of jumps in terms of linear operators / Sh. Zviadadze // Український математичний журнал. — 2015. — Т. 67, № 12. — С. 1649–1657. — Бібліогр.: 21 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT zviadadzesh determinationofjumpsintermsoflinearoperators
first_indexed	2025-07-14T20:23:04Z
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fulltext	UDC 517.5 Sh. Zviadadze (Javakhishvili Tbilisi State Univ., Georgia) DETERMINATION OF JUMPS IN TERMS OF LINEAR OPERATORS* ВИЗНАЧЕННЯ СТРИБКIВ У ТЕРМIНАХ ЛIНIЙНИХ ОПЕРАТОРIВ A theorem of Lukács [J. reine und angew. Math. – 1920. – 150. – S. 107 – 112] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. Móricz [Acta math. hung. – 2003. – 98. – P. 259 – 262] proved a similar theorem for the rectangular partial sums of double conjugate trigonometric Fourier series. We consider analogs of the Móricz theorem for generalized Cesáro means and for positive linear means. In the present paper we prove a similar theorem in terms of linear operators satisfying certain conditions. Теорема Лукаша [J. reine und angew. Math. – 1920. – 150. – S. 107 – 112] стверджує, що частиннi суми спряжених рядiв Фур’є перiодичної функцiї f , iнтегровної за Лебегом, розбiгаються з логарифмiчною швидкiстю в точках розриву першого роду функцiї f . Морiч [Acta math. hung. – 2003. – 98. – P. 259 – 262] довiв подiбну теорему для прямокутних частинних сум двiчi спряжених тригонометричних рядiв Фур’є. Розглянуто теореми, що аналогiчнi теоремi Морiча для узагальнених середнiх Чезаро та для позитивних лiнiй- них середнiх. У цiй статтi доведено аналогiчну теорему в термiнах лiнiйних операторiв, що задовольняють певнi умови. 1. Introduction. Let f be a 2π-periodic Lebesgue integrable function. The Fourier trigonometric series of the function f is defined by a0 2 + ∞∑ i=1 (ai cos ix+ bi sin ix), (1.1) where ai = 1 π π∫ −π f(x) cos ixdx and bi = 1 π π∫ −π f(x) sin ixdx are the Fourier coefficients of f . The conjugate series of (1.1) is defined by ∞∑ i=1 (ai sin ix− bi cos ix). (1.2) Let S̃k(f ;x) be the k th partial sum of series (1.2). Lukács [4] proved the following theorem. Theorem 1.1. If f ∈ L(−π, π] and for some point x ∈ (−π, π], there exists a number dx(f) such that lim t→0+ 1 t t∫ 0 \|f(x+ s)− f(x− s)− dx(f)\|ds = 0, then lim k→+∞ S̃k(f ;x) ln k = −dx(f) π . * Supported by Shota Rustaveli National Science Foundation (grant No. 12/25). c© SH. ZVIADADZE, 2015 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 1649 1650 SH. ZVIADADZE R. Riad [10] proved an analogous theorem in terms of the conjugate Walsh series. F. Móricz [5, 6] generalized Lukács’s theorem in terms of the Abel – Poisson means and proved estimate of the partial derivative of the Abel – Poisson mean of an integrable function at those points where it is smooth. Pinsky [9] generalized Fourier partial sums by using a family of convolusion operators with some classes of kernels. Q. Shi and X. Shi [11] discuss about the concentration factor methods for determination of jumps in terms of spectral data. Dansheng Yu, P. Zhou and S. Zhou [14] show how jumps can be determined by the higher order partial derivatives of the of its Abel – Poisson means. We [17, 18] examine the analogous theorems for the generalized Cesáro means, introduced by Akhobadze [1 – 3], as well as positive regular linear means, and consider [19] Lukás theorem for the functions and series introduced by Taberski [12, 13] as well as generalized Cesáro and positive regular linear means. Some results of this paper were announced in [17, 18]. P. Zhou and S. P. Zhou [15] proved an analogous theorem in terms of the linear operators which satisfy some certain conditions. F. Móricz [7] examined Lukács theorem for double trigonometric series. F. Móricz and W. R. Wade [8] generalized Lukács theorem for double Walsh series. We [21] generalized Móricz’s theorem and we proved that conditions in Móricz’s theorem is the best option for indices not to be dependent on each other. Also we considered analogues of these theorems for generalized Cesáro means and positive linear means matrix of which satisfy necessary conditions of regularity. 2. Determination of jumps in terms of linear operators. Let (m(k)), (n(k)) be a nonnegative sequences of real numbers such that lim k→+∞ m(k) = lim k→+∞ n(k) = +∞. Suppose there are given two sequences of 2π-periodic, odd and Lebesgue integrable functions Gk and Hk for which the following is true \|Gk(t)\| = O(m(k)), \|Hk(t)\| = O(n(k)) for all t, (2.1) \|Gk(t)\| = O(1/t), \|Hk(t)\| = O(1/t) t ∈ (0;π], (2.2) π∫ 0 Gk(t)dt ' lnm(k), π∫ 0 Hk(t)dt ' lnn(k). (2.3) Suppose that it is given two variable function f, 2π-periodic in both variables and Lebesgue integrable on (−π;π]2. Let’s consider the following linear operator: Fkj(f ;x, y) = 1 π2 π∫ −π π∫ −π f(u, v)Gk(u− x)Hj(v − y)dudv. (2.4) ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 DETERMINATION OF JUMPS IN TERMS OF LINEAR OPERATORS 1651 Let ϕ(x, y, u, v) := f(x+ u, y + v)− f(x− u, y + v)− f(x+ u, y − v) + f(x− u, y − v)− dxy(f), where dxy(f) is a number and Ψ(x, y, s, t) := s∫ 0 t∫ 0 \|ϕ(x, y, u, v)\|dudv. Theorem 2.1. Let f ∈ L(−π;π]2 and suppose that for a point (x, y) ∈ (−π;π]2 lim s,t→0+ Ψ(x, y, s, t)(ts)−1 = 0, (2.5) Ψ(x, y, s, t) = O(min{s, t}), 0 < s, t ≤ π. (2.6) Then for all sequences of convolution type operators Fkj which are defined by (2.4) where kernels satisfy conditions (2.1) – (2.3) the following equality is valid: lim k,j→+∞ Fkj(f ;x, y) lnm(k) lnn(j) = dxy(f) π2 . (2.7) Proof. Let us consider Fkj(f ;x, y), by changing of variables in the integral we get Fkj(f ;x, y) = 1 π2 π∫ 0 π∫ 0 ϕ(x, y, u, v)Gk(u)Hj(v)dudv+ + dxy(f) π2 π∫ 0 π∫ 0 Gk(u)Hj(v)dudv = A1(k, j) +A2(k, j). (2.8) By (2.5) for every ε > 0 we can choose δ such that Ψ(x, y, δ, δ)/δ2 < ε. (2.9) According the definition of m(k) and n(j) we can choose k and j such that 1/m(k), 1/n(j) < δ. Therefore A1(k, j) = 1 π2  1/m(k)∫ 0 + δ∫ 1/m(k) + π∫ δ   1/n(j)∫ 0 + δ∫ 1/n(j) + π∫ δ × ×ϕ(x, y, u, v)Gk(u)Hj(v)dudv = 3∑ r,s=1 Brs. (2.10) Throughout, we use C to stand for an absolute positive constant, which may have different values in different occurrences. ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 1652 SH. ZVIADADZE Note that Brs and Bsr, s, r ∈ {1, 2, 3}, r 6= s, can be estimated similarly. By (2.1) and (2.9) we have \|B11\| ≤ Cm(k)n(j) π2 1/m(k)∫ 0 1/n(j)∫ 0 \|ϕ(x, y, u, v)\|dudv = O(1). (2.11) Using (2.2) and integration by parts with respect to v we obtain \|B12\| ≤ Cm(k) 1/m(k)∫ 0 δ∫ 1/n(j) \|ϕ(x, y, u, v)\| v dudv ≤ Cε+ Cε δ∫ 1/n(j) dv v = O(lnn(j)). (2.12) By (2.1), (2.2), (2.6) and integration by parts with respect to v we get \|B13\| ≤ Cm(k) 1/m(k)∫ 0 π∫ δ \|ϕ(x, y, u, v)\| v dudv ≤ ≤ Cm(k)  π∫ 0 1/m(k)∫ 0 \|ϕ(x, y, u, t)\|dudt+ π∫ δ 1 v2 π∫ 0 1/m(k)∫ 0 \|ϕ(x, y, u, t)\|dudtdv  = = Cm(k)  1 π Ψ ( x, y, 1 m(k) , π ) + Ψ ( x, y, 1 m(k) , π ) π∫ δ dv v2  = O(1) (2.13) and \|B22\| ≤ C δ∫ 1/m(k)  δ∫ 1/n(j) \|ϕ(x, y, u, v)\|dv v  du u . (2.14) Furthermore, integration by parts with respect to v gives δ∫ 1/n(j) \|ϕ(x, y, u, v)\|dv v ≤ 1 δ δ∫ 0 \|ϕ(x, y, u, t)\|dt+ δ∫ 1/n(j) 1 v2 v∫ 0 \|ϕ(x, y, u, t)\|dtdv. Now by using (2.9) and integration by parts with respect to u we have δ∫ 1/m(k) 1 δ δ∫ 0 \|ϕ(x, y, u, t)\|dt+ δ∫ 1/n(j) 1 v2 v∫ 0 \|ϕ(x, y, u, t)\|dtdv  du u ≤ ≤ 1 δ2 δ∫ 0 δ∫ 0 \|ϕ(x, y, s, t)\|dsdt+ 1 δ δ∫ 0 δ∫ 1/n(j) 1 v2 v∫ 0 \|ϕ(x, y, s, t)\|dtdvds+ ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 DETERMINATION OF JUMPS IN TERMS OF LINEAR OPERATORS 1653 + δ∫ 1/m(k) δ∫ 1/n(j) 1 v2u2 v∫ 0 u∫ 0 \|ϕ(x, y, s, t)\|dsdtdudv ≤ ≤ ε+ ε δ∫ 1/n(j) dv v + ε δ∫ 1/m(k) δ∫ 1/n(j) dudv vu . By (2.14) and the last estimations we get \|B22\| ≤ Cε lnm(k) lnn(j), (2.15) where C is a fixed positive constant. Analogously, using once more integration by parts with respect to u and (2.6) we obtain \|B23\| ≤ C δ∫ 1/m(k) π∫ δ \|ϕ(x, y, u, v)\| uv dudv ≤ ≤ C δ δ∫ 1/m(k)  π∫ 0 \|ϕ(x, y, u, v)\|dv  du u ≤ ≤ C δ2 δ∫ 0 π∫ 0 \|ϕ(x, y, s, v)\|dsdv + C δ δ∫ 1/m(k) 1 u2 u∫ 0 π∫ 0 \|ϕ(x, y, s, v)\|dsdvdu = = C Ψ(x, y, δ, π) δ2 + C δ δ∫ 1/m(k) Ψ(x, y, u, π) u2 du = O(lnm(k)) (2.16) and \|B33\| ≤ 4 π4δ2 π∫ 0 π∫ 0 \|ϕ(x, y, u, v)\|dudv = O(1). (2.17) Finally by (2.10) – (2.17) we have lim k,j→+∞ A1(k, j)/(lnm(k) lnn(k)) = 0. (2.18) Now consider A2(k, j). By (2.3) we get I2(k) = dxy(f) π2 π∫ 0 π∫ 0 Gk(u)Hj(v)dudv = dxy(f) π2 π∫ 0 Gk(u)du π∫ 0 Hj(v)dv ' ' dxy(f) π2 lnm(k) lnn(j). If we combine (2.18) and the last estimation it completes the proof of theorem. ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 1654 SH. ZVIADADZE 3. Applications. The question arises naturally: which kind of kernel satisfies conditions (2.1) – (2.3)? First of all let us note that the conjugate Dirichlet kernel satisfies above mentioned conditions. Indeed assume that m(k) = k and n(j) = j. We know that \|D̃k(t)\| ≤ k for all t, \|D̃k(t)\| ≤ 2/t, 0 < t ≤ π, (see [16], Chap. II, (5.11)) and π∫ 0 D̃k(u)du ' ln k, k → +∞, where D̃k(t) denotes the conjugate Dirichlet kernel. In this case get Moricz’s [6] result. Analogously generalized Cesáro mean of the conjugate Dirichlet kernel satisfies (2.1) – (2.3) conditions (see proof of Theorem 2.1 in [20] or proof of Theorem 2.2 in [21]). Thus we get the author’s result [21]. In [21] we generalized Moricz’s theorem in case of matrix summability. It is easy to see that in the case if 4th dimensional matrix can be represented as a product of two dimensional matrixes, then Theorem 2.1 generalizes our [21] result. Even more we can make sure that generalized de la Vallée Poussin mean of the conjugate Dirichlet kernel satisfies conditions (2.1) – (2.3). In [15] by authors’ was obtained Lukacs type theorem where kernel satisfies conditions (2.1) – (2.4) from [15]. Authors also have shown that generalized de la Vallée Poussin mean of the conjugate Dirichlet kernel satisfies (2.1) – (2.4) conditions in [15]. But (2.1) – (2.3) conditions are different from (2.1) – (2.4) and we introduced different, self-contained proof. Let S̃kj(f ;x, y) be a rectangular partial sum of the double conjugate trigonometric Fourier series and let Vm,n,k,s(f ;x, y) = 1 (k + 1)(s+ 1) m∑ i=m−k n∑ j=n−s S̃ij(f ;x, y), be generalized de la Vallée Poussin mean of the rectangular partial sum of the double conjugate trigonometric Fourier series where 0 ≤ k ≤ m and 0 ≤ s ≤ n. It is easy to verify that Vm,n,0,0(f ;x, y) = S̃mn(f ;x, y), Vm,n,m,n(f ;x, y) = σ̃mn(f ;x, y), where σ̃mn(f ;x, y) denotes (C,1,1) means of the rectangular partial sums of conjugate trigonometric Fourier series. By definition of the generalized de la Vallée Poussin sums we get Vm,n,k,s(f ;x, y) = 1 π2 π∫ −π π∫ −π f(u, v)K̃m(u− x)K̃n(v − y)dudv, where K̃m(t) = 1 k + 1 m∑ i=m−k D̃i(t). ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 DETERMINATION OF JUMPS IN TERMS OF LINEAR OPERATORS 1655 Corollary 3.1. Let f ∈ L(−π;π]2 and suppose that for a point (x, y) ∈ (−π;π]2 (2.5) and (2.6) hold. Then lim m,n→+∞ Vm,n,k,s(f ;x, y) lnm lnn = dxy(f) π2 . Proof. We will show that (2.1), (2.2) and (2.3) conditions are true for generalized de la Vallée Poussin mean of the conjugate Dirichlet kernel. Suppose that Gm(t) = K̃m(t) and Hn(t) = K̃n(t) then by the definition of generalized de la Vallée Poussin mean of the conjugate Dirichlet kernel and the following estimation \|D̃k(t)\| ≤ k for all t (see [16], Chap. II, (5.11)) and by the formula for the sum of the terms of an arithmetic sequence we have \|K̃m(t)\| ≤ 1 k + 1 m∑ i=m−k \|D̃i(t)\| ≤ 1 k + 1 m∑ i=m−k i = = 1 k + 1 m− k +m 2 (m− (m− k) + 1) = 2m− k 2 ≤ m, for all t ∈ [0;π], and condition (2.1) holds. By the estimation \|D̃i(t)\| ≤ 2/t, t ∈ (0;π] (see [16], Chap. II, (5.11)) we obtain \|K̃m(t)\| ≤ 1 k + 1 m∑ i=m−k \|D̃i(t)\| ≤ 1 k + 1 m∑ i=m−k 2 t = 2 t . Thus condition (2.2) holds. Now consider π∫ 0 K̃m(u)du = 1 k + 1 m∑ i=m−k π∫ 0 D̃i(u)du = 1 k + 1 m∑ i=m−k Ui. It is well known that for any ε > 0 there exists number N = N(ε) such that, for all i > N we get 1− ε < Ui ln i < 1 + ε. (3.1) Consider following sum: 1 k + 1 m∑ i=m−k Ui = 1 k + 1 ( N∑ i=m−k Ui + m∑ i=N+1 Ui ) = I1 + I2. It is easy to see that I1 ≤ max i≤N \|Ui\| ·N. Also by the right-hand side inequality of (3.1) we have I2 ≤ 1 + ε k + 1 m∑ i=N+1 ln i ≤ (1 + ε) lnm k + 1 (m−N) ≤ (1 + ε) lnm. Therefore we get ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 1656 SH. ZVIADADZE lim m→+∞ 1 lnm π∫ 0 K̃m(u)du ≤ 1. (3.2) If for the same ε we choose M such that 2/M < ε, then we get following lower estimation of I2 : I2 = 1 k + 1 m∑ i=N+1 Ui = 1 k + 1 m/M∑ i=N+1 Ui + 1 k + 1 m∑ i=m/M+1 Ui = J1 + J2. By the left-hand side inequality of (3.1) we obtain J1 ≥ 1− ε k + 1 m/M∑ i=N ln i ≥ 1− ε k + 1 lnN m/M∑ i=N ≥ 1− ε k + 1 lnN = O(1). Reasoning analogously as in previous estimation we get J2 ≥ 1− ε k + 1 m∑ i=m/M+1 ln i ≥ 1− ε k + 1 (lnm− lnM)  m∑ i=m−k − m/M∑ i=m−k  = L1 − L2. It is easy to see that L1 = (1− ε)(lnm− lnM). On the other hand L2 = o(lnm), indeed L2 ≤ 1− ε k + 1 (lnm− lnM) · 2 · (m M − (m− k) ) ≤ k k + 1 (lnm− lnM) 2 M < < ε(lnm− lnM) = o(lnm). Thus we obtain lim m→+∞ 1 lnm π∫ 0 K̃m(u)du ≥ 1. From the (3.2) and last estimation we conclude that K̃m(u) satisfies condition (2.3). Corollary 3.1 is proved. In all above examined examples kernels Gm and Hm are the same type, and it’s integrals have the same order (logarithmic). Advantage of above considered integral operator and general kernel is that we can consider kernels with different order. Consider matrices (bki) and (cjs): bki = 1 if i = [ k1/ ln ln k ] , 0 if i 6= [ k1/ ln ln k ] , and cjs = 1 if s = j, 0 if s 6= j. ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 DETERMINATION OF JUMPS IN TERMS OF LINEAR OPERATORS 1657 We construct matrix (akjis) where akjis = bki · cjs. Let cite an example. Consider σkj(f ;x, y) = +∞∑ i=0 +∞∑ s=0 akjisS̃is(f ;x, y) = = 1 π2 π∫ −π π∫ −π f(u, v) +∞∑ i=0 bkiD̃i(u− x) +∞∑ s=0 cjsD̃s(v − y)dudv = = 1 π2 π∫ −π π∫ −π f(u, v)D̃[k1/ ln ln k](u− x)D̃j(v − y)dudv. It is easy to see that in this case Gm(k)(t) = D̃[k1/ ln ln k] and m(k) = [k1/ ln ln k]. 1. Akhobadze T. On generalized Cesáro summability of trigonometric Fourier series // Bull. Georg. Acad. Sci. – 2004. – 170. – P. 23 – 24. 2. Akhobadze T. On the convergence of generalized Cesáro means of trigonometric Fourier series I // Acta math. hung. – 2007. – 115. – P. 59 – 78. 3. Akhobadze T. On the convergence of generalized Cesáro means of trigonometric Fourier series II // Acta math. hung. – 2007. – 115. – P. 79 – 100. 4. Lukács F. Über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe // J. reine und angew. Math. – 1920. – 150. – S. 107 – 112. 5. Móricz F. Determination of jumps in terms of Abel – Poisson means // Acta math. hung. – 2003. – 98. – P. 259 – 262. 6. 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Soc. math. pol. Ser. I. Prace math. – 1974. – 17. – P. 499 – 512. 14. Yu D. S., Zhou P., Zhou S. On determinatioin of jumps in terms of Abel – Poisson mean of Fourier series // J. Math. Anal. and Appl. – 2008. – 341. – P. 12 – 23. 15. Zhou P., Zhou S. P. More on determination of jumps // Acta math. hung. – 2008. – 118. – P. 41 – 52. 16. Zigmund A. Trigonometric series. – Cambridge Univ. Press, 1959. – Vol. 1. 17. Zviadadze Sh. On the statement of F. Lukács // Bull. Georg. Acad. Sci. – 2005. – 171. – P. 411 – 412. 18. Zviadadze Sh. On a Lukács theorem for regular linear means of conjugate trigonometric Fourier series // Bull. Georg. Acad. Sci. – 2006. – 173. – P. 435 – 438. 19. Zviadadze Sh. On generalizations of a theorem of Ferenc Lukács // Acta math. hung. – 2009. – 122, № 1-2. – P. 105 – 120. 20. Zviadadze Sh. On some properties of conjugate trigonometric Fourier series // Georg. Math. J. – 2013. – 20, № 2. – P. 397 – 413. 21. Zviadadze Sh. On some properties of double conjugate trigonometric Fourier series // Acta math. hung. – 2012. – 134, № 4. – P. 452 – 417. Received 13.02.14 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12

Determination of jumps in terms of linear operators

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