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  • 1.
    Baramidze, Lasha
    et al.
    Department of Mathematics, Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Sharp Hp- Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 242Article in journal (Refereed)
    Abstract [en]

    We prove and discuss some new Hp-Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out

  • 2.
    Blahota, I.
    et al.
    Institute of Mathematics and Computer Sciences, University of Nyìıregyhàza.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Two-sided Estimates of the Lebesgue Constants with respect to Vilenkin Systems and Applications2018In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 60, no 1, p. 17-34Article in journal (Refereed)
    Abstract [en]

    In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

  • 3.
    Blahota, István
    et al.
    Institute of Mathematics and Computer Sciences, College of Nyíregyháza.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the Nörlund means of Vilenkin-Fourier series2015In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 65, no 4, p. 983-1002Article in journal (Refereed)
    Abstract [en]

    We prove and discuss some new (H p ,L p )-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q k : k ⩾ 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results.In the special cases of general Nörlund means t n with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q k : k ⩾ 0} in an appropriate way.

  • 4.
    Blahota, István
    et al.
    Institute of Mathematics and Computer Sciences, University of Nyíregyháza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A note on maximal operators of Vilenkin-Nörlund means2016In: Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, ISSN 1786-0091, E-ISSN 1786-0091, Vol. 32, no 12, p. 203-213Article in journal (Refereed)
    Abstract [en]

    In this paper we prove and discuss some new (Hp;Lp)-type inequalities of weighted maximal operators of Vilenkin - Nörlund means with non-increasing coeffcients. These results are the best possible in a special sense. As applications, both some well-known and new results are pointed out in the theory of strong convergence of Vilenkin - Nörlund means with non-increasing coeffcients

  • 5.
    Blahota, István
    et al.
    Institute of Mathematics and Computer Sciences, College of Nyíregyháza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the (C, α)-means with respect to the Walsh system2014In: Analysis Mathematica, ISSN 0133-3852, E-ISSN 1588-273X, Vol. 40, no 3, p. 161-174Article in journal (Refereed)
    Abstract [en]

    In our main result we prove strong convergence theorems for Cesàro means (C, α) on the Hardy spaces H1/(1+α), where 0 < α < 1. © 2014 Akadémiai Kiadó, Budapest, Hungary.

  • 6.
    Blahota, István
    et al.
    Institute of Mathematics and Computer Sciences, College of Nyíregyháza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Strong convergence theorem for Vilenkin-Fejér means2014In: Publicationes mathematicae (Debrecen), ISSN 0033-3883, E-ISSN 2064-2849, Vol. 85, no 1-2, p. 181-196Article in journal (Refereed)
    Abstract [en]

    As main result we prove strong convergence theorems of Vilenkin- Fejér means when 0 < p ≤ 1/2.

  • 7.
    Buchukuri, Tengiz
    et al.
    A. Razmadze Mathematical Institute, Tbilisi State University.
    Dudachava, Roland
    A. Razmadze Mathematical Institute, Tbilisi State University.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Laplace–Beltrami equation on hypersurfaces and Γ-convergence2017In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 40, no 13, p. 4637-4657Article in journal (Refereed)
    Abstract [en]

    A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface C with the boundary is investigated. The main objective is to trace what happens in Γ-limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ-limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space Rn.

  • 8.
    Memić, Nacima
    et al.
    Department of mathematics, University of Sarajevo.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A note on the maximal operators of Vilenkin-Nörlund means with non-increasing coefficients2016In: Studia scientiarum mathematicarum Hungarica (Print), ISSN 0081-6906, E-ISSN 1588-2896, Vol. 53, no 4, p. 545-556Article in journal (Refereed)
    Abstract [en]

    In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H1/(1+α) to the space weak-L1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H1/(1+α) to the space L1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

  • 9.
    Nagy, Károly
    et al.
    College of Nyiregyhaza, Institute of Mathematics and Computer Science, Nyiregyhaza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Approximation by Walsh-Marcinkiewicz means on the Hardy space H-2/32014In: Kyoto Journal of Mathematics, ISSN 2156-2261, E-ISSN 2154-3321, Vol. 54, no 3, p. 641-652Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Marcinkiewicz means in terms of the modulus of continuity on the Hardy space H-2/3.

  • 10.
    Nagy, Károly
    et al.
    College of Nyiregyhaza, Institute of Mathematics and Computer Science, Nyiregyhaza, Institute of Mathematics and Computer Sciences, College of Nyíregyháza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Strong convergence theorem for Walsh-Marcinkiewicz means2016In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 19, no 1, p. 185-195Article in journal (Refereed)
    Abstract [en]

    It is known that the maximal operator ofWalsh-Marcinkiewicz means is bounded from the dyadic martingale Hardy space Hp to the space Lp for p > 2/3 and the condition p > 2/3 is essential. In the case p = 2/3 the boundedness of the maximal operator does not hold. This means that the investigation of the maximal operator at the endpoint case p = 2/3 plays an important role. The main aim of this paper is to prove a strong convergence theorem for the Walsh-Marcinkiewicz means on the Hardy space H2/3.

  • 11.
    Nagy, Károly
    et al.
    College of Nyiregyhaza, Institute of Mathematics and Computer Science, Nyiregyhaza, Institute of Mathematics and Computer Sciences, University of Nyíregyháza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The Walsh–Kaczmarz–Marcinkiewicz means and Hardy spaces2016In: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 149, no 2, p. 346-374Article in journal (Refereed)
    Abstract [en]

    It is known that the maximal operator σ κ,∗ (f):=sup n∈P |σ κ n (f)| σκ,∗(f):=supn∈P|σnκ(f)| is bounded from the dyadic Hardy space H p Hp into the space L p Lp for p>2/3 p>2/3 [6]. Moreover, Goginava and Nagy showed that σ κ,∗ σκ,∗ is not bounded from the Hardy space H 2/3 H2/3 to the space L 2/3 L2/3 [9]. The main aim of this paper is to investigate the case 0

  • 12.
    Nagy, Károly
    et al.
    College of Nyiregyhaza, Institute of Mathematics and Computer Science, Nyiregyhaza.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Walsh-Marcinkiewicz means and Hardy spaces2014In: Central European Journal of Mathematics, ISSN 1895-1074, E-ISSN 1644-3616, Vol. 12, no 8, p. 1214-1228Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space Hp, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space Hp, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means

  • 13.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A Sharp Boundedness Result Concerning Some Maximal Operators of Vilenkin–Fejér Means2016In: Mediterranean Journal of Mathematics, ISSN 1660-5446, E-ISSN 1660-5454, Vol. 13, no 4, p. 1841-1853Article in journal (Refereed)
    Abstract [en]

    In this paper, we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fejér means in this subspace is bounded from the Hardy space Hp to the space Lp for all 0 < p ≤ 1/2. Moreover, we prove that the result is in a sense sharp

  • 14.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Maximal Operators of Vilenkin–Nörlund Means2015In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 2, no 1, p. 76-94Article in journal (Refereed)
    Abstract [en]

    In this paper we prove and discuss some new (H p ,weak−L p ) type inequalities of maximal operators of Vilenkin–Nörlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin–Nörlund means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

  • 15.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On an approximation of 2-dimensional Walsh–Fourier series in martingale Hardy spaces2018In: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 9, no 1, p. 137-150Article in journal (Refereed)
    Abstract [en]

    In this paper, we investigate convergence and divergence of partial sums with respect to the 2-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh-Fourier series. We also show that these conditions are in a sense necessary and suffcient. 

  • 16.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. RUDN University, Moscow, Russia; UiT The Arctic University of Norway.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. School of Informatics, Engineering and Mathematics, The University of Georgia.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the Nörlund logarithmic means with respect to Vilenkin system in the Martingale Hardy Space H12018In: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 154, no 2, p. 289-301Article in journal (Refereed)
    Abstract [en]

    We prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space H1.

  • 17. Simon, Ilona
    et al.
    Memić, Nacima
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Strong convergence of two-dimensional Vilenkin-Fourier series2016In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, no 4, p. 485-500Article in journal (Refereed)
    Abstract [en]

    We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space inline image to the space inline image for inline image We also prove that the sequence in the denominator cannot be improved.

  • 18.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University.
    A note on the norm convergence by Vilenkin-Fejér means2014In: Georgian Mathematical Journal, ISSN 1072-947X, E-ISSN 1572-9176, Vol. 21, no 4, p. 511-517Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to find necessary and sufficient conditions for the convergence of Fejér means in terms of the modulus of continuity on the Hardy spaces Hp when 0 < p ≤ 1/2.

  • 19.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Approximation by walsh-kaczmarz-fejér means on the hardy space2014In: Acta Mathematica Scientia, ISSN 0252-9602, E-ISSN 1003-3998, Vol. 34, no 5, p. 1593-1602Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Fejér means in the terms of the modulus of continuity on the Hardy spaces Hp, when 0

  • 20.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series2015Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The classical theory of Fourier series deals with decomposition of afunction into sinusoidal waves. Unlike these continuous waves the Vilenkin(Walsh) functions are rectangular waves. Such waves have already been usedfrequently in the theory of signal transmission, multiplexing, filtering,image enhancement, codic theory, digital signal processing and patternrecognition. The development of the theory of Vilenkin-Fourier series hasbeen strongly influenced by the classical theory of trigonometric series.Because of this it is inevitable to compare results of Vilenkin series tothose on trigonometric series. There are many similarities between thesetheories, but there exist differences also. Much of these can be explainedby modern abstract harmonic analysis, which studies orthonormal systems from thepoint of view of the structure of a topological group.In this PhD thesis we discuss, develop and apply this fascinating theoryconnected to modern harmonic analysis. In particular we make new estimationsof Vilenkin-Fourier coefficients and prove some new results concerningboundedness of maximal operators of partial sums. Moreover, we derivenecessary and sufficient conditions for the modulus of continuity so thatnorm convergence of the partial sums is valid and develop new methods toprove Hardy type inequalities for the partial sums with respect to theVilenkin systems. We also do the similar investigation for the Fej\'er means.Furthermore, we investigate some N\"orlund means but only in the case when their coefficients are monotone. Some well-know examples of N\"orlund means areFej\'er means, Ces\`aro means and N\"orlund logarithmic means. In addition, weconsider Riesz logarithmic means, which are not example of N\"orlundmeans. It is also proved that these results are the best possible in aspecial sense. As applications both some well-known and new results arepointed out.This PhD is written as a monograph consisting of four Chapters:Preliminaries, Fourier coefficients and partial sums of Vilenkin-Fourierseries on martingale Hardy spaces, Vilenkin-Fej\'er means onmartingale Hardy spaces, Vilenkin-N\"orlund means on martingale Hardyspaces. It is based on 15 papers with the candidate as author or coauthor,but also some new results are presented for the first time.In Chapter 1 we first present some definitions and notations, which arecrucial for our further investigations. After that we also define somesummabilitity methods and remind about some classical facts and results. We investigate some well-known results and prove new estimates for the kernels of these summabilitity methods, which are very important to prove our main results. Moreover, we define martingale Hardy spaces and construct martingales, which help us to prove sharpness of our main results in the later chapters.Chapter 2 is devoted to present and prove some new and known results about Vilenkin-Fourier coefficients and partial sums of martingales in Hardy spaces.First, we show that Fourier coefficients of martingales are not uniformly bounded when $0

  • 21.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the convergence of Fejér means of Walsh-Fourier series in the space H p2016In: Journal of Contemporary Mathematical Analysis, ISSN 1068-3623, Vol. 51, no 2, p. 90-102Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to find necessary and sufficient conditions for a modulus of continuity of a martingale F ∈ Hp, for which the Fejér means of Walsh-Fourier series converge in Hp-norm, when 0 < p ≤ 1/2.

  • 22.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. University of Georgia, Tbilisi, Georgia.
    On the Convergence of Partial Sums with Respect to Vilenkin System on the Martingale Hardy Spaces2018In: Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), ISSN 1068-3623, Vol. 53, no 5, p. 294-306Article in journal (Refereed)
    Abstract [en]

    In this paper, we derive characterizations of boundedness of subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces Hp when 0 < p < 1. Moreover, we find necessary and sufficient conditions for the modulus of continuity of martingales f ∈ Hp, which provide convergence of subsequences of partial sums on the martingale Hardy spaces Hp. It is also proved that these results are the best possible in a special sense. As applications, some known and new results are pointed out. 

  • 23.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the maximal operatiors of Riesz logarithmic means of Cilenkin-Fourier series2014In: Studia scientiarum mathematicarum Hungarica (Print), ISSN 0081-6906, E-ISSN 1588-2896, Vol. 51, no 1, p. 105-120Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to investigate (H-p, L-p) and (H-p, L-p,L-infinity) type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series

  • 24.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the maximal operators of Walsh-Kaczmarz-Nörlund means2015In: Academia Paedagogica Nyiregyhaziensis. Acta Mathematica, ISSN 0866-0182, Vol. 31, no 2, p. 259-271Article in journal (Refereed)
    Abstract [en]

    The main aim of this paper is to investigate (Hp;Lp,∞) type inequalities for maximal operators of Nörlund means with monotone coefficients of one-dimensional Walsh-Kaczmarz system. By applying this results we conclude a.e. convergence of such Walsh-Kaczmarz-Nörlund means

  • 25.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the partial sums of Vilenkin-Fourier series2014In: Journal of Contemporary Mathematical Analysis, ISSN 1068-3623, Vol. 49, no 1, p. 23-32Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces Hp, when 0 < p ≤ 1.

  • 26.
    Tephnadze, George
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the partial sums of Walsh-Fourier series2015In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 141, no 2, p. 227-242Article in journal (Refereed)
    Abstract [en]

    We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.

1 - 26 of 26
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