Approximation by Vilenkin-Nörlund means in Lebesgue spaces. (English) Zbl 07919545
Duduchava, Roland (ed.) et al., Tbilisi analysis and PDE seminar. Extended abstracts of the 2020–2023. Seminar talks. Cham: Birkhäuser. Trends Math., Res. Perspect. Ghent Anal. PDE Cent. 7, 11-19 (2024).
Summary: In this paper we improve and complement a result by F. Móricz and A. H. Siddiqi [J. Approx. Theory 70, No. 3, 375–389 (1992; Zbl 0757.42009)]. In particular, we prove that their estimate of the Nörlund means with respect to the Vilenkin system holds also without their additional condition. Moreover, we prove a similar approximation result in Lebesgue spaces for any \(1\leq p<\infty\).
For the entire collection see [Zbl 07851841].
For the entire collection see [Zbl 07851841].
MSC:
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 42B30 | \(H^p\)-spaces |
| 42A10 | Trigonometric approximation |
References:
| [1] | G.N. Agaev, N.Ya. Vilenkin, G.M. Dzhafarly, A.I. Rubinshtein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups (Baku, Ehim, 1981) (in Russian) · Zbl 0588.43001 |
| [2] | N. Anakidze, N. Areshidze, L. Baramidze, Approximation by Nörlund means with respect to Vilenkin system in Lebesgue spaces. Acta Math. Hungar. 172, 529-542 (2024) · Zbl 07857915 |
| [3] | N. Areshidze, G. Tephnadze, Approximation by Nörlund means with respect to Walsh system in Lebesgue spaces. Math. Inequal. Appl. 27(1), 137-147 (2024) |
| [4] | L. Baramidze, L.E. Persson, G. Tephnadze, P. Wall, Sharp \(H_p-L_p\) type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications. J. Inequal. Appl. 2016, paper no. 242, 20 pp. (2016) · Zbl 1352.42037 |
| [5] | D. Baramidze, L.-E. Persson, H. Singh, G. Tephnadze, Some new results and inequalities for subsequences of Nörlund logarithmic means of Walsh-Fourier series. J. Inequal. Appl. 2022, paper no. 30, 13 pp. (2022) · Zbl 1506.42036 |
| [6] | D. Baramidze, L. Baramidze, L.-E. Persson, G. Tephnadze, Some now restricted maximal operators of Fejér means of Walsh-Fourier series. Banach J. Math. Anal. 17(4), 75 (2023) · Zbl 1532.42030 |
| [7] | D. Baramidze, N. Nadirashvili, L.-E. Persson, G. Tephnadze, Some weak-type inequalities and almost everywhere convergence of Vilenkin-Nörlund means. J. Inequal. Appl. 2023, paper no. 66, 17 pp. (2023) · Zbl 1532.42031 |
| [8] | D. Baramidze, L.-E. Persson, K. Tangrand, G. Tephnadze, \((H_p-L_p)\) type inequalities for subsequences of Nörlund means of Walsh-Fourier series. J. Inequal. Appl., paper no. 52, 13 pp. (2023) · Zbl 1532.42029 |
| [9] | I. Blahota, K. Nagy, Approximation by \(\Theta \)-means of Walsh-Fourier series. Anal. Math. 44(1), 57-71 (2018) · Zbl 1399.42074 |
| [10] | I. Blahota, G. Tephnadze, On the \((C,\alpha )\)-means with respect to the Walsh system. Anal. Math. 40, 161-174 (2014) · Zbl 1313.42083 |
| [11] | I. Blahota, G. Tephnadze, A note on maximal operators of Vilenkin-Nörlund means. Acta Math. Acad. Paed. Nyireg. 32, 203-213 (2016) · Zbl 1363.42044 |
| [12] | I. Blahota, G. Tephnadze, R. Toledo, Strong convergence theorem of \((C,\alpha )\)-means with respect to the Walsh system. Tohoku Math. J. 67(4), 573-584 (2015) · Zbl 1338.42035 |
| [13] | I. Blahota, L.E. Persson, G. Tephnadze, On the Nörlund means of Vilenkin-Fourier series. Czech. Math J. 65(4), 983-1002 (2015) · Zbl 1374.42054 |
| [14] | I. Blahota, K. Nagy, G. Tephnadze, Approximation by Marcinkiewicz \(\Theta \)-means of double Walsh-Fourier series. Math. Inequal. Appl. 22(3), 837-853 (2019) · Zbl 1425.42030 |
| [15] | S. Fridli, On the rate of convergence of Cesaro means of Walsh-Fourier series. J. Approx. Theory 76(1), 31-53 (1994) · Zbl 0784.42015 |
| [16] | S. Fridli, P. Manchanda, A. Siddiqi, Approximation by Walsh-Nörlund means. Acta Sci. Math. 74(3-4), 593-608 (2008) · Zbl 1199.42124 |
| [17] | G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems. J. Approx. Theory 124(1), 25-43 (2003) · Zbl 1032.43003 |
| [18] | U. Goginava, Marcinkiewicz-Fejer means of \(d\)-dimensional Walsh-Fourier series. J. Math. Anal. Appl. 307(1), 206-218 (2015) · Zbl 1070.42019 |
| [19] | U. Goginava, Maximal operators of Walsh-Nörlund means on the dyadic Hardy spaces. Acta Math. Hungar. 169(1), 171-190 (2023) · Zbl 1538.42056 |
| [20] | B.I. Golubov, A.V. Efimov, V.A. Skvortsov, Walsh Series and Transforms (Kluwer Academic Publishers Group, Dordrecht, 1991) · Zbl 0785.42010 |
| [21] | N. Memić, An estimate of the maximal operator of the Nörlund logarithmic means with respect to the character system of the group of 2-adic integers on the Hardy space \(H_1\). Bull. Iranian Math. Soc. 48(6), 3381-3391 (2022) · Zbl 1537.42033 |
| [22] | N. Memić, L.E. Persson, G. Tephnadze, A note on the maximal operators of Vilenkin-Nörlund means with non-increasing coefficients. Stud. Sci. Math. Hung. 53(4), 545-556 (2016) · Zbl 1399.42079 |
| [23] | C.N. Moore, Summable Series and Convergence Factors, Summable Series and Convergence Factors (Dover Publications, New York, 1966) · Zbl 0142.30704 |
| [24] | F. Móricz, A. Siddiqi, Approximation by Nörlund means of Walsh-Fourier series. J. Approx. Theory 70(3), 375-389 (1992) · Zbl 0757.42009 |
| [25] | N. Nadirashvili, L.-E. Persson, G. Tephnadze, F. Weisz, Vilenkin-Lebesgue points and almost everywhere convergence of Vilenkin-Fejér means and applications. Mediterr. J. Math. 19(5), 239 (2022) · Zbl 1501.42006 |
| [26] | K. Nagy, Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series. Anal. Math. 36(4), 299-319 (2010) · Zbl 1240.42133 |
| [27] | K. Nagy, Approximation by Nörlund means of Walsh-Kaczmarz-Fourier series. Georgian Math. J. 18(1), 147-162 (2011) · Zbl 1210.42043 |
| [28] | K. Nagy, Approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions. Math. Inequal. Appl. 15(2), 301-322 (2012) · Zbl 1243.42038 |
| [29] | K. Nagy, G. Tephnadze, Walsh-Marcinkiewicz means and Hardy spaces. Cent. Eur. J. Math. 12(8), 1214-1228 (2014) · Zbl 1300.42003 |
| [30] | K. Nagy, G. Tephnadze, Approximation by Walsh-Marcinkiewicz means on the Hardy space. Kyoto J. Math. 54(3), 641-652 (2014) · Zbl 1311.42070 |
| [31] | K. Nagy, G. Tephnadze, Strong convergence theorem for Walsh-Marcinkiewicz means. Math. Inequal. Appl. 19(1), 185-195 (2016) · Zbl 1338.42037 |
| [32] | K. Nagy, G. Tephnadze, The Walsh-Kaczmarz-Marcinkiewicz means and Hardy spaces. Acta Math. Hung. 149(2), 346-374 (2016) · Zbl 1399.42081 |
| [33] | J. Pál, P. Simon, On a generalization of the concept of derivative. Acta Math. Hung. 29, 155-164 (1977) · Zbl 0345.42011 |
| [34] | L.E. Persson, G. Tephnadze, P. Wall, On the maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21(1), 76-94 (2015) · Zbl 1311.42071 |
| [35] | L.E. Persson, G. Tephnadze, P. Wall, Some new \((H_p,L_p)\) type inequalities of maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients. J. Math. Inequal. 9(4), 1055-1069 (2015) · Zbl 1329.42030 |
| [36] | L.E. Persson, G. Tephnadze, P. Wall, On the Nörlund logarithmic means with respect to Vilenkin system in the martingale Hardy space \(H_1\). Acta Math. Hung. 154(2), 289-301 (2018) · Zbl 1399.42082 |
| [37] | L.E. Persson, G. Tephnadze, G. Tutberidze, On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces. Oper. Matrices 14(1), 283-294 (2020) · Zbl 1465.42031 |
| [38] | L.-E. Persson, F. Schipp, G. Tephnadze, F. Weisz, An analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. J. Fourier Anal. Appl. 28, 48, 1-29 (2022) · Zbl 1492.42032 |
| [39] | L.E. Persson, G. Tephnadze, F. Weisz, Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series (Birkhäuser/Springer, 2022), book manuscript · Zbl 1512.42042 |
| [40] | F. Schipp, Certain rearrangements of series in the Walsh series. Mat. Zametki 18, 193-201 (1975) · Zbl 0349.42013 |
| [41] | F. Schipp, W.R. Wade, P. Simon, J. Pál, Walsh Series. An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol, 1990) · Zbl 0727.42017 |
| [42] | P. Simon, Strong convergence of certain means with respect to the Walsh-Fourier series. Acta Math. Hungar. 49(3-4), 425-431 (1987) · Zbl 0643.42020 |
| [43] | P. Simon, On the Cesáro summability with respect to the Walsh-Kaczmarz system. J. Approx. Theory 106(2), 249-261 (2000) · Zbl 0987.42021 |
| [44] | G. Tephnadze, Fejér means of Vilenkin-Fourier series. Stud. Sci. Math. Hung. 49(1), 79-90 (2012) · Zbl 1265.42099 |
| [45] | G. Tephnadze, On the maximal operators of Vilenkin-Fejér means on Hardy spaces. Math. Inequal. Appl. 16(2), 301-312 (2013) · Zbl 1263.42008 |
| [46] | G. Tephnadze, On the maximal operators of Vilenkin-Fejér means. Turk. J. Math. 37, 308-318 (2013) · Zbl 1278.42037 |
| [47] | G. Tephnadze, On the maximal operators of Walsh-Kaczmarz-Fejér means. Period. Math. Hung. 67(1), 33-45 (2013) · Zbl 1299.42097 |
| [48] | G. Tephnadze, Approximation by Walsh-Kaczmarz-Fejér means on the Hardy space. Acta Math. Sci. 34(5), 1593-1602 (2014) · Zbl 1324.42043 |
| [49] | G. Tephnadze, The one-dimensional martingale hardy spaces and partial sums and Fejér means with respect to Walsh system. Mem. Differential Equations Math. Phys. 88, 109-158 (2023) · Zbl 1522.42058 |
| [50] | G. Tutberidze, Modulus of continuity and boundedness of subsequences of Vilenkin-Fejer means in the martingale Hardy spaces. Geo. Math. J. 29(1), 153-162 (2022) · Zbl 1482.42072 |
| [51] | N.Y. Vilenkin, On the theory of lacunary orthogonal systems (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 13, 245-252 (1949) · Zbl 0033.11202 |
| [52] | N.Y. Vilenkin, On the theory of Fourier integrals on topological groups. Mat. Sbornik N. S. 30(72), 233-244 (1952) · Zbl 0047.35403 |
| [53] | N.Y. Vilenkin, On a class of complete orthonormal systems. Amer. Math. Soc. Trans. 28(2), 1-35 (1963) · Zbl 0125.34304 |
| [54] | F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis (Springer, Berlin, 1994) · Zbl 0796.60049 |
| [55] | F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series. Bolyai Soc. Math. Stud. 5, 353-367 (1996) · Zbl 0866.42019 |
| [56] | A. Zygmund, Trigonometric Series, vol. 1 (Cambridge University Press, 1959) · Zbl 0085.05601 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
