On the Nörlund means of Vilenkin-Fourier series

István Blahota; Lars-Erik Persson; Giorgi Tephnadze

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 983-1002
  • ISSN: 0011-4642

Abstract

top
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.

How to cite

top

Blahota, István, Persson, Lars-Erik, and Tephnadze, Giorgi. "On the Nörlund means of Vilenkin-Fourier series." Czechoslovak Mathematical Journal 65.4 (2015): 983-1002. <http://eudml.org/doc/276158>.

@article{Blahota2015,
abstract = {We prove and discuss some new $( H_\{p\},L_\{p\})$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\lbrace q_\{k\}\colon k\ge 0\rbrace $. 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 $\lbrace q_\{k\}\colon k\ge 0\rbrace $ in an appropriate way.},
author = {Blahota, István, Persson, Lars-Erik, Tephnadze, Giorgi},
journal = {Czechoslovak Mathematical Journal},
keywords = {Vilenkin system; Vilenkin group; Nörlund means; martingale Hardy space; maximal operator; Vilenkin-Fourier series; strong convergence; inequality},
language = {eng},
number = {4},
pages = {983-1002},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Nörlund means of Vilenkin-Fourier series},
url = {http://eudml.org/doc/276158},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Blahota, István
AU - Persson, Lars-Erik
AU - Tephnadze, Giorgi
TI - On the Nörlund means of Vilenkin-Fourier series
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 983
EP - 1002
AB - We prove and discuss some new $( H_{p},L_{p})$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\lbrace q_{k}\colon k\ge 0\rbrace $. 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 $\lbrace q_{k}\colon k\ge 0\rbrace $ in an appropriate way.
LA - eng
KW - Vilenkin system; Vilenkin group; Nörlund means; martingale Hardy space; maximal operator; Vilenkin-Fourier series; strong convergence; inequality
UR - http://eudml.org/doc/276158
ER -

References

top
  1. Blahota, I., 10.1023/A:1026769207159, Acta Math. Hung. 89 (2000), 15-27. (2000) Zbl0973.42020MR1912235DOI10.1023/A:1026769207159
  2. Blahota, I., Relation between Dirichlet kernels with respect to Vilenkin-like systems, Acta Acad. Paedagog. Agriensis, Sect. Mat. (N.S.) 22 (1994), 109-114. (1994) Zbl0882.42017
  3. Blahota, I., Gát, G., 10.1007/s10496-008-0001-z, Anal. Theory Appl. 24 (2008), 1-17. (2008) Zbl1164.42022MR2422455DOI10.1007/s10496-008-0001-z
  4. Blahota, I., Tephnadze, G., 10.1007/s10476-014-0301-9, Anal. Math. 40 (2014), 161-174. (2014) Zbl1313.42083MR3240221DOI10.1007/s10476-014-0301-9
  5. Blahota, I., Tephnadze, G., 10.5486/PMD.2014.5896, Publ. Math. Debrecen 85 (2014), 181-196. (2014) MR3231514DOI10.5486/PMD.2014.5896
  6. Fujii, N., A maximal inequality for H 1 -functions on a generalized Walsh-Paley group, Proc. Am. Math. Soc. 77 (1979), 111-116. (1979) MR0539641
  7. G{á}t, G., 10.1016/S0021-9045(03)00075-3, J. Approx. Theory 124 (2003), 25-43. (2003) Zbl1032.43003MR2010779DOI10.1016/S0021-9045(03)00075-3
  8. G{á}t, G., 10.1007/BF01872107, Acta Math. Hung. 61 (1993), 131-149. (1993) Zbl0805.42019MR1200968DOI10.1007/BF01872107
  9. Gát, G., Goginava, U., Almost everywhere convergence of ( C , α ) -means of quadratical partial sums of double Vilenkin-Fourier series, Georgian Math. J. 13 (2006), 447-462. (2006) Zbl1107.42006MR2271060
  10. Gát, G., Goginava, U., 10.1007/s10114-005-0648-8, Acta Math. Sin., Engl. Ser. 22 (2006), 497-506. (2006) MR2214371DOI10.1007/s10114-005-0648-8
  11. Gát, G., Nagy, K., On the logarithmic summability of Fourier series, Georgian Math. J. 18 (2011), 237-248. (2011) Zbl1221.42049MR2805978
  12. Goginava, U., 10.1007/s10476-010-0101-9, Anal. Math. 36 (2010), 1-31. (2010) MR2606574DOI10.1007/s10476-010-0101-9
  13. Goginava, U., Maximal operators of Fejér-Walsh means, Acta Sci. Math. 74 (2008), 615-624. (2008) Zbl1199.42127MR2487936
  14. Goginava, U., The maximal operator of Marcinkiewicz-Fejér means of the d -dimensional Walsh-Fourier series, East J. Approx. 12 (2006), 295-302. (2006) MR2252557
  15. Goginava, U., The maximal operator of the ( C , α ) means of the Walsh-Fourier series, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 26 (2006), 127-135. (2006) Zbl1121.42020MR2388683
  16. Goginava, U., Almost everywhere convergence of subsequence of logarithmic means of Walsh-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 21 (2005), 169-175. (2005) Zbl1093.42018MR2162613
  17. Goginava, U., 10.1006/jath.2001.3632, J. Approx. Theory 115 (2002), 9-20. (2002) Zbl0998.42018MR1888974DOI10.1006/jath.2001.3632
  18. Moore, C. N., Summable Series and Convergence Factors, Dover Publications, New York (1966). (1966) Zbl0142.30704MR0201863
  19. Móricz, F., Siddiqi, A. H., 10.1016/0021-9045(92)90067-X, J. Approx. Theory 70 (1992), 375-389. (1992) Zbl0757.42009MR1178380DOI10.1016/0021-9045(92)90067-X
  20. Nagy, K., Approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions, Math. Inequal. Appl. 15 (2012), 301-322. (2012) Zbl1243.42038MR2962234
  21. Nagy, K., Approximation by Nörlund means of Walsh-Kaczmarz-Fourier series, Georgian Math. J. 18 (2011), 147-162. (2011) Zbl1210.42043MR2787349
  22. Nagy, K., Approximation by Cesàro means of negative order of Walsh-Kaczmarz-Fourier series, East J. Approx. 16 (2010), 297-311. (2010) Zbl1216.42006MR2789336
  23. Nagy, K., 10.1007/s10476-010-0404-x, Anal. Math. 36 (2010), 299-319. (2010) Zbl1240.42133MR2738323DOI10.1007/s10476-010-0404-x
  24. Pál, J., Simon, P., 10.1007/BF01896477, Acta Math. Acad. Sci. Hung. 29 (1977), 155-164. (1977) Zbl0345.42011MR0450884DOI10.1007/BF01896477
  25. Schipp, F., Rearrangements of series in the Walsh system, Math. Notes 18 (1976), 701-706 translation fromMat. Zametki 18 (1975), 193-201. (1975) MR0390633
  26. Simon, P., 10.1007/s006050070004, Monatsh. Math. 131 (2000), 321-334. (2000) MR1813992DOI10.1007/s006050070004
  27. Simon, P., 10.1006/jmaa.2000.6732, J. Math. Anal. Appl. 245 (2000), 52-68. (2000) Zbl0987.42022MR1756576DOI10.1006/jmaa.2000.6732
  28. Simon, P., Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27 (1984), 87-101. (1984) Zbl0586.43001MR0823096
  29. Simon, P., Weisz, F., 10.1016/j.jat.2007.05.004, J. Approx. Theory 151 (2008), 1-19. (2008) Zbl1143.42032MR2403893DOI10.1016/j.jat.2007.05.004
  30. Tephnadze, G., On the maximal operators of Riesz logarithmic means of Vilenkin-Fourier series, Stud. Sci. Math. Hung. 51 (2014), 105-120. (2014) Zbl1299.42098MR3188506
  31. Tephnadze, G., 10.3103/S1068362314010038, J. Contemp. Math. Anal. 49 23-32 Russian (2014). (2014) MR3237573DOI10.3103/S1068362314010038
  32. Tephnadze, G., 10.1007/s10474-013-0361-5, Acta Math. Hung. 142 (2014), 244-259. (2014) Zbl1313.42086MR3158862DOI10.1007/s10474-013-0361-5
  33. Tephnadze, G., On the maximal operators of Vilenkin-Fejér means on Hardy spaces, Math. Inequal. Appl. 16 (2013), 301-312. (2013) Zbl1263.42008MR3060398
  34. Tephnadze, G., On the maximal operators of Vilenkin-Fejér means, Turk. J. Math. 37 (2013), 308-318. (2013) Zbl1278.42037MR3040854
  35. Tephnadze, G., A note on the Fourier coefficients and partial sums of Vilenkin-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 28 (2012), 167-176. (2012) Zbl1289.42084MR3048092
  36. Tephnadze, G., Fejér means of Vilenkin-Fourier series, Stud. Sci. Math. Hung. 49 (2012), 79-90. (2012) Zbl1265.42099MR3059789
  37. Tephnadze, G., The maximal operators of logarithmic means of one-dimensional Vilenkin-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 27 (2011), 245-256. (2011) Zbl1265.42100MR2880697
  38. Vilenkin, N. J., On a class of complete orthonormal systems, Am. Math. Soc. Transl. Ser. (2), 28 (1963), 1-35 translation fromIzv. Akad. Nauk SSSR, Ser. Mat. 11 (1947), 363-400. (1947) Zbl0036.35601MR0154042
  39. Weisz, F., 10.1023/B:AMHU.0000028241.87331.c5, Acta Math. Hung. 103 (2004), 139-176. (2004) Zbl1060.42021MR2047878DOI10.1023/B:AMHU.0000028241.87331.c5
  40. Weisz, F., 10.1023/A:1014364010470, Anal. Math. 27 (2001), 141-155. (2001) Zbl0992.42016MR1834858DOI10.1023/A:1014364010470
  41. Weisz, F., 10.1007/BF02205221, Anal. Math. 22 (1996), 229-242. (1996) Zbl0866.42020MR1627638DOI10.1007/BF02205221
  42. Weisz, F., 10.1007/BFb0073448, Lecture Notes in Mathematics 1568 Springer, Berlin (1994). (1994) Zbl0796.60049MR1320508DOI10.1007/BFb0073448

NotesEmbed ?

top

You must be logged in to post comments.