A Boas-type theorem for $\alpha$-monotone functions

  • M. Dyachenko
  • A. Mukanov
  • E. Nursultanov


We define the class of $\alpha$-monotone functions using fractional integrals.  For such functions we prove a Boas-type result on the summability of the Fourier coefficients.


Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988.

Bergh, J. and Löfström, J., Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.

Boas, Jr., R. P., Integrability theorems for trigonometric transforms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 38, Springer-Verlag New York Inc., New York, 1967.

Booton, B., General monotone sequences and trigonometric series, Math. Nachr. 287 (2014), no. 5-6, 518–529. http://dx.doi.org/10.1002/mana.201200297

Cossar, J., A theorem on Cesàro summability, J. London Math. Soc. 16 (1941), 56–68.

Dyachenko, M. and Tikhonov, S., Integrability and continuity of functions represented by trigonometric series: coefficients criteria, Studia Math. 193 (2009), no. 3, 285–306. http://dx.doi.org/10.4064/sm193-3-5

Dyachenko, M. I., Trigonometric series with generalized-monotone coefficients, Izv. Vyssh. Uchebn. Zaved. Mat. (1986), no. 7, 39–50.

Dyachenko, M. I., Nursultanov, E. D., and Zhantakbayeva, A. M., Hardy-Littlewood type theorems, Eurasian Math. J. 4 (2013), no. 2, 140–143.

Dyachenko, M. I., Piecewise monotone functions of several variables and a theorem of Hardy and Littlewood, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 6, 1156–1170, English translation: Math. USSR-Izv. 39 (1992), no. 3, 1113–1128.

Dyachenko, M. I., The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients, Izv. Vyssh. Uchebn. Zaved. Mat. (2008), no. 5, 38–47, English translation: Russian Math. (Iz. VUZ) 52 (2008), no. 5, 32–40. http://dx.doi.org/10.3103/S1066369X08050046

Dyachenko, M. I. and Nursultanov, E. D., The Hardy-Littlewood theorem for trigonometric series with α-monotone coefficients, Mat. Sb. 200 (2009), no. 11, 45–60, English translation: Sb. Math. 200 (2009), no. 11-12, 1617–1631. http://dx.doi.org/10.1070/SM2009v200n11ABEH004053

Hardy, G. H. and Littlewood, J. E., Notes on the theory of series (XIII): Some new properties of Fourier constants, J. London Math. Soc. S1-6 (1931), no. 1, 3–9. http://dx.doi.org/10.1112/jlms/s1-6.1.3

Kopezhanova, A. N., Nursultanov, E. D., and Persson, L.-E., On inequalities for the Fourier transform of functions from Lorentz spaces, Mat. Zametki 90 (2011), no. 5, 785–788, English translation: Math. Notes 90 (2011), no. 5-6, 767–770. http://dx.doi.org/10.1134/S0001434611110150

Liflyand, E. and Tikhonov, S., Two-sided weighted Fourier inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 341–362.

Nursultanov, E. and Tikhonov, S., Net spaces and boundedness of integral operators, J. Geom. Anal. 21 (2011), no. 4, 950–981. http://dx.doi.org/10.1007/s12220-010-9175-7

Nursultanov, E. D., Interpolation properties of some anisotropic spaces and Hardy-Littlewood type inequalities, East J. Approx. 4 (1998), no. 2, 243–275.

Nursultanov, E. D., Net spaces and inequalities of Hardy-Littlewood type, Mat. Sb. 189 (1998), no. 3, 83–102. http://dx.doi.org/10.1070/SM1998v189n03ABEH000309

Sagher, Y., An application of interpolation theory to Fourier series, Studia Math. 41 (1972), 169–181.

Sagher, Y., Some remarks on interpolation of operators and Fourier coefficients, Studia Math. 44 (1972), 239–252.

Sagher, Y., Integrability conditions for the Fourier transform, J. Math. Anal. Appl. 54 (1976), no. 1, 151–156.

Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional integrals and derivatives: Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.

Simonov, B. and Tikhonov, S., Norm inequalities in multidimensional Lorentz spaces, Math. Scand. 103 (2008), no. 2, 278–294. http://dx.doi.org/10.7146/math.scand.a-15080

Tikhonov, S., Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326 (2007), no. 1, 721–735. http://dx.doi.org/10.1016/j.jmaa.2006.02.053

Zygmund, A., Trigonometric series. 2nd ed. Vol. II, Cambridge University Press, New York, 1959.

How to Cite
Dyachenko, M., Mukanov, A., & Nursultanov, E. (2017). A Boas-type theorem for $\alpha$-monotone functions. MATHEMATICA SCANDINAVICA, 120(1), 39-58. https://doi.org/10.7146/math.scand.a-25503