BMO Space and its relation with wavelet theory

Subject Areas : Applied Mathematics

M. Ghanbari ¹

1 - Department of Mathematics, Islamic Azad University, Farahan-Branch, Farahan, Iran.

Received: 2015-10-05 Accepted : 2015-10-05 Published : 2012-12-01

Keywords: BMO space, wavelets, Orthonormal system, Rademacher system, Walsh system, Haar system,

Abstract :

The aim of this paper is a) if Σak2< ∞then Σak rk(x) is inBMO that{rk(x)} is Rademacher system. b) P1k=1 ak!nk (x) 2 BMO; 2k  nk < 2k+1that f!n(x)g is Walsh system. c) If jakj < 1k then P1k=1 ak!k(x) 2 BMO.

References:

[1] L. Grafakos, Modern Fourier Analysis, Second Edition, Graduate Texts in Math.,
No. 250, Springer, New York, 2008.
[2] L. Grafakos, L. Liu, D. Yang, Maximal function characterizations of Hardy spaces
on RD-spaces and their applications, Sci. China Ser. A 51 (2008), 2253-2284.
[3] Dziubanski, G. Garrigos, T. Martnez, J. L. Torrea, J. Zienkiewicz, BMO spaces
related to Schrodinger operators with potentials satisfying a reverse Holder in-
equality, Math. Z. 249 (2005), 329-356.
[4] E. Hernftndez, G. Weiss, A First Course on wavelet, University Autonoma of
Madrid, Washington University in St. Louis, 1996
[5] B. S. Kashin, A. A. Saakyan, Orthogonal Series, Transl. Mth. Monographs, vol
75, AMS, Providence, 1989
[6] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and
Oscillatory Integrals, Princeton University Press, Princeton, New Jersey 1993
[7] M. T. Lacey, E. Terwilleger, B. D. Wick, Remarks on product VMO, Proc. Amer.
Math. Soc. 134 (2006), 465-474.
[8] O. Blasco, S. Pott, Dyadic BMO on the bidisk, Rev. Mat. Iberoamericana 21
(2005), , 483-510.