It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of we...It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.展开更多
Norlund logarithmic means of multiple Walsh-Fourier series acting from space L In^d-1 L （[0, 1）d）, d≥1 into space weak - LI（[0,1）^d） are studied. The maximal Orlicz space such that the Norlund logarithmic means...Norlund logarithmic means of multiple Walsh-Fourier series acting from space L In^d-1 L （[0, 1）d）, d≥1 into space weak - LI（[0,1）^d） are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.展开更多
The main aim of this paper is to prove that for any 0 〈 p≤ 2/3 there exists a martingale f E Hp such that Marcinkiewicz Fejer means of the two-dimensional conjugate Walsh Fourier series of the martingale f is not un...The main aim of this paper is to prove that for any 0 〈 p≤ 2/3 there exists a martingale f E Hp such that Marcinkiewicz Fejer means of the two-dimensional conjugate Walsh Fourier series of the martingale f is not uniformly bounded in the space Lp.展开更多
Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedur...Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.展开更多
In this paper, the notion of p-wavelet packets on the positive half-line P＋ is introduced. A new method for constructing non-orthogonal wavelet packets related to Walsh functions is developed by splitting the wavelet...In this paper, the notion of p-wavelet packets on the positive half-line P＋ is introduced. A new method for constructing non-orthogonal wavelet packets related to Walsh functions is developed by splitting the wavelet subspaces directly instead of using the lowpass and high-pass filters associated with the multiresolution analysis as used in the classical theory of wavelet packets. Further, the method overcomes the difficulty of constructing non-orthogonal wavelet packets of the dilation factor p 〉 2.展开更多
A well-known result for Vilenkin systems is the fact that for all 1 〈 p 〈 oc the n-th partial sums of Fourier series of all functions in the space Lp converge to the function in LP-norm. This statement can not be ge...A well-known result for Vilenkin systems is the fact that for all 1 〈 p 〈 oc the n-th partial sums of Fourier series of all functions in the space Lp converge to the function in LP-norm. This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups, but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.展开更多
基金This paperwas written while theauthorwasresearching at Humboldt University in Berlin supported by Alexandervon Humboldt Foundation.This research was also supported by the Hungarian Scientific Research Funds (OTKA) NoF0 1 963 3 and by the Foundation
文摘It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.
基金The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. M 36511/2001, T 048780by the Szechenyi fellowship of the Hungarian Ministry of Education Szo 184/200
文摘Norlund logarithmic means of multiple Walsh-Fourier series acting from space L In^d-1 L （[0, 1）d）, d≥1 into space weak - LI（[0,1）^d） are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.
文摘The main aim of this paper is to prove that for any 0 〈 p≤ 2/3 there exists a martingale f E Hp such that Marcinkiewicz Fejer means of the two-dimensional conjugate Walsh Fourier series of the martingale f is not uniformly bounded in the space Lp.
文摘Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.
文摘In this paper, the notion of p-wavelet packets on the positive half-line P＋ is introduced. A new method for constructing non-orthogonal wavelet packets related to Walsh functions is developed by splitting the wavelet subspaces directly instead of using the lowpass and high-pass filters associated with the multiresolution analysis as used in the classical theory of wavelet packets. Further, the method overcomes the difficulty of constructing non-orthogonal wavelet packets of the dilation factor p 〉 2.
基金Supported by project TAMOP-4.2.2.A-11/1/KONV-2012-0051
文摘A well-known result for Vilenkin systems is the fact that for all 1 〈 p 〈 oc the n-th partial sums of Fourier series of all functions in the space Lp converge to the function in LP-norm. This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups, but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.
