In the present paper, we introduce a new difference sequence space rqB(u,p) by using the Riesz mean and the B-difference matrix. We show rqB(u,p) is a complete linear metric space and is linearly isomorphic to the space l(p). We have also computed its ?-, ?- and ?-duals. Furthermore, we have constructed the basis of rqB(u,p) and characterize a matrix class (rqB(u, p), l?).

In the present paper, we characterize r q (u, p) : f ∞ , r q (u, p) : f and r q (u, p) : f 0 ; where f ∞ , f and f 0 denotes, respectively, the spaces of almost bounded sequences, almost convergent sequences and almost sequences converging to zero, where the space r q (u, p) of non-absolute type have recently been introduced by Neyaz and Hamid (see, [13]).

In this paper, we introduce the concept of periodic Gabor frames on non-Archimedean fields of positive characteristic. We first establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame for [Formula: see text]. Then, we present some equivalent characterizations of Parseval Gabor frames on non-Archimedean fields by means of some fundamental equations in the time domain. Finally, potential applications of Gabor frames on non-Archimedean fields are also discussed.

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum normalΛ={}0,rfalse/N+2ℤ, where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

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