Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in $$L^2({\mathbb {K}})$$ L 2 ( K )
- PDF / 317,624 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 111 Downloads / 200 Views
Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in L2 (K) O. Ahmad1
· N. A. Sheikh1 · M. A. Ali1
Received: 2 December 2019 / Accepted: 22 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we introduce the structure of nonuniform nonhomogeneous dual wavelet frames over non-Archimedean fields. A characterization of nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces over non-Archimedean fields is obtained. Keywords Dual wavelet frame · Fourier transform · Sobolev Space Mathematics Subject Classification 42C15 · 42C40 · 42C30
1 Introduction Non-Archimedean fields have been used for the global space-time theory in order to unify both microscopic and macroscopic physics. Some problems occurred with the practical applications of the classical fields R and C because in physical sciences, there are absolute limitations on measurements such as Plank time, Plank length, Plank mass etc. The use of real time and space-time coordinates in mathematical physics leads to some problems with the Archimedean axiom on the microscopic level. According to the Archimedean axiom “any given segment on the line can be surpassed by the successive addition of a smaller segment along the same line”. This means that, we can measure the arbitrary small distances but a measurement of distances smaller than the Planck length is impossible. Volovich proposes to base physics on coalition of non-Archimedean normed fields and the classical fields. The pseries fields and p-adic fields are best known non-Archimedean normed fields. As p → ∞, many of the fundamental properties of p-adic analysis approach their counterparts in classi-
B
O. Ahmad [email protected] N. A. Sheikh [email protected] M. A. Ali [email protected]
1
Department of Mathematics, National Institute of Technology, Srinagar, Jammu and Kashmir 190006, India
123
O. Ahmad et al.
cal analysis. Thus, p-adic analysis could provide a bridge from microscopic to macroscopic physics. It is well known that one of the important factor behind the stable decomposition of a signal for analysis or transmission is related to the type of representation used for its spanning set (representation system). A careful choice of the spanning set enables us to solve a variety of analysis tasks. During the last two decades, many researchers have contributed in the designing and time-frequency analysis of these representation systems for the various spaces, namely, finite and infinite abelian groups, Euclidean spaces, locally compact abelian groups, etc.(see [4,5,13,19] and references therein). The theory of frames and wavelets plays a significant role in the interpretation of data obtained by analyzing the signal with respect to the above mentioned systems. The redundancy of a frame offers more flexibility to accommodate specific design requirements. In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and
Data Loading...
