Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators

作者：Khrennikov A Y^*; Shelkovich V M; van der Walt Jan Harm

来源：Journal of Fourier Analysis and Applications, 2013, 19(6): 1323-1358.

DOI：10.1007/s00041-013-9304-3

• Adeles
• Multiresolution analysis
• Wavelets
• Infinite tensor products of Hilbert spaces
• Adelic fractional operator

In our previous paper, the Haar multiresolution analysis (MRA) in was constructed, where is the adele ring. Since is the infinite tensor product of the spaces , p=a,2,3,aEuro broken vertical bar, the adelic MRA has some specific properties different from the corresponding finite-dimensional ones. Nevertheless, this infinite-dimensional MRA inherits almost all basic properties of the finite-dimensional case. In this paper we derive explicit formulas for bases in V (j) , , and for the wavelet bases generated by the above-mentioned adelic MRA. In view of the specific properties of the adelic MRA, there arise some technical problems in the construction of wavelet bases. These problems were solved with the aid of the operator formalization of the process of generation of wavelet bases. We study the spectral properties of the fractional operator introduced by S. Torba and W.A. ZA(0)A +/- iga-Galindo. We prove that the constructed wavelet functions are eigenfunctions of this fractional operator. This paper, as well as our previous paper, introduces new ideas to construct different infinite-dimensional MRAs. Our results can be used in the theory of adelic pseudo-differential operators and equations over the ring of adeles and in adelic models in physics.

• 出版日期2013-12