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Journal article
TWO-DIMENSIONAL SHANNON TYPE EXPANSIONS VIA ONE-DIMENSIONAL AFFINE AND WAVELET LATTICE ACTIONS
Colloquium mathematicum, v 157(1)
01 Jan 2019
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Abstract
It is rather unexpected, but true, that it is possible to construct reproducing formulae and orthonormal bases of L-2(R-2) just by applying the standard onedimensional wavelet action of translations and dilations to the first variable x(1) of the generating function psi(x(1), x(2)), psi is an element of L-2(R-2), i.e., by making use of building blocks
psi(u,s) (x(1), x(2)) = s(-1/2)(x(1)-u/s, x(2)), where u is an element of R, s > 0,
in the case of reproducing formulae, and
psi(k,m) (x(1),x(2)) = 2(-k/2)psi(x(1)-2(k)m/2(k),x(2)), where k, m is an element of Z,
in the case of orthonormal bases. It is possible to compensate the fact that the second variable x(2) is not acted upon by a careful selection of the generating function psi. Shannon wavelet tiling of the time-frequency plane R-2 is a standard illustration of orthogonality and completeness phenomena corresponding to the Shannon wavelet,
chi(2(k)m,2(k)(m+1)]((x))chi(2)-k(I)(xi), k, m is an element of Z, I = -(1/2, 1] boolean OR (1/2,1],
with x representing time and xi frequency. In our current context, of the wavelet action restricted to the first coordinate of R-2, it is substituted by a phase space tiling of R-4 with unbounded, hyperboloid type blocks of the form
chi(2(k)m,2(k)(m+1)]((x1)) Sigma(n,l) chi(2)-k(ID(n,l))((xi 1))chi(n, n+1]((x2))chi(l,l+1]((xi 2)), k, m is an element of Z,
where I-r = 2(-r) I, r >= 1, and D:Z x Z -> N is a bijection, an additional parameter of the generating function, needed for the lift from L-2(R) to L-2(R-2). The variables x(1), x(2) are the coordinates of position, and the variables xi(1),xi(2) of momentum.
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Details
- Title
- TWO-DIMENSIONAL SHANNON TYPE EXPANSIONS VIA ONE-DIMENSIONAL AFFINE AND WAVELET LATTICE ACTIONS
- Creators
- K. Nowak - Drexel UniversityM. Pap - University of Pecs
- Publication Details
- Colloquium mathematicum, v 157(1)
- Publisher
- Ars Polona-Ruch
- Number of pages
- 14
- Grant note
- European Union - European Social Fund; European Social Fund (ESF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Computer Science
- Web of Science ID
- WOS:000469877600006
- Scopus ID
- 2-s2.0-85068902139
- Other Identifier
- 991019169614604721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics