Research output: Contribution to journal › Article › peer-review
Minimal splines and wavelets. / Dem'yanovich, Yu K.
In: Vestnik St. Petersburg University: Mathematics, Vol. 41, No. 2, 01.06.2008, p. 88-101.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Minimal splines and wavelets
AU - Dem'yanovich, Yu K.
PY - 2008/6/1
Y1 - 2008/6/1
N2 - This paper is dedicated to the memory of the prominent mathematician S. G. Mikhlin. Here, Mikhlin's idea of approximation relations is used for construction of wavelet resolution in the case of spline spaces of zero height. These approximation relations allow one to establish the embedding of the spline spaces corresponding to nested grids. Systems of functionals which are biorthogonal to the basic splines are constructed using the relations; then the systems obtained are used for wavelet decomposition. It is established that, for a fixed pair of grids of which one is embedded into the other and for an arbitrary fixed (on the coarse grid) spline space, there exists a continuum of spline spaces (on the fine grid) which contain the aforementioned spline space on the coarse grid. The wavelet decomposition of such embedding is given and the corresponding formulas of decomposition and formulas of reconstruction are deduced. The space of (A, φ)-splines is introduced with three objects: the full chain of vectors, prescribed infinite grid on real axis and the preassigned vector-function φ with m + 1 components (m is called the order of the splines). Under certain assumptions, the splines belong to the class C m - 1. The gauge relations between the basic splines on the coarse grid and the basic splines on the fine grid are deduced. A general method for construction of a biorthogonal system of functionals (to basic spline system) is suggested. In this way, a chain of nested spline spaces is obtained, and the wavelet decomposition of the chain is discussed. The spaces and chains of spaces are completely classified in the terms of manifolds. The manifold of spaces considered is identified with the manifold of complete sequences of points of the direct product of an interval on the real axis and the projective space ℙ m; the manifold of nested spaces is identified with the manifold of nested sequences of points of the direct product mentioned above.
AB - This paper is dedicated to the memory of the prominent mathematician S. G. Mikhlin. Here, Mikhlin's idea of approximation relations is used for construction of wavelet resolution in the case of spline spaces of zero height. These approximation relations allow one to establish the embedding of the spline spaces corresponding to nested grids. Systems of functionals which are biorthogonal to the basic splines are constructed using the relations; then the systems obtained are used for wavelet decomposition. It is established that, for a fixed pair of grids of which one is embedded into the other and for an arbitrary fixed (on the coarse grid) spline space, there exists a continuum of spline spaces (on the fine grid) which contain the aforementioned spline space on the coarse grid. The wavelet decomposition of such embedding is given and the corresponding formulas of decomposition and formulas of reconstruction are deduced. The space of (A, φ)-splines is introduced with three objects: the full chain of vectors, prescribed infinite grid on real axis and the preassigned vector-function φ with m + 1 components (m is called the order of the splines). Under certain assumptions, the splines belong to the class C m - 1. The gauge relations between the basic splines on the coarse grid and the basic splines on the fine grid are deduced. A general method for construction of a biorthogonal system of functionals (to basic spline system) is suggested. In this way, a chain of nested spline spaces is obtained, and the wavelet decomposition of the chain is discussed. The spaces and chains of spaces are completely classified in the terms of manifolds. The manifold of spaces considered is identified with the manifold of complete sequences of points of the direct product of an interval on the real axis and the projective space ℙ m; the manifold of nested spaces is identified with the manifold of nested sequences of points of the direct product mentioned above.
UR - http://www.scopus.com/inward/record.url?scp=77949284075&partnerID=8YFLogxK
U2 - 10.3103/S1063454108020039
DO - 10.3103/S1063454108020039
M3 - Article
AN - SCOPUS:77949284075
VL - 41
SP - 88
EP - 101
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 2
ER -
ID: 49712405