Semi-Orthogonal Binary Spline Wavelets in Incompressible Fluid Dynamics

Abstract:

Jia and coworkers [1] have shown that with =M_N and  ̃=M_1as a pair of locally supported refinable functions, one can construct a function, _N(N being an odd integer) given by _N≔∑_(j=0)^N▒〖((-1)^j)/2 [M_(N+1) (j)+M_N (j+1) ] M_N (2∙-j)〗. Here M_N is a binary spline function of degree N. For r =0, 1, 2, …, N-1, the set {2^(j/2) _N^((r) ) (2^j∙-j);j,k ϵ Z} is a Riesz basis for L_2 (R). This base involves the first N-1 derivatives of the generating function and therefore is useful for dynamical systems with derivative constraints.

• Periodical: Applied Mechanics and Materials (Volumes 249-250)
• Pages: 164-169
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.249-250.164
• Online since: December 2012
• Authors: Steve Wai Chan, King Fai Lai, Mansoor Syed
• Keywords: Binary Spline Wavelets, Divergence Free Test Functions, Higher Order Finite Element Method, Navier-Stokes Equation, Riesz Basis, Wiener-Hopf Factorization

References

[1] R Jia, J Wang, D Zhou, Compactly Supported Wavelet for Sobolev Space, Applied and Computational Harmonic Analysis, 15, 224-241, (2003).

DOI: 10.1016/j.acha.2003.08.003

[2] O Christensen, Frame perturbation, Proc. Amer. Math. Soc. 123, 1217-1220, (1995).

[3] A Chorin, Numerical solution of the Navier-Stokes equations, Mathematics of Computation, vol. 22, 104, 754-762, (1968).

DOI: 10.1090/s0025-5718-1968-0242392-2

[4] S Zaglmayr, Higher Order Finite Element Methods for Electromagnetic Field Computation, Dissertation, Linz, (2006).

[5] O Christensen, P Casazza, Approximation of the Inverses Frame Operator and applications to the Gabor Frame, www. framerc. org/papers/ApproximationOfTheInverseFrameOperator Frames containing a Riesz basis and Approximation, www. braunschweig. de/Medien. /christensen-lindner2. pdf.

DOI: 10.1007/978-3-0348-8272-9_7

[6] C van der Mee, G Rodriguez, S Seatzu, Spectral factorization of bi-infinite multi-index block Toeplitz matrices, Linear Algebra and its Applications, 343-344, 355-380, (2002).

DOI: 10.1016/s0024-3795(01)00441-4

[7] T Ehrhart, C van der Mee, L Rodman, I Spitkovsky, Factorization in Weighted Wiener Matrix Algebra on Linearly Ordered Abelian Groups, Integral Equations and Operator Theory, 58, 65-86, (2007).

DOI: 10.1007/s00020-007-1491-3

[8] I Gohberg, S Goldberg, M Kaashoek, Classes of Linear Operators, vol. II, in Operator Theory, Advances and Applications, vol. 63, Birkhauser, Basel, (1993).

DOI: 10.1007/978-3-0348-8558-4_9

[9] T Goodman, C Micchelli, G Rodriguez, S Seatzu, On the Cholesky factorization of the Gram matrix of locally supported functions, BIT 34, (1994).

DOI: 10.1007/bf01737164

[10] K Lai, M Syed, S Chan, R Castillo, Towards Building a Duplex Wireless Undersea Acoustic Communication Link I, submitted for publication.

[11] C deBoor, R Jia, A Pinkus, Structure of Invertible Bi-infinite Totally Positive Matrices, Linear Algebra and Its Applications, 47: 41-55, (1982).

DOI: 10.1016/0024-3795(82)90225-7