## Rocky Mountain Journal of Mathematics

### Construction of new multiple knot B-spline wavelets

#### Abstract

This paper deals with construction of non-uniform multiple knot B-spline wavelet basis functions (with minimal support). These wavelets are semi-orthogonal on a bounded interval. A large family of multiple knot B-spline wavelets is presented that gives a variety of basis functions with explicit formulas and locally compact supports. Moreover, the structure of this wavelet is conceptually simple and easy to implement. Finally, some examples of multiple knot B-spline wavelets are also presented.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 5 (2017), 1463-1495.

Dates
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.rmjm/1506045621

Digital Object Identifier
doi:10.1216/RMJ-2017-47-5-1463

Mathematical Reviews number (MathSciNet)
MR3705761

Zentralblatt MATH identifier
1376.41005

Keywords
Multiple knot B-spline wavelet

#### Citation

Esmaeili, Maryam; Tavakoli, Ali. Construction of new multiple knot B-spline wavelets. Rocky Mountain J. Math. 47 (2017), no. 5, 1463--1495. doi:10.1216/RMJ-2017-47-5-1463. https://projecteuclid.org/euclid.rmjm/1506045621

#### References

• M. Bertram, Single-knot wavelets for non-uniform B-splines, Comp. Aided Geom. Design 22 (2005), 849–-864.
• C.K. Chui and E. Quak, Wavelets on a bounded interval, in Numerical methods in approximation theory, D. Braess and L.L. Schumaker, eds., Birkhäuser Verlag, Basel, 1992.
• C. Cohen, Numerical analysis of wavelet methods, Volume 32, Elsevier, North-Holland, 2003.
• I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, PA, 1992.
• J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. IEEE Inter. Conf. Neural Networks IV, 1995, 1942–-1948.
• T. Lyche, K. Mørken and E. Quak, Theory and algorithms for non-uniform spline wavelets, in Multivariate approximation and applications, N. Dyn, D. Leviatan, D. Levin, and A. Pinkus, eds., Cambridge University Press, Cambridge, 2001.
• S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^{2}(\mathbb{R})$, Trans. Amer. Math. Soc. 315 (1989), 69–87.
• Y. Meyer, Ondelettes et Opréteurs, Hermann, Paris, 1990.
• F. Mirafzali and A. Tavakoli, Hydroelastic analysis of fully nonlinear water waves with floating elastic plate via multiple knot B-splines, Appl. Ocean Res. 51 (2015), 171-–180.
• G. Plonka, Generalized spline wavelets, Constr. Approx. 12 (1996), 127–155.
• F. Pourakbari and A. Tavakoli, Modification of multiple knot B-spline wavelet for solving $($partially$)$ Dirichlet boundary value problem, Adv. Appl. Math. Mech. 4 (2012), 799–820.
• E. Quak and N. Weyrich, Algorithms for spline wavelet packets on an interval, BIT 37 (1997), 76–95.
• L.L. Schumaker, Spline functions: Basic theory, Third edition, Cambridge University Press, Cambridge, 2007.
• A. Tavakoli and M. Esmaeili, Construction of dual multiple knot B-spline wavelets on the interval, in preparation.
• K. Urban, Wavelet methods for elliptic partial differential equations, G.H. Golub, A.M. Stuart and E. Suli, eds., Oxford University Press, Inc., New York, 2009.