Abstract
The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.
Similar content being viewed by others
References
D. Berthold, W. Hoppe and B. Silbermann, The numerical solution of the generalized airfoil equation, J. Integral Equations Appl. 4 (1992) 309–336.
M.R. Capobianco and W. Themistoclakis, On the boundedness of de la Vallée Poussin operators, East J. Approx. 7(4) (2001) 417–444.
C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, eds. D. Braess and L.L. Schumaker (Birkhäuser, Basel, 1992) pp. 53–75.
A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmonic Anal. 1 (1993) 54–81.
I. Daubechies, Ten Lectures on Wavelets, CBMS Lecture Series, Vol. 61 (SIAM, Philadelphia, PA, 1992).
B. Fischer and J. Prestin, Wavelets based on orthogonal polynomials, Math. Comp. 66(220) (1997) 1593–1618.
B. Fischer and W. Themistoclakis, Orthogonal polynomial wavelets, Numer. Algorithms 30 (2002) 37–58.
T. Kilgore and J. Prestin, Polynomial wavelets on the interval, Constr. Approx. 12 (1996) 95–110.
T. Kilgore, J. Prestin and K. Selig, Polynomial wavelets and wavelet packet bases, Studia Sci. Math. Hungar. 33 (1997) 419–431.
T. Lyche, K. Mørken and E. Quak, Theory and algorithms for nonuniform spline wavelets, in: Multivariate Approximation Theory and Applications, eds. N. Dyn, D. Leviatan, D. Levin and A. Pinkus (Cambridge Univ. Press, Cambridge, 2001) pp. 152–187.
S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315(1) (1989) 69–87.
Y. Meyer, Ondelettes sur l’intervalle, Rev. Mat. Iberoamericana 7 (1992) 115–133.
C.A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmonic Anal. 1 (1994) 391–401.
P. Nevai, Mean convergence of Lagrange interpolation III, Trans. Amer. Math. Soc. 282 (1984) 669–698.
G. Plonka, K. Selig and M. Tasche, On the construction of wavelets on the interval, Adv. Comput. Math. 4 (1995) 357–388.
J. Prestin and K. Selig, Interpolating and orthonormal trigonometric wavelets, in: Signal and Image Representation in Combined Spaces, eds. J. Zeevi and R. Coifman (Academic Press, San Diego, 1998) pp. 201–255.
G. Steidl, Fast radix-p discrete cosine transform, Appl. Algebra Engrg. Comm. Comput. 3 (1992) 39–46.
G. Szegö, Orthogonal Polynomials, revised ed., AMS Colloquium Pubblications, Vol. XXIII (Amer. Math. Soc., New York, 1959).
M. Tasche, Polynomial wavelets on [−1,1], in: Approximation Theory, Wavelets and Applications, ed. S.P. Singh (Kluwer, Dordrecht, 1995) pp. 49–70.
W. Themistoclakis, Trigonometric wavelet interpolation in Besov spaces, Facta Univ. (Nis) Ser. Math. Inform. 14 (1999) 49–70.
W. Themistoclakis, Some interpolating operators of de la Vallée Poussin type, Acta Math. Hungar. 84(3) (1999) 221–235.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by R.-Q. Jia
Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.
AMS subject classification
65D05, 65T60
Rights and permissions
About this article
Cite this article
Capobianco, M.R., Themistoclakis, W. Interpolating polynomial wavelets on [−1,1]. Adv Comput Math 23, 353–374 (2004). https://doi.org/10.1007/s10444-004-1828-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-004-1828-2