## Real Analysis Exchange

### Wavelet Analysis in Spaces of Slowly Growing Splines Via Integral Representation

Valery A. Zheludev

#### Abstract

In this paper we consider polynomial splines with equidistant nodes which may grow as O$|x|^s$. We present an integral representation of such splines with a distribution kernel where exponential splines are used as basic functions. By this means we characterize splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form the biorthogonal basis. We suggest a scheme of wavelet analysis in the spaces of growing splines based on integral representation of the splines. The key point of that scheme is the refinement equation for the exponential splines which contains only two terms. We construct the so called exponential wavelets. We establish conditions for a spline to be a basic wavelet which enable us to form a library of such wavelets. We give formulas for the decomposition of a spline into a weak orthogonal sum of the sparse-grid spline and an element of the corresponding wavelet space. Reconstruction formulas are presented which permit the use of arbitrary bases of spline and wavelet spaces.

#### Article information

Source
Real Anal. Exchange, Volume 24, Number 1 (1998), 229-262.

Dates
First available in Project Euclid: 23 March 2011

https://projecteuclid.org/euclid.rae/1300906026

Mathematical Reviews number (MathSciNet)
MR1691749

Zentralblatt MATH identifier
0940.42018

#### Citation

Zheludev, Valery A. Wavelet Analysis in Spaces of Slowly Growing Splines Via Integral Representation. Real Anal. Exchange 24 (1998), no. 1, 229--262. https://projecteuclid.org/euclid.rae/1300906026

#### References

• A. Aldroubi, and M. Unser, Families of multiresolution and wavelet spaces with optimal properties, Numer. Funct. Anal. Optimiz. 14(1993), 417–446.
• G. Battle, A block spin construction of ondelettes. Part I. Lemarié functions, Comm. Math. Phys. 110(1987), 601–615.
• C. de Boor, R. DeVore, and A. Ron, Approximation from shift-invariant subspaces of $L_2($R$^s)$, Trans. Amer. Math. Soc., 341 (1994), 787–806.
• C. K. Chui and J. Z. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc.330 (1992), 903–915.
• C. K. Chui and J. Z. Wang, Computational and algorithmic aspects of cardinal spline-wavelets, Appr. Th. Appl., 9 (1993), 53–75.
• C. K. Chui An introduction to wavelets Acad.Press, San Diego CA, 1992.
• N. Dyn, Subdivision schemes in computer-aided geometric design, in Advances in Numerical Analysis II, Wavelets, Subdivision Algorithms and Radial Functions, W.A. Light (ed.), Oxford University Press,(1992) 36–104.
• M. Kamada, K. Toraici and Riochi Mori, Periodic splines orthonormal bases, J. Approx. Th. 55, (1988), 27–34.
• Y. W. Koh, S. L. Lee and H. H. Tan, Periodic orthogonal splines and wavelets, Appl. and Comp. Harm. Anal., 1995, 201-218.
• P. G. Lemarié, Ondelettes à localization exponentielle, J. de Math. Pure et Appl., 67 (1988) 227–236.
• I. J. Schoenberg, Cardinal spline interpolation, CBMS, 12, SIAM, Philadelphia, 1973.
• I. J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart.Appl. Math., 4 (1946), 45-99, 112–141.
• Yu. N. Subbotin, On the relation between finite differences and the corresponding derivatives, Proc. Steklov Inst. Math., 78 (1965), 24–42.
• A. H. Zemanian, Distribution theory and transform analysis, McGraw-Hill, New York, 1965.
• V. A. Zheludev, Spline Harmonic Analysis and Wavelet Bases, Proc. Symp.Appl.Math., 48 (ed. W.Gautcshi), AMS 1994, 415–419.
• V. A. Zheludev, Periodic splines and wavelets, in Contemporary Mathematics, 190, Mathematical Analysis, Wavelets & Signal Processing, (M.E.H.Ismail, M.Z.Nashed, A.I.Zayed, A.F.Ghaleb Eds.), Amer. Math. Soc., Providence, RI, pp. 339-354, 1995.
• V. A. Zheludev, Integral representation of slowly growing equidistant splines and spline wavelets, Tel Aviv University, The School of Mathematical Sciences, Technical Report 5-96, (1996).
• V. A. Zheludev Integral representation of slowly growing equidistant splines, Approximation Theory and Applications, to appear.