## Real Analysis Exchange

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

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

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1691749

**Zentralblatt MATH identifier**

0940.42018

**Subjects**

Primary: 42C5 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

**Keywords**

wavelets splines integral representation distribution

#### 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