On the construction of wavelets on a bounded interval

This paper presents a general approach to a multiresolution analysis and wavelet spaces on the interval Our method is based on the Chebyshev transform correspond ing shifts and the discrete cosine transform DCT For the wavelet analysis of given functions e cient decomposition and reconstruction algorithms are proposed using fast DCT algorithms As examples for scaling functions and wavelets polynomials and trans formed splines are considered Introduction Recently several constructions of wavelets on a bounded interval have been presented Most of these approaches are based on the theory of cardinal wavelets The simplest con struction consists in the trivial extension of functions f R by setting f x for x R n These functions can be analyzed by means of cardinal wavelets But in general this extension produces discontinuities at x as well as x which are re ected by large wavelet coe cients for high levels near the endpoints and even if f is smooth on Thus the regularity of f is not characterized by the decay of wavelet coe cients Another simple solution often adapted in image analysis consists in the even periodic extension f of f R If f C then f C R But in general if f C then the derivative of f has discontinuities at the integers The smoothness of f is again not characterized by the decay of wavelet coe cients In Meyer has derived orthonormal wavelets on by restricting Daubechies scal ing functions and wavelets to and orthonormalizing their restrictions by the Gram Schmidt procedure This idea led to numerical instabilities such that further investigations of wavelets on a bounded interval were necessary see We are interested in wavelet methods on a bounded interval which can exactly analyze the boundary behaviour of given functions Up to now three methods are known to solve this problem The often used rst method is based on special boundary and interior scaling functions as well as wavelets see such that numerical problems at the boundaries can be reduced Then the bases of sample and wavelet spaces do not consist in shifts of single functions The second method see works with two generalized di lation operations since the classical dilation is not applicable for functions on a bounded interval A third wavelet construction on the interval I rst proposed in is based on Chebyshev polynomials Both scaling functions and wavelets are polynomials which satisfy certain interpolation properties As shown in this polynomial wavelet ap proach can be considered as generalized version of the well known wavelet concept which is based on shift invariant subspaces of the weighted Hilbert space L w I with respect to the Chebyshev shifts see where w denotes the Chebyshev weight The objective of this paper is a new general approach to multiresolution of L w I and to wavelets on the interval I based on the ideas in As known the Fourier transform and shift invariant subspaces of L R are essential tools for the construction of cardinal multiresolution and wavelets see Analogously the nite Fourier transform and shift invariant subspaces of L lead to a uni ed approach to periodic wavelets see This concept can be transferred to the Hilbert space L of even periodic functions using the shift operator Sa F F a F a a R for F L The isomorphism between L and L w I can be exploited in order to construct new sample and wavelet spaces in L w I Using fast algorithms of dis crete cosine transforms DCT e cient frequency based algorithms for decomposition and reconstruction are proposed As special scaling functions and wavelets we consider algebraic polynomials and transformed splines It is remarkable that our decomposition algorithm for polynomial wavelets needs less multiplications up to a certain level than the fast decomposition algorithm for cubic spline wavelets on proposed in The outline of our paper is as follows In Section we brie y introduce the Chebyshev transform related shifts and the DCT In Section we analyze shift invariant subspaces of L w I The scalar product of functions from shift invariant subspaces can be simpli ed to a nite sum by means of the so called bracket product In Section we consider a nonstationary multiresolution of L w I consisting of shift invariant subspaces Vj j N generated by shifts of scaling functions j The required conditions for the multiresolution of L w I and their consequences for the scaling functions j are analyzed in detail In Section we introduce the wavelet space Wj j N as the orthogonal complement of Vj in Vj Then Wj is a shift invariant subspace generated by shifts of the wavelet j Using the two scale symbol of j and the bracket product of j and j the wavelet j is characterized in Theorem Section provides fast numerically stable decomposition and reconstruction algorithms based on fast DCT algorithms In Section we present polynomial wavelets on I see Finally in Section we adapt the theory of periodic splines to the interval I with respect to the Chebyshev nodes Note that the transformed spline wavelets are supported on small subintervals of I The examples show that periodic multiresolutions of L with even scaling functions j can be transformed into a multiresolution of L w I Chebyshev Transform and Shifts In this section we introduce the Chebyshev transform and corresponding shifts and we examine their relations to the even shifts of periodic even functions For more details on Chebyshev shifts we refer to Throughout this paper we consider the interval I and the Chebyshev weight w x x for x Let L w I be the weighted Hilbert space of all measurable functions f I R with the property Z