A Riesz basis of wavelets and its dual with quintic deficient splines

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1 Note di Matematica 25, n. 1, 2005/2006, A Riesz basis of wavelets and its dual with quintic deficient splines F. Bastin Department of Mathematics B37, University of Liège, B-4000 Liège, Belgium f.bastin@ulg.ac.be Abstract. In this note, the dual of the Riesz basis of quintic splines wavelets obtained in [1] is explicitly constructed. Keywords: Wavelets, quintic deficient splines, Riesz basis, dual basis MSC 2000 classification: primary 42C40, secondary 41A15 Introduction It is well known that for any natural number m, the cardinal B-spline N m+1 = χ [0,1] χ [0,1] (m + 1factors can be used as a scaling function to construct orthogonal and biorthogonal bases of wavelets in L 2 (R, with different properties (see for example [3], [8]. But, in approximation theory for instance, other splines are also very popular: the deficient splines (see some recent results in [5], [9]. In the paper [1], one can find a direct approach of the problem of the explicit construction of scaling functions, multiresolution analysis and wavelets with symmetry properties and compact support, involving deficient splines of degree 5 and regularity 3. Other results can also be found in [6], [7]. The present paper is a continuation of [1]. It gives an explicit construction of the dual basis of the deficient splines wavelets basis obtained in [1]. The dual is also generated by two wavelets, which are deficient splines with symmetry properties and exponential decay. 1 Definitions, notations, deficient spline wavelets For m N, the set of deficient splines of degree 2m + 1 is the set V 0 = {f L 2 (R : f [k,k+1] = P (2m+1 k, k Z and f C m+1 (R}. For m = 1, it is the set of classical cardinal cubic splines. For m = 2 we denote it as the set of deficient quintic splines

2 56 F. Bastin and, in this note, we only consider this case. In this section, we recall the explicit and direct construction of a Riesz basis of wavelets consisting of deficient splines wavelets with compact support and symmetry property of [1]. 1 Proposition. The following functions ϕ a andϕ s x x5 if x [0, 1] 9 ϕ a (x = 8 (x (x (x if x [1, 2] (3 x (3 x5 if x [2, 3] 0 if x < 0 or x > 3 x x5 if x [0, 1] 57 ϕ s (x = (x (x if x [1, 2] (3 x (3 x5 if x [2, 3] 0 if x < 0 or x > 3 are respectively antisymmetric and symmetric with respect to 3 2 and the family constitutes a Riesz basis of V 0. For every Z we define {ϕ a (. k, k Z} {ϕ s (. k, k Z} V = {f L 2 (R : f(2. V 0 }. 2 Proposition. The sequence V ( Z is an increasing sequence of closed sets of L 2 (R and V = {0}, V = L 2 (R. Z Moreover, the functions ϕ a, ϕ s satisfy the following scaling relation ϕ s (2ξ ϕ s (ξ = M 0 (ξ ϕ a (2ξ ϕ a (ξ Z where M 0 (ξ is the matrix (called filter matrix M 0 (ξ = e 3iξ/ cos( ξ 3ξ cos( 2 9i(sin( ξ 3ξ 2 + sin( 2 i(11 sin( 3ξ sin( ξ 3ξ 2 7 cos( cos( ξ 2 For every Z, we denote by W the orthogonal complement of V in V +1. Using standard techniques of Fourier analysis in the context of wavelets, one obtains the following result..

3 Wavelets with deficient splines 57 3 Proposition. A function f belongs to W 0 if and only if there exists p, q L 2 loc, 2π periodic such that f(2ξ = p(ξ ϕ s (ξ + q(ξ ϕ a (ξ and ( p(ξ M 0 (ξ W (ξ q(ξ ( p(ξ + π + M 0 (ξ + π W (ξ + π q(ξ + π = 0 a.e. where M 0 is the filter matrix obtained in Proposition 2 and W (ξ is the matrix ( ωs (ξ ω m (ξ W (ξ = ω m (ξ ω a (ξ with ω a (ξ = ω s (ξ = ω m (ξ = ϕ a (ξ + 2lπ 2 = ϕ s (ξ + 2lπ 2 = cos ξ 385 cos(2ξ cos ξ + 53 cos(2ξ ϕ s (ξ + 2lπ ϕ a (ξ + 2lπ = i sin ξ ( cos ξ Theorem. There exists deficient splines wavelets with support in [0, 5] and symmetry properties (with respect to 5/2. More precisely, there exists real numbers verifying p (s, q (s, p (a, q (a, = 0,..., 7 p (s = p (s 7, q(s = q (s 7, p(a = p (a 7, q(a = q (a 7, = 0, 1, 2, 3 such that the family {ψ s (. k : k Z} {ψ a (. k : k Z} constitutes a Riesz basis of W 0, where ψ s (2ξ = ψ a (2ξ = 7 =0 7 =0 p (s e iξ ϕ s (ξ + p (a e iξ ϕ s (ξ + 7 =0 7 =0 q (s e iξ ϕ a (ξ q (a e iξ ϕ a (ξ.

4 58 F. Bastin Explicit values of the coefficients can be found in [1]. It follows that the family {2 /2 ψ s (2. k :, k Z} {2 /2 ψ a (2. k :, k Z} ((* constitutes a Riesz basis of L 2 (R of deficient splines wavelets with compact support and symmetry properties. The symmetry properties can be written as follows ψ s (ξ = e 5iξ ψs ( ξ, ψ a (ξ = e 5iξ ψa ( ξ. Here are pictures of ϕ s, ϕ a and of ψ s, ψ a (up to a multiplicative constant 2 The dual basis The following result is classical in the context of frames and Riesz basis (see for example [2], [4]. 5 Proposition. If f m (m N is a Riesz basis of an Hilbert space H, there exists a unique sequence g m (m N of elements of H such that < f m, g k >= δ km for every m, k N. More precisely one has g m = S 1 f m, m N where S is the frame operator S : H H f m=1 < f, f m > f m.

5 Wavelets with deficient splines 59 The sequence g m (m N is also a Riesz basis and is called the dual Riesz basis of f m (m N. It also satisfies for every f H. f = m=1 < f, f m > g m = m=1 < f, g m > f m Now, we want to give an explicit construction of the dual basis of the Riesz basis (*. But before doing so, let us install some notations and let us also briefly recall some additional properties concerning the wavelet basis (*. We denote by W ψ the matrix similar to W (see Proposition 3 but defined using the functions ψ a, ψ s instead of ϕ a, ϕ s, i.e. ( ωψs (ξ ω ψs,ψ W ψ (ξ = a (ξ ω ψs,ψ a (ξ ω ψa (ξ where ω ψa (ξ = ψ a (ξ + 2lπ 2, ω ψs (ξ = ω ψs,ψ a (ξ = These functions have the following properties. ψ s (ξ + 2lπ ψ a (ξ + 2lπ. ψ s (ξ + 2lπ Property. The functions ω ψa, ω ψs, ω ψs,ψ a are 2π- periodic trigonometric polynomials such that and ω ψa (ξ c > 0, ω ψs (ξ c > 0, ω ψa ( ξ = ω ψa (ξ, ω ψs ( ξ = ω ψs (ξ ω ψs,ψ a (ξ = ω ψs,ψ a (ξ = ω ψs,ψ a ( ξ for every ξ R. There are also A, B > 0 such that A det(w ψ (ξ B, ξ R. Proof. The proof is direct, using the support and the symmetry properties of the functions ψ a, ψ s and the Riesz condition satisfied by the basis (*. QED Since the wavelets of different levels are orthogonal to each other (that is to say, the spaces W and W are orthogonal if, it suffices to consider one scale (say, = 0 to construct the dual. That s the reason why we present the construction of the dual as follows.

6 60 F. Bastin 6 Theorem. The functions ψ 1, ψ 2 defined as ψ 1 (ξ = α 1 (ξ ψ a (ξ + β 1 (ξ ψ s (ξ, ψ 2 (ξ = α 2 (ξ ψ a (ξ + β 2 (ξ ψ s (ξ where α 1 (ξ = ω ψ s (ξ det(w ψ (ξ, β 1(ξ = ω ψ s,ψ a (ξ det(w ψ (ξ, α 2 (ξ = β 1 (ξ = β 1 (ξ, β 2 (ξ = ω ψ a (ξ det(w ψ (ξ are such that the family of functions { } 2 /2 ψi (2. k : i = 1, 2;, k Z is the dual basis of the basis of wavelets (*. Proof. First, we look for a function ψ 1 in W 0 such that < ψ a (. k, ψ 1 > L 2 (R= δ 0k and < ψ s (. k, ψ 1 > L 2 (R= 0 for every k Z. Since {ψ s (. k : k Z} {ψ a (. k : k Z} constitute a Riesz basis of W 0, we look in fact for 2π-periodic and L 2 loc functions α 1, β 1 such that ψ 1 (ξ = α 1 (ξ ψ a (ξ + β 1 (ξ ψ s (ξ and such that < e ik. ψa, α 1 ψa +β 1 ψs > L 2 (R= 2πδ 0k and < e ik. ψs, α 1 ψa +β 1 ψs > L 2 (R= 0 for every k Z. The last equalities are equivalent to 2π 0 e (α ikξ 1 (ξω ψa (ξ + β 1 (ξω ψs,ψ a (ξ dξ = 2πδ 0k 2π 0 e (α ikξ 1 (ξω ψs,ψ a (ξ + β 1 (ξω ψs (ξ dξ = 0, k Z hence also to { α1 (ξω ψa (ξ + β 1 (ξω ψs,ψ a (ξ = 1 α 1 (ξω ψs,ψ a (ξ + β 1 (ξω ψs (ξ = 0. Using matrices, this can be rewritten as ( ωψs (ξ ω ψs,ψ a (ξ ω ψs,ψ a (ξ ω ψa (ξ ( β1 (ξ α 1 (ξ ( β1 (ξ = W ψ (ξ α 1 (ξ = ( 0 1.

7 Wavelets with deficient splines 61 The solutions of this system is ( ( β1 (ξ 1 ωψs,ψ = a (ξ α 1 (ξ det(w ψ (ξ ω ψs (ξ = ( 1 ωψs,ψ a (ξ det(w ψ (ξ ω ψs (ξ. We proceed exactly in the same way to find a function ψ 2 in W 0 such that < ψ a (. k, ψ 2 >= 0 and < ψ s (. k, ψ 2 >= δ 0k for every k Z. In this case, the final system is ( ( β2 (ξ 1 W ψ (ξ = α 2 (ξ 0 which gives the solutions. Since the spaces W and W are orthogonal if, we obtain, for,, k, k Z: and < 2 /2 ψ a (2. k, 2 /2 ψ1 (2. k >= δ δ kk < 2 /2 ψ s (2. k, 2 /2 ψ1 (2. k >= 0 < 2 /2 ψ a (2. k, 2 /2 ψ2 (2. k >= 0 < 2 /2 ψ s (2. k, 2 /2 ψ1 (2. k >= δ δ kk hence the conclusion. QED 7 Proposition. The functions ψ 1, ψ 2 are deficient splines with exponential decay and symmetry properties ( ψ 1, ψ 2 are respectively antisymmetric and symmetric relatively to 5/2. Proof. By construction, these functions are deficient splines. Their explicit expressions in terms of the Fourier transforms of the wavelets ψ a, ψ s and the form of the coefficients α i, β i give the exponential decay and the symmetry properties. QED Here are pictures of an approximation of the dual functions ψ 1, ψ 2 (up to a constant factor.

8 62 F. Bastin Acknowledgements. The author is grateful to Professors P. Butzer and P. Wodyllo for discussions during the meeting in Strobl, Austria, May References [1] F. Bastin, P. Laubin: Quintic deficient splinewavelets, Bull. Soc. Roy. Sc. Liège, Vol 71 (3, 2002, [2] Wavelets: mathematics and applications, Studies in advanced mathematics, CRC 1994, edited by John J. Benedetto, Michael W. Frazier. [3] C.K. Chui, J.Z. Wang: On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc., 330, 2, 1992, [4] I. Daubechies: Ten lectures on wavelets, CBMS 61, Siam, [5] O. Gilson, O. Faure, P. Laubin: Quasi-optimal convergenceusing interpolation by nonuniform deficient splines, to appear in the Journal of Concrete and Applicable Mathematics [6] T.N.T. Goodman, S.L. Lee: Wavelets of multiplicity r, Trans.Amer. Math. Soc. 342, 1, 1994, [7] T.N.T. Goodman, S.L. Lee, W.S. Tang: Wavelets in wanderingsubspaces, Trans. Amer. Math. Soc., 338, 2, 1993, [8] Y. Meyer: Ondelettes et opérateurs, I,II,III, Hermann, Paris, [9] S.S. Rana, Y.P. Dubey: Best error bounds of deficient quintic splines interpolation, Indian J. Pure Appl. Math., 28, 1997,