Perturbations of the Haar wavelet

Abstract

Let m ∈ Z+ be given. For any e > 0 we construct a function f{e} having the following properties: (a) f{e} has support in [−e, 1 + e]. (b) f{e} ∈ Cm(−∞,∞). (c) If h denotes the Haar function and 0 < δ <∞, then ‖f−h‖Lδ(R) ≤ (1+2δ)1/δ(2e)1/δ . (d) f{e} generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to 1 as e→ 0. Let H be a Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖ := 〈·, ·〉1/2. Let Z+ denote the natural numbers. A sequence {fn, n ∈ Z+} ⊂ H is called a frame if there are constants A and B such that for every f ∈ H A‖f‖2 ≤ ∑ n∈Z+ |〈f, fn〉| ≤ B‖f‖2. The constants A and B are called bounds of the frame. If only the right-hand inequality is satisfied for all f ∈ H, then {fn, n ∈ Z+} is called a Bessel sequence with bound B. A frame is called exact, or a Riesz basis, if upon the removal of any single element of the sequence, it ceases to be a frame. However, not every frame is a Riesz basis: as is well known, a sequence {fn, n ∈ Z+} ⊂ H is a Riesz basis if and only if it is the image of an orthonormal basis under a bounded invertible linear operator U : H → H ([1, 11]). Clearly for an orthonormal basis both frame bounds equal 1. In [5] it is shown that adding a Bessel sequence with a small bound to a Riesz basis transforms the original basis into another Riesz basis. It is also shown how frame bounds for the new basis are obtained from frame bounds of the original basis. Given an arbitrary positive integer m, in the present paper we use these results to perturb the Haar function into a function f{e} ∈ Cm(−∞,∞) with support in [−e, 1 + e] (thus having good time and frequency localization) that preserves the symmetry of the Haar wavelet and generates an affine Riesz basis in L2(−∞,∞). The lack of orthogonality precludes the use of the fast wavelet transform. On the other hand the functions f{e} are given explicitly in terms of cardinal B-splines. Other wavelets do not have a closed form representation and have to be obtained recursively, using the cascade algorithm [4, 10]. Since frame bounds are given Received by the editors March 18, 1996 and, in revised form, June 21, 1996. 1991 Mathematics Subject Classification. Primary 42C99; Secondary 41A05, 46C99.