Perturbations of the Haar wavelet

Abstract. Let m 2 Z+ be given. For any "> 0 we construct a function ff"g having the following properties: (a) ff"g has support in ["; 1 + "]. (b) ff"g 2 Cm(1;1). (c) If h denotes the Haar function and 0 < <1, then kff"ghkL(R) (1+2 )1=(2")1= . (d) ff"g generates an ane Riesz basis whose frame bounds (which are given explicitly) converge to 1 as "! 0. Let H be a Hilbert space with inner product h; i and norm k k: = h; i1=2. Let Z+ denote the natural numbers. A sequence ffn; n 2 Z+g H is called a frame if there are constants A and B such that for every f 2