Piecewise linear interpolation revisited: BLaC-wavelets

The central issue of the present note is the BLaC operator, a ”Blending of Linear and Constant” approach. Several properties are proved, e.g., its positivity and the reproduction of constant functions. Starting from these results, error estimates in terms of ω1 and ω2 are given. Furthermore, we present the degree of approximation in the bivariate tensor product case. This is applicable to image compression. [2000 MSC] 41A15, 41A25, 41A36, 41A63 Dedicated to Professor Dr. Gh. Coman on the occasion of his 70th birthday. 1. Definitions and properties BLaC-wavelets (”Blending of Linear and Constant wavelets”) were introduced by G. P. Bonneau, S. Hahmann and G. Nielson around 1996 and constitute a tool to compromise between the perfect locality of Haar wavelets and the better regularity of linear wavelets. This compromise is realized by means of a parameter 0 < ∆ ≤ 1 that will appear in the sequel. First we introduce some notations. For the real parameter 0 < ∆ ≤ 1 consider the scaling function φ∆ : R → [0, 1] given by φ∆(x) :=  x ∆ , 0 ≤ x < ∆, 1, ∆ ≤ x < 1, − 1 ∆ · (x− 1−∆), 1 ≤ x < 1 + ∆, 0, else. Remark 1.1. The two extreme situations are obtained for ∆ = 1 and ∆ → 0, when φ∆ reduces to B-spline functions of first order, also called hat-functions, and to piecewise constant functions, respectively. The gap in between can be smoothly covered by letting ∆ be in the interval (0, 1]. Furthermore, for i = −1, . . . , 2 − 1, n ∈ N, we define by dilatation and translation of φ∆ the following family of (fundamental) functions: (1) φi (x) := φ∆(2 x− i), x ∈ [0, 1]. 1Alfréd Haar was born in 1885 in Budapest and died 1933 in Szeged. Until after World War I he had also a chair at the University of Cluj (then Kolozsvár). More about his biography can be found on the following site: http://www-history.mcs.st-andrews.ac.uk/Mathematicians .