Dyadic wavelets and refinable functions on~a~half-line

V. Yu. Protasov ; Yu. A. Farkov

Sbornik. Mathematics, Tome 197 (2006) no. 10, pp. 1529-1558

Résumé

For an arbitrary positive integer $n$ refinable functions on the positive half-line $\mathbb R_+$ are defined, with masks that are Walsh polynomials of order $2^n-1$. The Strang-Fix conditions, the partition of unity property, the linear independence, the stability, and the orthonormality of integer translates of a solution of the corresponding refinement equations are studied. Necessary and sufficient conditions ensuring that these solutions generate multiresolution analysis in $L^2(\mathbb R_+)$ are deduced. This characterizes all systems of dyadic compactly supported wavelets on $\mathbb R_+$ and gives one an algorithm for the construction of such systems. A method for finding estimates for the exponents of regularity of refinable functions is presented, which leads to sharp estimates in the case of small $n$. In particular, all the dyadic entire compactly supported refinable functions on $\mathbb R_+$ are characterized. It is shown that a refinable function is either dyadic entire or has a finite exponent of regularity, which, moreover, has effective upper estimates. Bibliography: 13 items.