Proof of a result relating to orthogonal scaling functions series expansions

In Zarowski the projection error behaviour of the scaling function series expansion was considered, and a closed form expression (which is valid when φ(t) satisfies certain moment constraints) for the error ε(t)=x(t)-a(t)(a(t) is the projection of x(t) onto subspace V_L of L²(R)) was given in Zarowski (2000) for x(t)={ts,t⩾0; 0,t<0 with s∈{0,1,2,...}. However, a proof of the validity of the stated expression for ε(t) was not given except for a few special values of s. In this paper we give a proof which is valid for all s∈{0,1,2,...}. Some miscellaneous facts about ε(t) are also presented