Expansion of Self-Similar Functions in the Faber–Schauder System

摘要

AbstractLet Ω =ANbe a space of right-sided infinite sequences drawn from a finite alphabetA= {0,1}, N = {1,2,…}. Let ρ(x, y)Σk=1∞|xk?yk|2?kbe a metric on Ω =AN, and μ the Bernoulli measure on Ω with probabilitiesp0,p1p0+p1= 1. Denote byB(x,ω) an open ball of radiusrcentered at ω. The main result of this paper$$mu (B(omega ,r))r + sumolimits_{n = 0}^infty {sumolimits_{j = 0}^{{2^n} - 1} {{mu _{n,j}}} } (omega )au ({2^n}r - j)$$μ(B(ω,r))r+∑n=0∞∑j=02n?1μn,j(ω)τ(2nr?j), where τ(x) = 2min {x,1 ?x}, 0 ≤ x ≤ 1, (τ(x) = 0, ifx< 0 orx$${mu _{n,j}}(omega ) = (1 - {p_{{omega _{n + 1}}}})prod _{k = 1}^n{p_{{omega _k}}} oplus {j_k}$$μn,j(ω)=(1?pωn+1)∏k=1npωk?jk,$$j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}$$j=j12n?1+j22n?2+...+jn. The family of functions 1,x, τ(2