Radial averaging operator from the origin

For continuous $F$ near zero, its radial average is

(AF)(Y)=\int_0^1F(tY)\,dt.

For $Y\ne0$ , this equals $Y^{-1}\int_0^YF(s)\,ds$ ; the integral on the unit interval also defines $AF(0)=F(0)$ .

Regularity and coefficients

If $F$ is smooth, differentiation on the compact integration interval gives $(AF)^{(m)}(Y)=\int_0^1t^mF^{(m)}(tY)\,dt$ , so the average is smooth through zero. If $F=\sum F_\alpha Y^\alpha$ is analytic, then $(AF)_\alpha=F_\alpha/(\alpha+1)$ . The apparent quotient singularity is removed by this integral representation.