Compactified approximation from endpoint asymptotics

Suppose a continuous function $F(\lambda )$ on $[0,\infty )$ has a known power-law tail $F(\lambda )\sim a{\lambda }^{-p}$. Put $u=\lambda /(1+\lambda )$ and $h(u)=(1+\lambda {)}^{p}F(\lambda )$. When $h$ extends continuously to $u=1$, polynomial approximation on $[0,1]$ gives a global approximation $F(\lambda )\approx (1+\lambda {)}^{-p}{h}_N(u)$ using only elementary operations. Taylor series at zero and the tail asymptotic expansion can constrain endpoint coefficients. Agreement at the two endpoints alone does not guarantee accuracy in the intervening interval; uniform error control or intermediate calibration is needed.