About window function

Question

In Charles.K.Chui's An introduction to wavelets, on Page54, the window function is a non-trivial function $w∈L^{2}(R)$ satisfying $tw(t)∈L^{2}(R)$. I want to ask how to understand the notion, and how the window function w(t) behaves as t→∞.

Answer 1

The notation means that $\int t^2| w(t) |^2 dt <\infty$. Actually this expresses the decay of $w$ pretty good; $t^2|w(t) |^2 \in o(1/t)$ i.e. $w(t) \in o(t^{-3/2})$.