3.5. Time-frequency localization.

One of our main motivations for studying wavelet transforms (or windowed Fourier transforms) is that they provide a time-frequency picture, with, hopefully, good localization properties in both variables. We have asserted several times that if ? itself is well localized in time and in frequency, then the frame generated by ? will share that property. In this section we want to make this vague statement more precise.

For the sake of convenience, we assume ? and to be symmetric (true if, e.g., ? is real and symmetric a good example is the Mexican hat function) ^[16]; then ? is centered around 0 in time and near ? ₀ in frequency (with, e.g., ? ₀ = . If ? is well localized in time and frequency, then ? _m,n will similarly be well localized around a ^m ₀ nb ₀ in time and around a ^{? m} ₀ ? ₀ in frequency. Intuitively speaking, ? f, ? _m,n ? then represents the "information content" in f near time a ^m ₀ nb ₀ and near the frequencies a ^{? m} ₀ ? ₀. If f itself is "essentially localized" on two rectangles in time-frequency space, meaning that, for some 0 < ? ₀ < ? ₁ < ?, 0 < T < ?,