2. The Continuous Wavelet Transform

The images of -functions under the continuous wavelet transform constitute a reproducing kernel Hilbert space (r.k.H.s.). Such r.k.H.s.'s occur and are useful in many different contexts. One of the simplest examples is the space of all bandlimited functions, discussed in §§2.1 and 2.2. In §2.3 we introduce the concept of band and time limiting; of course no nonzero function can be strictly time-limited (i.e., for t outside [−T,T]) and band-limited ( for ), but one can still introduce time-and-band-limiting operators. We present a short review of the beautiful work of Landau, Pollak, and Slepian on this subject. We then switch to the continuous wavelet transform: the resolution of the identity in §2.4 (with a proof of (1.3.1)), the corresponding r.k.H.s. in §2.5. In §2.6 we briefly show how the one-dimensional results of the earlier sections can be extended to higher dimensions. In §2.7 we draw a parallel with the continuous windowed Fourier transform. In §2.8 we show how a different kind of time-and-band-limiting operator can be built from the continuous windowed Fourier transform or from the wavelet transform. Finally, we comment in §2.9 on the “zoom-in” property of the wavelet transform.