<title>Pyramidal Riesz products associated with subband coding and self-similarity</title>

Multiscale Riesz Products and their Support Properties

“…In this paper we shall not investigate the properties of measures stemming from the full n flexibility of their pyramidal structure, but essentially only for a constant sequence 1 = 2 = · · ·. However, for an arbitrary sequence { n }, the authors in [2] proved that there exists a continuous measure µ α ∈ M(T) which is a weak- * limit point of the sequence {P N }, and that µ α is absolutely continuous if and only if |α n | 2 < ∞.…”

“…Proof. To simplify notation, we shall let α (1) = α, α (2) = β, P (1) n = P n (H, R, α) and P (2) n = P n (H, R, β). We shall use the functions H (R n−1 t) as our sequence of random variables H n , and singularity will be proven if we show that…”

“…The Fourier and wavelet approaches were first combined in [2] by introducing the concept of a pyramidal Riesz product The term "pyramidal" is used since P N depends on the choice of n at each level n, and since various sequences { n } correspond to paths down a binary tree. This is an infinite analogue of wavelet packet algorithms.…”

“…Section 2 gives an overview of our point of view and sets up some general notation. Section 4 is devoted to generalizing the results obtained in [2], using trigonometric polynomials as generating functions. In Section 3 we focus on multiscale Riesz products using piecewise constant generating functions, which allows us to obtain measures of thin support, as explained in Section 5.…”

Waveform and Transceiver Designs for Smart Radio

“…This section attempts to outline the contemporary works which apply wavelets with either goal. For a purely mathematical perspective the work by Benedetto [2] provides a theoretical foundation for secure communication.…”