Density in L2(Γ,μ) of Certain Families of Functions on LCA Groups Related to the Multi-Channel Sampling Problem

JH-singularity and JH-regularity of multivariate stationary processes over LCA groups

“…Therefore, one could start with Theorem 3.4 and try to derive the results of Section 2 from it. For special classes of measures M such an approach was taken by several authors (see [11], [14], [8]). However, it does not seem to be easy to infer Theorem 2.2 from Theorem 3.4.…”

“…In the case p = q = 1 another description of measures µ with Z(µ; F) = L 2 1,1 (µ) was obtained in [8,Theorem 3.5]. To illustrate the usefulness of Corollary 4.1 we mention that the assertions of Examples 4.4 and 4.6 as well as of Proposition 4.6 of [8] are its straightforward consequences, whereas some computations were needed to infer them from [8, Theorem 3.5].…”

“…Lloyd [11] introduced a measure ν by setting ν(B) := ∞ k=−∞ µ(B + 2kπ) for any Borel subset B of R. With its aid he computed the orthoprojection in L 2 (µ) onto sp{e ik • : k ∈ Z} and derived from this necessary and sufficient conditions for (1.2). Lloyd's approach was generalized to multivariate processes in [14] and applied to a problem of multichannel sampling in [8]. Another approach first taken by Yaglom [21] in a more or less explicit form consists in restricting the measure µ to each interval [2πk, 2π(k + 1)), k ∈ Z, shifting these restrictions to [0, 2π) and then adding them to obtain a measureμ.…”

“…Section 3 is devoted to some consequences of Theorem 2.2. In Section 4 we discuss a problem which arises in multichannel sampling and was solved in [8] using Lloyd's approach. It seems to us that from Theorem 2.2 one can deduce a simpler and more lucid solution of that problem.…”