Among the topics are tight and full spark Chebyshev frames with real entries and worst-case coherence analysis, spectral properties of an operator polynomial with coefficients in a Banach algebra, the invertibility of graph translation and support of Laplacian Fiedler vectors, p-Riesz bases in quasi-shift invariant spaces, and a matrix characterization of boundary representation of positive matrices in the

Hardy space. ([umlaut] Ringgold, Inc., Portland, OR)

It is well-known that H[mathematical expression not reproducible], where [H.sup.q] (U) denotes the

Hardy space on U.

Motivated by [17, 20, 37], etc, in this paper, we use the area integral function SL associated with the operator L to define the Herz-type

Hardy space [H[??].sup.[alpha],p.sub.q,L]([R.sup.n]).

The

Hardy space [H.sup.p], 1 [less than or equal to] p < [infinity], consists of those functions f in H(D) for which [mathematical expression not reproducible] where dm is the normalized arc length measure on T, where T = {z [member of] C : |z| = 1}.

The rearrangement- invariant

Hardy space associated with x, [H.sub.x], consists of those f [member of] S'(R)/P such that

Recently, Ky [20] introduced a new Musielak-Orlicz

Hardy space [H.sup.[phi]]([R.sup.n]), via the grand maximal function, and established its atomic characterization.

Ueki, "Weighted composition operators from the weighted Bergman space to the weighted

Hardy space on the unit ball," Applied Mathematics and Computation, vol.

Given a Schwartz function [phi] [member of] S([R.sup.d]) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and supp [phi] [subset] [[0, 1/2].sup.d], the local

Hardy space [h.sub.p]([R.sup.d]) (0 < p [less than or equal to] [infinity]) consists of all tempered distributions f for which

They are everywhere defined in some special cases on the classical

Hardy Space [H.sup.2] (the case when [[beta].sub.n] = 1 for all n).

For p [member of][1, [infinity]) the

Hardy space [H.sup.p]([[PI].sub.+]) consists of all f [member of] H([[PI].sub.+]) such that

An elementary inquiry, based on examples and counterexamples, of some qualitative properties of doubly orthogonal systems of analytic functions on domains in [C.sup.n] leads to a better understanding of the deviation from the classical

Hardy space of the disk setting.

For instance the classical theorem of Beurling [3] on the structure of analytically invariant subspaces of the

Hardy space on the torus has been influential in many areas of modern mathematics, ranging from the dilation theory of a contraction, to interpolation problems in function theory or to the probabilistic analysis of time series.