in this paper, let $l$ be a completeresiduated lattice, and let {bf set} denote the category of setsand mappings, $lf$-{bf pos} denote the category of $lf$-posets and$lf$-monotone mappings, and $lf$-{bf cslat}$(sqcup)$, $lf$-{bfcslat}$(sqcap)$ denote the category of $lf$-completelattices and $lf$-join-preserving mappings and the category of$lf$-complete lattices and $lf$-meet-preserving mapping...

Several techniques are known to compute a new orthogonal polynomial φk+1 of degree k + 1 from Lk := span{φ0, ..., φk} in case of (discrete) orthogonality on the real line. In the Arnoldi approach one chooses Φk ∈ Lk and makes xΦk orthogonal against φ0, ..., φk. By taking as Φk a linear combination of φk and the kernel (or GMRES) polynomial ψk(x) = ∑k j=0 φj(0)φj(x), one needs to orthogonalize o...

where we take formally £_i(x)=0, po(x) — l, and have incorporated suitable constant factors in the pn{x). Conversely [2] we may pass from (1.2) to the existence of at least one spectral function \[/(x) with respect to which (1.1) holds. The position is much less clear in respect to orthogonal polynomials in several variables, for which work has been devoted mainly to the approach which starts w...

An orthogonality space is a set X together with symmetric and irreflexive binary relation ⊥, called the relation. A block partition of maximal mutually orthogonal elements X, decomposition collection subsets each which complement union others. (X,⊥) normal if any gives rise to unique space. The one-dimensional subspaces Hilbert equipped usual provides motivating example. Together maps that are,...

Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V...