The approximation of bivariate functions by sums of univariate ones using the L1-metric

1982 ◽  
Vol 25 (2) ◽  
pp. 173-181 ◽  
Author(s):  
W. A. Light ◽  
J. H. McCabe ◽  
G. M. Phillips ◽  
E. W. Cheney
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Approximation Problem ◽  
The One ◽  
Image Position ◽  
Special Case ◽  
Bivariate Functions

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

scholarly journals Pythagorean Orthogonality in a Normed Linear Space

1958 ◽  
Vol 9 (4) ◽  
pp. 168-169
Author(s):  
Hazel Perfect
Keyword(s):  
Euclidean Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Image Position ◽  
Pythagorean Orthogonality

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

scholarly journals On Exposed and Farthest Points in Normed Linear Spaces

1971 ◽  
Vol 12 (3) ◽  
pp. 301-308 ◽  
Author(s):  
M. Edelstein ◽  
J. E. Lewis
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Nonempty Subset ◽  
Linear Spaces ◽  
Farthest Points ◽  
Normed Linear Spaces ◽  
Farthest Point ◽  
Image Position

Let S be a nonempty subset of a normed linear space E. A point s0 of S is called a farthest point if for some x ∈ E, . The set of all farthest points of S will be denoted far (S). If S is compact, the continuity of distance from a point x of E implies that far (S) is nonempty.

scholarly journals On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajracharya ◽  
Vishnu Narayan Mishra
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Isosceles Orthogonality ◽  
Complete Proof ◽  
Underlying Space ◽  
Real Inner Product Space ◽  
Special Case ◽  
Real Normed Linear Space ◽  
Pythagorean Orthogonality

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

Minimal projections in Lp-spaces

1985 ◽  
Vol 97 (1) ◽  
pp. 127-136 ◽  
Author(s):  
E. J. Halton ◽  
W. A. Light
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Minimal Projection ◽  
Proper Subspace ◽  
Linear Map ◽  
Lp Spaces ◽  
Minimal Projections ◽  
Image Position

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the numberhas some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case

A Note on Best Simultaneous Approximation in Normed Linear Spaces

1976 ◽  
Vol 19 (3) ◽  
pp. 359-360 ◽  
Author(s):  
Arne Brøndsted
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Convex Functions ◽  
Existence And Uniqueness ◽  
Simultaneous Approximation ◽  
Normed Linear Space ◽  
Present Note ◽  
Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

Best Possible Nets in a Normed Linear Space1

1975 ◽  
Vol 18 (1) ◽  
pp. 45-48 ◽  
Author(s):  
L. L. Keener
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Sufficient Condition ◽  
Bounded Set ◽  
Image Position

In this note we examine the question of the existence of a best possible N-net for a bounded set in a normed linear space. A sufficient condition for existence is given which leads to easy proofs of some of the standard results. The pertinent reference here is the paper by Garkavi [1].Let E be a normed linear space and let M be a bounded set in E. Any system of N points in E will be called an N-net. For a given M and the net SN = {y1, y2,…, yN} defineand

scholarly journals The Set of all Generalized Limits of Bounded Sequences

1957 ◽  
Vol 9 ◽  
pp. 79-89 ◽  
Author(s):  
Meyer Jerison
Keyword(s):  
Linear Space ◽  
Linear Functional ◽  
Normed Linear Space ◽  
Bounded Sequence ◽  
Real Numbers ◽  
General Element ◽  
Linear Operation ◽  
Image Position ◽  
Generalized Limits ◽  
Bounded Sequences

Let M be the normed linear space whose general element, x, is a bounded sequenceof real numbers, and ‖x‖ = l.u.b. |ξn|. Let T denote the linear operation (of norm 1) defined by Tx = (ξ2, ξ3, … , ξn+1,…). A generalized limit is a linear functional ϕ on M which satisfies the conditions.

Symmetric powers, metabelian Lie powers and torsion in groups

1995 ◽  
Vol 118 (3) ◽  
pp. 449-466 ◽  
Author(s):  
Ralph Stöhr
Keyword(s):  
Finite Order ◽  
Free Group ◽  
Torsion Subgroup ◽  
Central Extensions ◽  
Torsion Element ◽  
Symmetric Powers ◽  
The One ◽  
Image Position ◽  
Special Case ◽  
Relatively Free Group

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free groupwhere Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

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