The Helly Problem and Best Approximation in a Space of Continuous Functions

Garkavi, A. L.

doi:10.1070/im1967v001n03abeh000573

Cited by 5 publications

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“…For X = IR, this was done by A. L. Gaxkavi [1], and for X = C, by the author [2]. G. M. Ustinov [3] announced, for real C(Q, X), assertions similar to the results of the present paper; however, he used a different terminology, different assumptions concerning the space X, and no vector measures.…”

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confidence: 49%

Existence of best approximation elements inC(Q,X)

Vlasov¹

1995

Math Notes

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ABSTRACT. Generalizing the result of A. L. Garkavi (the case X = R) and his own previous result concerning X = C), the author characterizes the existence subspaces of finite codimension in the space C(Q, X) of continuous functions on a bicompact space Q with values in a Banach space X, under some assumptions concerning X. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form where /Jl E C(Q, X)* are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).We characterize the existence subspaces of finite codimension in the space of continuous functions on a bicompact space Q with values in a Banach space X. For X = IR, this was done by A. L. Gaxkavi [1], and for X = C, by the author [2]. G. M. Ustinov [3] announced, for real C(Q, X), assertions similar to the results of the present paper; however, he used a different terminology, different assumptions concerning the space X, and no vector measures. Our exposition is directly based on the methods of [2], which turn out to be appropriate in the abstract case as well.In the present paper we use the following notation: Q is an infinite bicompact space, X is a Banach space, E is the a-algebra of Borel subsets of Q, C = C(Q,X) is the space of continuous mappings g: Q --* X equipped with the norm Ilgll = sup, QIIg(*)ll, =d B = B(Q, X) is the space of uniform limits of simple functions g(t) = Ez~xe~(t) ( t E Q, zi E X, ei E E, X, is the characteristic function ofa set e, and the sum is finite) equipped with the same norm; thus, C is a subspace of B. We consider C and B as real (complex) spaces if X is real (complex)) The dual space C* is the space of vector measures /~ on E with values in X* equipped with the norm I1 11 = I.IQ = var(~, Q) = sup II~e~ll x" : ei e ~, e~ n ej = e (i # j), n e N ;here the measure I~1 is assumed to be finite and regular and (g,/~) = fQ (g(t), d/~) is the Gavurin integral defined for simple functions by the formula (9, #) = E(xi, I~ei) and extended by continuity to the entire B [4]. Let us note the inequalities Angle brackets axe usually omitted in the sequel. The function /J E L 1 (Q, E, U,I, x*) defined by the relation V e E E ~e=~dl~l= f ~(t)l~l (dt) (the Bochner integral) is called the Radon-Nikodym demvative of the vector measure # with respect to its variation I~,1 and is denoted by d~t/dlg I [5]. We also need art analog of the Radon-Nikodym derivative I The symbol ( ) is used for brevity to express a parMlel assertion.

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confidence: 49%

Existence of best approximation elements inC(Q,X)

Vlasov¹

1995

Math Notes

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“…In the real case, a similar theorem was established by Gaxkavi [2] (for other versions, also see [4]). …”

Section: #} Is a Basis In L • I = Q \ Q' = {Tlt2tq} And Qmentioning

confidence: 69%

“…We consider finite-codimensional Chebyshev subspaces in the complex space C(Q), where Q is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real space C(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex space C(Q) (with the same Q).KEY WORDS: Chebyshev subspace, spaces of continuous functions, best approximation elements.The present paper is closely related to [1][2][3][4] and directly continues the author's paper [5]. A set M in a normed space X is called a proziminal set, or an ezis~ence set, if for any x 6 X the set PMX = {y 6 M : Q'(I) is the set of limit (isolated) points of Q (we assume that Q is infinite); E is the a-algebra of Borel subsets of Q; e and E (possibly, with subscripts) always stand for elements of E; OA is the boundary, .4 the interior, and IAI the caxdinality of a set A; we write v << # on A if a measure v is absolutely continuous with respect to # on a set A (if A = Q, we simply write v << #).…”

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Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

Vlasov

1997

Math Notes

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ABSTRACT. We consider finite-codimensional Chebyshev subspaces in the complex space C(Q), where Q is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real space C(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex space C(Q) (with the same Q).KEY WORDS: Chebyshev subspace, spaces of continuous functions, best approximation elements.The present paper is closely related to [1][2][3][4] and directly continues the author's paper [5]. A set M in a normed space X is called a proziminal set, or an ezis~ence set, if for any x 6 X the set PMX = {y 6 M : Q'(I) is the set of limit (isolated) points of Q (we assume that Q is infinite); E is the a-algebra of Borel subsets of Q; e and E (possibly, with subscripts) always stand for elements of E; OA is the boundary, .4 the interior, and IAI the caxdinality of a set A; we write v << # on A if a measure v is absolutely continuous with respect to # on a set A (if A = Q, we simply write v << #). We need the following theorem. Theorem 1. For a t~nite-codimensional subspace L in the complex or reaJ space C(Q) to be an existence set, it is necessary and sut~cient that the following conditions be satist~ed:(la) for any measure # 6 L • , there exists a continuous Radon-Nikodym derivative dp/d [#[; (lb) for any #, v 9 L • , the set Sv \ S~ is closed; (lc) for any p, v 9 L • , the measure v is absolutely continuous with respect to g on S~.This theorem is due to Garkavi [2] in the real case and the author [5] in the complex case. The following lemma describes some properties of the Radon-Nikodym derivative (see [6, w Lemma 1. 1) /f dp/dv = f, that is, pe = Lf(t)dv for all e 9 ~, then Isle --Llf(t)ldlvl and dv/dg = 1If on {t 9 Sv : f(t) 7:= 0}. /f, moreover, dv/dA = T, then dp/dA = dg /dv, dv/dA = f~o, that is, = L f dv = L f':

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“…The following theorem was established by Garkavi [3] in the real case and by Vlasov [4] for the complex CðQÞ: See also [5,6].…”

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confidence: 94%

On Existence Theorems for Finite-Codimensional Subspaces in C(Q)

Vlasov

2002

Journal of Approximation Theory

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Some variants of the theorem on the existence of best approximation elements in a finite-codimensional subspace of the complex space CðQÞ of continuous functions are given. In approximation theory, existence subspaces of finite codimension have been studied since the 1960s. The following theorem was established by Garkavi [3] in the real case and by Vlasov [4] for the complex CðQÞ: See also [5,6]. Theorem A. In order that a finite-codimensional subspace L & CðQÞ be an existence set, it is necessary and sufficient that the following conditions be satisfied:The purpose of this note is to formulate the existence theorem in its different variants.

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The Helly Problem and Best Approximation in a Space of Continuous Functions

Abstract: A scheme to generate the twc-mode field entangled states is proposed, based on the Raman interaction of an effective twc-level atom with the pump and Stokes fields. The degree of the entanglement between the two field modes in the entangled states is also studied.

Cited by 5 publications

References 2 publications

Existence of best approximation elements inC(Q,X)

Existence of best approximation elements inC(Q,X)

Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

On Existence Theorems for Finite-Codimensional Subspaces in C(Q)

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