On approximation of affine Baire-one functions

Lukeš, Jaroslav; Malý, Jan; Netuka, Ivan; Smrčka, Martin; Spurný, Jiří

doi:10.1007/bf02787408

Cited by 15 publications

(10 citation statements)

References 21 publications

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“…(Recall that f is a Baire-one function, or a function of the first Baire class, if it is a pointwise limit of a sequence of continuous functions.) As was shown in [8,Corollary 6.4], f U is even a pointwise limit of a bounded sequence of functions from H(U ). We denote the space of all pointwise limits of bounded sequences from H(U ) as H 1 …”

Section: Introductionmentioning

confidence: 81%

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On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Spurný

2006

Potential Anal

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Let H 1 (U ) denote the space of all pointwise limits of bounded sequences from H(U ), where H(U ) consists of all continuous functions on the closure U of a bounded open set U ⊂ R m that are harmonic on U . It is shown that the space H 1 (U ) is a lattice in the natural ordering if and only if the set ∂ reg U of all regular points of U is an F σ -set. Mathematics Subject Classifications (2000): 31B05, 54C30.

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Section: Introductionmentioning

confidence: 81%

“…This concludes the proof of claim. P According to claim, Corollary 6.2 and Corollary 6.4 of[8], the function f U belongs to H 1 (U ). Now we verify that for |f U | there does not exist the least upper bound in H 1 (U ).…”

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confidence: 91%

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Spurný

2006

Potential Anal

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“…Let f be the characteristic function of {x}. By (f), H U f = f on U and thus f is an upper semicontinuous H(U)-affine function on U (see [15,Corollary 6.2] or Lemma 3.10(b) below). By [24,Theorem 4.5(iii)] there exists a decreasing sequence {h n } of functions from H(U) such that h n → f .…”

Section: Theorem 22 Let U Be a Bounded Open Set Inmentioning

confidence: 97%

Complementability of Spaces of Harmonic Functions

Spurný

2008

Potential Anal

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Let U be a bounded open set of the Euclidean space R d and let H(U) denote the space of all real-valued continuous functions on U that are harmonic on U. We present a sufficient condition on the set ∂ reg U of all regular points of U that ensures that H(U) is complemented in C U . We also present examples showing that this condition is not necessary. The proof of the positive result is based upon a general result on complementability of a simplicial function space in a C(K) space.

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“…We refer the reader to [17,Corollary 6.2] for the proof that T f ∈ A bf (X) for any bounded Borel function on X. If f is a bounded Borel function defined on a Borel subset K of X, we set T f = T f , where f = f on K and 0 elsewhere.…”

Section: Remark 34mentioning

confidence: 99%