### Citations

297 |
Real and Abstract Analysis.
- Hewitt, Stromberg
- 1969
(Show Context)
Citation Context ...h respect to a positive measure : T ! R (abbreviated, m ) if for every " > 0 there is a = (") > 0 such that for all A 2 T with (A) we have jm(A)j ": Since m hassnite variation jmj (see =-=[10]-=-, Theorem 19.13 (v)), the condition m yields (1.2) jm(A)j "+ jmj (T ) (A) for all A 2 T and " > 0; that represents the case of (1.1) when X = T ; f = m; q = 1, p = and = jmj (T )=: In tur... |

239 |
Banach lattices and positive operators
- Schaefer
- 1974
(Show Context)
Citation Context ...") > 0 such that (2.1) kT (f)k " kfk+ (") Z K jf j d; whenever f 2 C(K): Proof. If T is weakly compact, then the set K = fjx0 T j : x0 2 E0; kx0k 1g is relatively weakly compact in C(K)0 (see =-=[19]-=-, p. 119); according to the Riesz representation theorem (see [10], p. 177), the functionals on a space C(K) can be viewed as Borel regular measures, so here modulus means variation. By a classical re... |

231 |
A relation between pointwise convergence of functions and convergence of functionals,
- Brezis, Lieb
- 1983
(Show Context)
Citation Context ...continuity is also instrumental in establishing the Radon-Riesz property for Lp-spaces with 1 p <1: See Corollary 3 below, which is a consequence of following result due to H. Brezis and E. H. Lieb =-=[4]-=-, about the "missing term" in Fatous Lemma: Theorem 2. . Let (fn)n be a sequence of functions in a space Lp() with p 2 [1;1); which veri es the following conditions: i) sup kfnk <1; ii) fn ! f almos... |

188 |
Absolutely summing operators
- Diestel, Jarchow, et al.
- 1995
(Show Context)
Citation Context ...intwise to a function f 2 C(K): Then kT (fn) T (f)k ! 0: For further developments related to Theorem 1 see our papers [14], [15], [16], [17], and the monograph of J. Diestel, H. Jarchow and A. Tonge =-=[7]-=-, Ch. 15. The property of absolute continuity is also instrumental in establishing the Radon-Riesz property for Lp-spaces with 1 p <1: See Corollary 3 below, which is a consequence of following resu... |

67 |
Partial Di¤erential Equations
- Evans
- 1998
(Show Context)
Citation Context ...uers approach [3] in terms of Choquet boundary) can be found in the monograph [1]. 4. Absolute Continuity and PDE There are many instances when the concept of absolute continuity appears in PDE (see =-=[8]-=-) but we shall restrict here to the remarkable theorem of F. Rellich concerning the compact embedding of Sobolev spaces. Theorem 9. Ifsis a bounded open subset of RN then the canonical injection i : ... |

48 |
Sur les applications linéaires faiblement compactes d’espaces du type C(K
- Grothendieck
- 1953
(Show Context)
Citation Context ...Riesz representation theorem (see [10], p. 177), the functionals on a space C(K) can be viewed as Borel regular measures, so here modulus means variation. By a classical result due to A. Grothendieck =-=[9]-=-, the relative weak compactness of K means that 3for every bounded sequence of Borel measurable functions fn : K ! R which is pointwise convergent to 0 we have (2.2) lim n!1 Z K fnd = 0; uniformly fo... |

41 |
Korovkin-Type Approximation Theory and its Applications, Walter de Gruyter,
- Altomare, Campiti
- 1994
(Show Context)
Citation Context ...ce kfk 1 f kfk 1 implies kfk T (1) T (f) kfk T (1) and thus kT (f)k kT (1)k kfk : Theorem 5. (P.P. Korovkin [12]). Consider the functions e0(x) = 1; e1(x) = x; e2(x) = x 2 in C(=-=[0; 1]-=-); and suppose there is given a sequence Tn : C([0; 1])! C([0; 1]) (n 2 N) of positive linear operators such that Tn(f)! f uniformly on [0; 1] for f 2 fe0; e1; e2g: Then Tn(f)! f uniformly on [0; 1] f... |

15 |
On convergence of linear positive operators in the space of continuous functions,”
- Korovkin
- 1953
(Show Context)
Citation Context ...0 there is a number = (") > 0 such that jf(s) f(t)j "+ (")d(s; t) for all s; t 2sand f 2 A: 3. Absolute Continuity and Approximation Theory We start with the beautiful result of P. P. Korovkin =-=[12]-=-, which put in a new perspective the whole subject of approximation in the case of continuous functions. In order to state this result we need a preparation. Suppose that E is a Banach lattice. A line... |

12 |
Real Analysis with Real Applications
- Davidson, Donsig
- 2002
(Show Context)
Citation Context ...(t) = nX k=0 n k tk(1 t)nkf (k=n) : In fact, Tn( (; t))(t) = t(1 t) n for all t 2 [0; 1]: This computation is part of Bernsteins classical proof of the Weierstrass Approximation Theorem. See =-=[6]-=-, pp. 290-292. Corollary 5. (Féjer Approximation Theorem). The Cesàro averages of the Fourier partial sums of a continuous function f of period 2 converge uniformly to f: 9Proof. We have to conside... |

8 | General topology - Arkhangel’skĭı - 1996 |

7 |
Introduction à l’Analyse Complexe
- Chabat
- 1990
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Citation Context ...d on the compact subsets of CN : Because they are holomorphic, a compactness principle due to P. Montel assures us that a subsequence should be uniformly convergent on each compact subset of CN . See =-=[5]-=-, p. 209. Taking into account the estimate (4.1), that subsequence should also verify lim sup j;k!1 jjfj fkjj2Hm1 = 0. The proof of Theorem 9 is done. Acknowledgement. This research was partially s... |

6 |
Absolute continuity in Banach space theory
- Niculescu
- 1979
(Show Context)
Citation Context ...all continuous real-valued functions de ned on a compact Hausdor¤ space K. The basic fact, which led to the concept of absolutely continuous operator, is as follows: Theorem 1. (C. P. Niculescu [15], =-=[16]-=-). Suppose that E is a Banach space. A bounded linear operator T 2 L(C(K); E) is weakly compact if and only if there exists a positive Borel measure on K such that for every " > 0 one cansnd a (") ... |

5 | The Hardy-Landau-Littlewood Inequalities with less
- Niculescu, Buse
(Show Context)
Citation Context ...ry of inequalities o¤ers many interesting applications where the concept of absolute continuity is instrumental. In particular this is the case of the famous Hardy-Landau-Littlewood inequalities. See =-=[18]-=-. The aim of this paper is to illustrate the usefulness of the notion of absolute continuity in other areas of mathematics such as Functional Analysis, Approximation Theory and PDE. In particular we s... |

4 |
Approximation and abstract boundaries
- Bauer
- 1978
(Show Context)
Citation Context ...eory above to encompass some spaces of di¤erentiable functions (for example, the Sobolev spaces). A nice account of the most signi cant developments in the Korovkin theory (including Bauers approach =-=[3]-=- in terms of Choquet boundary) can be found in the monograph [1]. 4. Absolute Continuity and PDE There are many instances when the concept of absolute continuity appears in PDE (see [8]) but we shall ... |

4 |
On uniform continuity and compactness in metric spaces
- Hueber
- 1981
(Show Context)
Citation Context ...mly continuous function f : R! R veri es an estimate of the form jf(x)j a jxj+ b: A characterization of the metric spaces on which every continuous function is also uniformly continuous appeared in =-=[11]-=-. In the special case when M is also compact, the role of the distance function in (2.6) can be taken by any separating function for M . Recall that a separating function is a nonnegative continuous f... |

4 |
Variation on a theorem of
- Lomeĺı, Garćıa
- 2006
(Show Context)
Citation Context ...kfk jTn(1)(t) 1j "Tn(1)(t) + (")Tn( (; t))(t) + kfk jTn(1)(t) 1j and the conclusion follows from our hypothesis. Theorem 6 is a variant of a recent result by H. E. Lomeli and C. L. Garcia =-=[13]-=- (based on a slightly di¤erent concept of separating function). In order to understand how Theorem 6 extends the Theorem of Korovkin, let us consider the case were M is a compact subset of RN ands(s; ... |

3 |
Opérateurs absolument continus
- Niculescu
- 1974
(Show Context)
Citation Context ... operator and (fn)n is a bounded sequence of functions in C(K) which converges pointwise to a function f 2 C(K): Then kT (fn) T (f)k ! 0: For further developments related to Theorem 1 see our papers =-=[14]-=-, [15], [16], [17], and the monograph of J. Diestel, H. Jarchow and A. Tonge [7], Ch. 15. The property of absolute continuity is also instrumental in establishing the Radon-Riesz property for Lp-space... |

2 |
Absolute Continuity and Weak
- Niculescu
- 1975
(Show Context)
Citation Context ...m) of all continuous real-valued functions de ned on a compact Hausdor¤ space K. The basic fact, which led to the concept of absolutely continuous operator, is as follows: Theorem 1. (C. P. Niculescu =-=[15]-=-, [16]). Suppose that E is a Banach space. A bounded linear operator T 2 L(C(K); E) is weakly compact if and only if there exists a positive Borel measure on K such that for every " > 0 one cansnd a... |

1 |
Niculescu, Operators of type A and local absolute continuity
- P
- 1985
(Show Context)
Citation Context ...n is a bounded sequence of functions in C(K) which converges pointwise to a function f 2 C(K): Then kT (fn) T (f)k ! 0: For further developments related to Theorem 1 see our papers [14], [15], [16], =-=[17]-=-, and the monograph of J. Diestel, H. Jarchow and A. Tonge [7], Ch. 15. The property of absolute continuity is also instrumental in establishing the Radon-Riesz property for Lp-spaces with 1 p <1: S... |