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## Approximation of distributions by convolutions in the Hausdorff metric (2000)

### Citations

348 |
Interpolation spaces
- Bergh, Löfström
- 1976
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Citation Context ... H \Lambda \Gamma ! f for any distribution f 2 X 2 H, then limn!1k_n j M+ i ffiX j\Lambdask ! 1: Proof of Corollary 2.1. Indeed, if X, Y are Banach spaces, according to the duality theorem (sf.,[11], =-=[12]-=-), if X " Y is dense in the both spaces X and Y , then (X T Y )\Lambdas= X\Lambdas+ Y \Lambda . From here and Lemmas 2.2 and 2.6, we obtain M (X " L1; L1) = (X " L1)ffi\Lambdas= Xffi\Lambdas+ M: Hence... |

3 | On approximation of functions by convolution operators in the Hausdorff metric, Dokl. Akad - Petukhov - 1993 |

3 |
Introduction to Hp Spaces (Cambridge Univ
- Koosis
- 1980
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Citation Context ...e represented as the sum of positive and negative distributions, belonging to the same space. At the same time, maximum of spaces embedded in !H1, possessing this property, is the space L log L (sf., =-=[16]-=-, Chapter 5). Thus, there is an understandable reason for the appearance of the space exp L = M(L log L; L1) in the statement of Theorem 4.1. Similarly, such the space for c0 is the space of continuou... |

2 |
On approximation of functions by singular integrals in the Hausdorff metric
- Petukhov
- 1988
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Citation Context ...nnection it is necessary to mention paper [2], where the problem of the approximation of, so-called, H-continuous functions by positive operators in the Hausdorff metrics was considered. In our paper =-=[3]-=-, we used other approach. We suggested to consider approximations by linear (integral) operators of the form Jn(f ) = Z b a f (t)K n(x; t)dt on the classical normed space of the functions essentially ... |

2 |
On the convergence of sequences of convolution operators in the Hausdorff metric. Algebra i Analiz
- Petukhov
- 1993
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Citation Context ...anonical graphs of the bounded function, invariant with respect to the change of a function on a set of measure zero, was introduced in [3]. The same definition without any change was used in [4] and =-=[5]-=- for integrable functions. Canonical graphs of unbounded functions are not compact sets. Therefore, to measure distance between unbounded functions it is necessary to compactify the axis of values by ... |

2 | On approximation of periodic distributions in the Hausdorff metric, Rossiisk. Akad. Nauk Dokl - Petukhov - 1994 |

2 |
Convergence of Fourier series for functions in the classes of Besov-- Lizorkin--Triebel
- Petukhov
- 1994
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Citation Context ... of convolution sequences In(f ) = f \Lambdas_n (0.1) on the classes of 2ss-periodic functions with summable pth power (1 ^ p ^ 1) and on the class B V (functions of bounded variation) were found. In =-=[7]-=-, the convergence of Fourier series of functions of the Besov--Lizorkin--Triebel classes was studied. The goal of this work is studding the convergence of operator sequences (0.1) on quasiBanach space... |

2 |
Theory of Function Spaces", Akademische Verlagsgesellschaft Geest & Portig
- Triebel
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Citation Context ...tributions; (b) For any ffi ? 0 limn!1k((1 \GammasO/ffi)_n) j X\Lambda k ! 1: The following statement follows immediately from Theorem 3.4 and from the fact that (! Hp)\Lambdas= B1=p\Gamma 11;1 (sf., =-=[18]-=-, Chapter 2). Corollary 3.1. Sequence (0.1) converges on the space !Hp (0 ! p ! 1) if and only if the following two conditions hold simultaneously: (a) Sequence (0.1) converges on the two distribution... |

1 |
2] P.P. Korovkin, Experiment of axiomatic construction of certain problems of the approximation theory of functions one variable, Uchenye Zapiski Kalininskogo gos
- Sendov, Kluwer
- 1990
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Citation Context ...r (complemented) graphs was generated in the sixties mainly due to works of Bulgarian mathematicians. The state of art of this subject in the end of the seventies is reflected in the book by B.Sendov =-=[1]-=-. The Hausdorff metrics has a number of attractive properties. In our opinion, the main of these properties is its naturalness for human eyes. Roughly speaking, functions are considered as close if th... |

1 |
Fourier series: a modern introduction", 2nd ed
- Edwards
- 1982
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Citation Context ...ributions (generalized functions) D on T is defined to be the set of all bounded linear functionals on C 1 . Principal facts, concerning periodic distributions, can be found in the book by R.Edwards (=-=[8]-=-, Chapter 12). Here we recall some of them. The space D can be identify with the set of formal trigonometric series Pn2Zcneinx, where Z is a set of integer numbers with coefficients of temperate growt... |

1 |
A.J.Lohwatter, "The Theory of Cluster Sets
- Collingwood
- 1966
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Citation Context ... trace of the closure on T. Now we implement this reasoning accurately. Let f 2 D , F (z) = f \LambdasPr(`). Let us recall that the cluster set of the function F at the point ` 2 T (see, for example, =-=[9]-=-) is defined to be the set C(F; `), consisting of those points ff 2 _R1 for which the sequence of complex numbers zn, satisfying conditions jznj ! 1, zn ! z0 (jz0j = 1, arg z0 = `), and limzn!z0 F (zn... |

1 |
Functional analysis," Pergamon Press
- Kantorovich, Akilov
- 1982
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Citation Context ...equires the proof. Let Y = Xffi\Lambda , Z = Y ffi\Lambda , h 2 D . Bearing in mind that for any Banach space B and b 2 B we have kb j Bk = supfhc; bi j kc j B\Lambda k ^ 1g (2.6) (see, for instance, =-=[10]-=-, Chapter 5) and by Lemma 2.4, we obtain a chain of the equalities kh j Zk = sup r!1 kh r j Zk = sup r!1 supfhh r; gi j g 2 ffiY ; kg j Y k ^ 1g = sup r!1 supfhh; g ri j g 2 ffiY ; kg j Y k ^ 1g = sup... |

1 |
Analogies entre les s 'eries trigonom'etriques et les s'eries sph'eriques. Ann. de l'Ecole Nor. Sup
- Kogbetliantz
- 1923
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Citation Context ...2) and Dn(t) is the Dirichlet kernel, Dn(t) :=s12 + nX *=1 cos *t! = sin(n + 1=2)t2 sin t=2 : By applying the Abel transform to (6.2) several times, we obtain the well-known Kogbetliantz formula (see =-=[13]-=- and [14], ChapterIII) Kffn (t) = =se it=2 2Affn sin t=2seint (1 \Gammase\Gamma it)m + mX k=1 Aff\Gamma kn+k 1(1 \Gammase\Gamma it)k + 1X *=1 Aff\Gamma m+1n+m+* e \Gamma *t (1 \Gammaseit)m !! ; (6.3) ... |

1 |
Trigonometric series," 2nd ed., Cambridge Univ
- Zygmund
- 1959
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Citation Context ...function jf j \Deltasmaxf0; log jf jg is integrable. We denote by exp L the space of continuous linear functionals on L log L. More detail information about these spaces can be found, for example, in =-=[14]-=- and [15]. Let # is a 2ss-periodic function, #(x) = ss \Gammasx, x 2 [0; 2ss). Obviously, # = S[P1 \Gammas1]. The sequence (0.1) is called converging on the two functions if it converges in the Hausdo... |