By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive...By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.展开更多
A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this pa...A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σnf → f （n → ∞） for functions f ∈ L^p, where p 〉 1 （Journal of Approximation Theory, 101（1）, 1-36, （1999）） and also the a.e. convergence σMnf → f （n → ∞） for functions f ∈ L^1 （Journal of Approximation Theory, 124（1）, 25-43, （2003））. The aim of this paper is to prove the a.e. relation limn→∞ σnf = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its ＂big elements＂ are not ＂too dense＂.展开更多
New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integrodifferential equations that are defined on arbitrary smooth closed contours...New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integrodifferential equations that are defined on arbitrary smooth closed contours of the complex plane.We carry out the convergence analysis in classical Hölder spaces.A numerical example is also presented.展开更多
The paperintroduces Herm ite-Fejértype(Herm ite type) interpolation ofhigherorder denoted by Smn (f)(Sm n(f)), and gives som e basic properties including expression form ulas, convergence relationship betw een ...The paperintroduces Herm ite-Fejértype(Herm ite type) interpolation ofhigherorder denoted by Smn (f)(Sm n(f)), and gives som e basic properties including expression form ulas, convergence relationship betw een Sm n(f) and Hmn(f) (Herm ite-Fejérinterpolation ofhigheror- der), and the saturation ofSmn(f).展开更多
Let f be a holomorphic function on the unit polydisc Dn,with Taylor expansion f(z) = ∞ |k|=0 akzk ≡ ∞ (k1+···+kn=0) (ak1,···,kn zk1 1znkn)where k = (k1, , kn) ∈ Z+n. The authors defin...Let f be a holomorphic function on the unit polydisc Dn,with Taylor expansion f(z) = ∞ |k|=0 akzk ≡ ∞ (k1+···+kn=0) (ak1,···,kn zk1 1znkn)where k = (k1, , kn) ∈ Z+n. The authors define generalized Hilbert operator on Dn by Hγ,n(f)(z) = ∞ |k|=0 i1,···,in≥0 ai1,···,in n j=1 Γ(γj + kj + 1)Γ(kj + ij + 1) Γ(kj + 1)Γ(kj + ij + γj + 2) zk,where γ ∈ Cn, such that R γj > -1, j = 1, 2, , n. An upper bound for the norm of the operator on Hardy spaces Hp(Dn) is found. The authors also present a Fejér-Riesz type ineq...展开更多
文摘By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.
基金the Hungarian National Foundation for Scientific Research(OTKA),Grant No.M36511/2001 and T 048780
文摘A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σnf → f （n → ∞） for functions f ∈ L^p, where p 〉 1 （Journal of Approximation Theory, 101（1）, 1-36, （1999）） and also the a.e. convergence σMnf → f （n → ∞） for functions f ∈ L^1 （Journal of Approximation Theory, 124（1）, 25-43, （2003））. The aim of this paper is to prove the a.e. relation limn→∞ σnf = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its ＂big elements＂ are not ＂too dense＂.
基金This research of Iurie Caraus was supported by a Fulbright Grant.The First author would like to thank the Department of Mathematics,North Carolina State University,and Dr.Zhilin Li for the support and the hospitality during his visitThe second author is partially supported by the US ARO grants 550694-MA,the AFSOR grant FA9550-09-1-0520,the US NSF grant DMS-0911434,the US NIH grant 096195-01,and the CNSF grant 11071123.
文摘New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integrodifferential equations that are defined on arbitrary smooth closed contours of the complex plane.We carry out the convergence analysis in classical Hölder spaces.A numerical example is also presented.
文摘The paperintroduces Herm ite-Fejértype(Herm ite type) interpolation ofhigherorder denoted by Smn (f)(Sm n(f)), and gives som e basic properties including expression form ulas, convergence relationship betw een Sm n(f) and Hmn(f) (Herm ite-Fejérinterpolation ofhigheror- der), and the saturation ofSmn(f).
基金Supported by the NNSF of China(10671115)grants from Specialized Research Fund for the doctoral program of Higher Education(20060560002)NSF of Guangdong Province(7300614)
文摘Let f be a holomorphic function on the unit polydisc Dn,with Taylor expansion f(z) = ∞ |k|=0 akzk ≡ ∞ (k1+···+kn=0) (ak1,···,kn zk1 1znkn)where k = (k1, , kn) ∈ Z+n. The authors define generalized Hilbert operator on Dn by Hγ,n(f)(z) = ∞ |k|=0 i1,···,in≥0 ai1,···,in n j=1 Γ(γj + kj + 1)Γ(kj + ij + 1) Γ(kj + 1)Γ(kj + ij + γj + 2) zk,where γ ∈ Cn, such that R γj > -1, j = 1, 2, , n. An upper bound for the norm of the operator on Hardy spaces Hp(Dn) is found. The authors also present a Fejér-Riesz type ineq...
