We introduce a modification of Kantorovich-type operators in polynomial weighted spaces of functions. Then we study some approximation properties of these operators. We give some inequalities for these operators by me...We introduce a modification of Kantorovich-type operators in polynomial weighted spaces of functions. Then we study some approximation properties of these operators. We give some inequalities for these operators by means of the weighted modulus continuity and also obtain a Voronovskaya-type theorem. Furthermore, in our paper show that the operators give better degree of approximation of functions belonging to weighted spaces than classical Szaisz- Kantorovich operators.展开更多
It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main ...It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main purpose of this paper is to give the sharp constants B（p,q,,,,n）for the above inequality for three cases：（i）p=1 and q=1;（ii）p=1 and 1q 6∞,or 1p 6∞and q=1;（iii）1p,q∞and 1p＋1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1/p＋1/q〉1.展开更多
Let k be a positive integer. For any positive integer x =∑i=0^∞xi2^i, where xi = 0, 1,we define the weight w（x） of x by w（x） ：= ∑i=0^∞xi. For any integer t with 0 〈 t 〈 2^k- 1, let St ：= {（a,b）∈ Z^2｜...Let k be a positive integer. For any positive integer x =∑i=0^∞xi2^i, where xi = 0, 1,we define the weight w（x） of x by w（x） ：= ∑i=0^∞xi. For any integer t with 0 〈 t 〈 2^k- 1, let St ：= {（a,b）∈ Z^2｜a＋b≡t（mod 2^k-1）,w（a）＋w（b）〈k,0≤a,b≤2^k-2}.This paper gives explicit formulas for cardinality of St in the cases of w（t） ≤ 3 and an upper bound for cardinality of St when w（t） = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w（t） ≤ 4.展开更多
文摘We introduce a modification of Kantorovich-type operators in polynomial weighted spaces of functions. Then we study some approximation properties of these operators. We give some inequalities for these operators by means of the weighted modulus continuity and also obtain a Voronovskaya-type theorem. Furthermore, in our paper show that the operators give better degree of approximation of functions belonging to weighted spaces than classical Szaisz- Kantorovich operators.
基金supported by National Natural Science Foundation of China(Grant Nos.11071250 and 11271162)
文摘It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main purpose of this paper is to give the sharp constants B（p,q,,,,n）for the above inequality for three cases：（i）p=1 and q=1;（ii）p=1 and 1q 6∞,or 1p 6∞and q=1;（iii）1p,q∞and 1p＋1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1/p＋1/q〉1.
基金supported partially by the National Science Foundation of China under Grant No.11371260the Youth Foundation of Sichuan University Jinjiang College under Grant No.QJ141308
文摘Let k be a positive integer. For any positive integer x =∑i=0^∞xi2^i, where xi = 0, 1,we define the weight w（x） of x by w（x） ：= ∑i=0^∞xi. For any integer t with 0 〈 t 〈 2^k- 1, let St ：= {（a,b）∈ Z^2｜a＋b≡t（mod 2^k-1）,w（a）＋w（b）〈k,0≤a,b≤2^k-2}.This paper gives explicit formulas for cardinality of St in the cases of w（t） ≤ 3 and an upper bound for cardinality of St when w（t） = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w（t） ≤ 4.
