In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a th...In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.展开更多
Let u(z) be a real_valued harmonic function in the unit disk, we say u(z)∈h p(1【p【+∞), if the p _mean M p(r,u)=12π∫ 2π 0|u(re i θ )| p d θ 1/p is bounded. M.Riesz showed ...Let u(z) be a real_valued harmonic function in the unit disk, we say u(z)∈h p(1【p【+∞), if the p _mean M p(r,u)=12π∫ 2π 0|u(re i θ )| p d θ 1/p is bounded. M.Riesz showed that if u(z)∈h p(1【p【+∞), then there exists a constant A p, depending only on p such that M p(r,v)≤A pM p(r,u), where v(z) is the conjugate harmonic function of u(z).When v(0)=0 and 1【p≤2, W.K.Hayman showed that A p can be given by pp-1 1/p . First, this paper shows that the constant pp-1 1/p can be changed by a smaller constant pp-1 2/p -1 1/2 . Next, if 1【p≤2, then there exists a constant θ 0∈2-p2pπ,π2p such that M p p(r,v)≤ Im p sin pπ2+ tg pπ2pM p p(r,u) for any analytic function f(z)=u(z)+ i v(z) in the unit disk, whenever -θ 0【 arg f(z)【π2.展开更多
In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex fu...In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex functions are derived to obtain the equivalent condition of the Stable Farkas Lemma, which is formulated by using the epigraph of the conjugates for the convex functions involved and turns out to be weaker than the classic Slater condition. Moreover, we get some necessary and sufficient conditions for stable duality results of the composite convex functions and present an example to illustrate that the monotonic increasing property of the outer convex function in the objective function is essential. Our main results in this paper develop some recently results.展开更多
The author gives a new proof of Attouch Brezis′ theorem concerned with the duality for the sum of convex functions in general Banach spaces, and gives also some sufficient conditions for the difference of two close...The author gives a new proof of Attouch Brezis′ theorem concerned with the duality for the sum of convex functions in general Banach spaces, and gives also some sufficient conditions for the difference of two closed convex sets to be closed in reflexive Banach spaces.展开更多
The authors give some sufficient conditions for the difference of two closed convex sets to be closed in general Banach spaces, not necessarily reflexive.
文摘In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.
文摘Let u(z) be a real_valued harmonic function in the unit disk, we say u(z)∈h p(1【p【+∞), if the p _mean M p(r,u)=12π∫ 2π 0|u(re i θ )| p d θ 1/p is bounded. M.Riesz showed that if u(z)∈h p(1【p【+∞), then there exists a constant A p, depending only on p such that M p(r,v)≤A pM p(r,u), where v(z) is the conjugate harmonic function of u(z).When v(0)=0 and 1【p≤2, W.K.Hayman showed that A p can be given by pp-1 1/p . First, this paper shows that the constant pp-1 1/p can be changed by a smaller constant pp-1 2/p -1 1/2 . Next, if 1【p≤2, then there exists a constant θ 0∈2-p2pπ,π2p such that M p p(r,v)≤ Im p sin pπ2+ tg pπ2pM p p(r,u) for any analytic function f(z)=u(z)+ i v(z) in the unit disk, whenever -θ 0【 arg f(z)【π2.
基金Supported by the Natural Science Foundation of China(Nos.11401533,11301484,11071180,11371273,11171247)Hong Kong Polytechnic University(G-YX1Q)+1 种基金the Research Grants Council of Hong Kong(No.Poly U 5306/11E)Scientific Research Foundation of Zhejiang Agriculture and Forestry University(No.2013FR080)
文摘In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex functions are derived to obtain the equivalent condition of the Stable Farkas Lemma, which is formulated by using the epigraph of the conjugates for the convex functions involved and turns out to be weaker than the classic Slater condition. Moreover, we get some necessary and sufficient conditions for stable duality results of the composite convex functions and present an example to illustrate that the monotonic increasing property of the outer convex function in the objective function is essential. Our main results in this paper develop some recently results.
文摘The author gives a new proof of Attouch Brezis′ theorem concerned with the duality for the sum of convex functions in general Banach spaces, and gives also some sufficient conditions for the difference of two closed convex sets to be closed in reflexive Banach spaces.
文摘The authors give some sufficient conditions for the difference of two closed convex sets to be closed in general Banach spaces, not necessarily reflexive.
