A double inequality for a generalized-Euler-constant function

The rate of convergence of the sequence , a>0, towards the generalized Euler?s constant , where γ(1) is the Euler-Mascheroni constant, is accurately estimated using the Euler-Maclaurin summation formula. The expression

     

Bounds for the generalized Marcum Q-function

In this paper we consider the generalized Marcum Q-function of order ν > 0 real, defined by

     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

On a generalization of Mittag-Leffler function and its properties

Let s and z be complex variables, Γ(s) the Gamma function, and for any complex ν the generalized Pochhammer symbol. The principal aim of the paper is to investigate the function

     

The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation

The well-posedness of the Cauchy problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local results for data in (s?−1/8) and the global well-posedness for data in are obtained if l=2. Moreover, for l=3, the problem is locally well-posed for data in H^s (s?1/4). The main idea is to use the Fourier restriction norm method.     

Polygon dissections and some generalizations of cluster complexes

Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex in terms of polygon dissections for such a group and any positive integer m. For m=1, is isomorphic to the cluster complex corresponding to W, defined in [S. Fomin, A.V. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977-1018]. We enumerate the faces of and show that the entries of its h-vector are given by the generalized Narayana numbers , defined in [C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005) 179-196]. We also prove that for any m?1 the complex is shellable and hence Cohen-Macaulay.     

Generalized sampling in shift-invariant spaces with multiple stable generators

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with ? stable generators. This is done in the light of the theory of frames in the product Hilbert space (? times). The generalized samples are expressed as the frame coefficients of an appropriate function in with respect to some particular frame in . Since any multiply stable generated shift-invariant space is the image of by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in . An example in the setting of the Hermite cubic splines is exhibited.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces

It is proved that the inequality δX(ε)?cεp, p?2, where δX is the modulus of convexity of X, is sufficient and necessary for the inequality

     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

On k-regular functions with values in a universal Clifford algebra

In this paper, we mainly study properties of nullsolutions of the operator Dk (k∈N^∗=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x₁,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.     

The logarithmic derivative of areally mean p-valent functions

Let denote the class of functions f which are areally mean p-valent and have k maximal growth directions in Δ={z:|z|<1}. In this paper we give an asymptotic formula of the mean square of the logarithmic derivative f′/f for with some restricted condition. This generalizes the corresponding result in [J. Anal. Math. 36 (1979) 36-43]. Also we construct a function which does not satisfy this asymptotic formula.     

On the ideal structure of some Banach algebras related to convolution operators on L(G)

Let G be a locally compact group and let p∈(1,∞). Let be any of the Banach spaces C_δ,p(G), PF_p(G), M_p(G), AP_p(G), WAP_p(G), UC_p(G), PM_p(G), of convolution operators on L^p(G). It is shown that PF_p(G)′ can be isometrically embedded into UC_p(G)′. The structure of maximal regular ideals of (and of MA_p(G)″, B_p(G)″, W_p(G)″) is studied. Among other things it is shown that every maximal regular left (right, two sided) ideal in is either dense or is the annihilator of a unique element in the spectrum of A_p(G). Minimal ideals of is also studied. It is shown that a left ideal M in is minimal if and only if , where Ψ is either a right annihilator of or is a topologically x-invariant element (for some x∈G). Some results on minimal right ideals are also given.