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Characterizations similar to (3.0.3) and (3.0.4) can be obtained for these spaces. One important thing when we research in complex analysis is to have a good variety of examples of functions which belong to the spaces we are working with. Because of that, the next section is devoted to explore the structure of Morrey spaces, characterizing for some typical classes of analytic functions C those functions in C which lie in the Morrey spaces, and paying attention to the differences and similarities with Hardy spaces and BM OA.

Among them, maps that preserve the disjointness of cozeroes defined between spaces of scalar- valued continuous functions on locally compact and compact spaces, as a generalization of the concept of homomorphism, have a long history in functional analysis in the context of rings, algebras, or vector lattices under several names such as Lamperti operators, separating maps, disjointness preserving operators, etc. (see, for example, [1, 2, 3, 4, 5, 7, 8, 13, 16]). In recent years, certain attention has been given to such maps when defined on spaces of vector-valued continuous functions (see, e.g., [10, 14]). However, we do not know much about disjointness preserving maps on vector- valued settings in comparison with scalar-valued contexts and something similar can be said with regard to (algebra) homomorphisms between vector-valued group algebras.

Besides, Chan and Shapiro studied in [5] the hypercyclicity of translations in the setting of Hilbert spaces of entire functions of “slow growth”; in [5] the posed question was whether translations in a “reasonable” space of entire functions are always hypercyclic, and it was shown that this is not true. For example, differentiation and translation operators in the Paley-Wiener space P W a are bounded but not hypercyclic; in particular, T w is an isometry of

Atomic decompositions, and even partial results of the same fashion, for func- tions in spaces of analytic functions are very useful in operator theory. In particular, they can be used to describe dual spaces [63] or to study basic questions such as the boundedness, the compactness or the Schatten class membership of concrete opera- tors [5, 7, 21, 29, 48, 54, 55, 69, 72]. In this study we will use Theorem 5.3.1 to describe those positive Borel measures µ on D such that the differentiation operator defined by D (n) ( f ) = f (n) for n ∈ N ∪ { 0 } is bounded from A p,q ω to the Lebesgue space

Our aim here is to obtain some results on unbounded analytic functions on some open subsets of infinite dimensional real Banach spaces. The proofs, which need some particular techniques, will follow some ideas used in the proofs of similar re- sults recently obtained in the complex case, most of them by the authors. See [1-3] and [8].

Proof of Theorem 1.8, necessary condition. We have already considered the case of growth functions satisfying the Dini condition in Remark 2.3. We assume now that ̺ is of upper type 1/n, so that we have the factorization of molecules. We adapt the proof given in [BGS]. Let b ∈ H 2 ( B n ). We assume that h

Let A be a selfadjoint operator on a complex Hilbert space (H ; h ., . i ) . The Gelfand map establishes a ∗ -isometrically isomorphism Φ between the set C (Sp (A)) of all continuous functions defined on the spectrum of A, denoted Sp (A) , an the C ∗ -algebra C ∗ (A) generated by A and the identity operator 1

In the following theorems we describe some basic properties of Paley-Wiener vectors and show that they share similar properties to those of the classical Paley- Wiener functions. The next theorem, whose proof can be found in [15], shows that the space P W ω (D) has properties (A) and (B). See also [14, 16]

ε}. In the case that (X, k · k) is a Banach space, then we will assume that the distance between two elements x, y ∈ X is given by d(x, y) := kx −yk. By L(X ) we denote the space of all continuous linear mappings from X into X. Let B be a fundamental family of bounded subsets of X. For every n ∈ N , B ∈ B let us define the continuous seminorm p n,B (T) := sup x∈B p n (T x) on L(X). Then the system

In this paper, we study the lineability of the set of unbounded holomorphic functions defined on a Banach space. To begin with, let us explain the main notation used in our work. Throughout the article, E will denote a complex Banach space and U will be an open subset of E. If x ∈ E and r > 0, then B (x, r) represents the open ball in E with center x and radius r. For each n ∈ N, P ( n E) denotes the

In this paper we have analyzed several functions proposed in the literature to simultaneously generalize weighted means and OWA operators. On the one hand, OWAWA operators do not generalize weighted means and OWA operators in the usual sense. On the other hand, the HWA operators are neither idempotent nor compensative while WOWA operators do not always provide the expected result. Due to the questionable behavior of these operators, we have imposed a condition to maintain the relationship among the weights and we have characterized the functions that satisfy this condition. However, the obtained functions have some problems in their definition and they are not monotonic. So, we can conclude that none of the analyzed functions is fully convincing.

13. Torra, V.: The weighted OWA operator. Int. J. Intell. Syst. 12(2), 153–166 (1997) 14. Torra, V.: On some relationships between the WOWA operator and the Choquet in- tegral. In: Proceedings of the Seventh International Conference on Information Pro- cessing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98). pp. 818–824. Paris (France) (July 1998)

An important aspect of the DPG method is the inherent guaranteed error control by the computed residuals. For early computational experiments with adaptivity based on these residuals, see [10]. In [4], an a posteriori error analysis is given that includes data approximation errors. When coupling the DPG method with other discretizations, the inherent residuals for the DPG approximation have to be combined with appropriate estimators. For the case of boundary elements, see [13, 14], and for the DPG method dealing with contact conditions, see [15]. In this paper, we do not specifically deal with a posteriori error estimation. Having set our heterogeneous formulation, we proceed to rewrite it by using the so-called trial-to-test operator (which maps the test space to the ansatz space). This is only done for the ultra-weak formulation. The whole system then transforms into one where spaces on the ansatz and test sides are identical. In this way, our heterogeneous variational formulation fits the Lax–Milgram framework just as in [14]. We prove coercivity under the condition that the trial-to-test operator is weighted by a sufficiently large constant. Then quasi-optimal convergence of a discretized version follows by standard arguments. When proving coerciv- ity we follow steps that are similar to the ones when studying the coupling of DPG with boundary elements. But whereas [14] analyzes only the Laplacian, here we set up the scheme and prove coercivity for a gen- eral second-order equation of reaction-advection-diffusion type. Throughout, we assume that our problem is uniformly well posed, i.e., we do not study variations for singularly perturbed cases as in [13]. Also note that, since coefficients are variable, transmission problems can be treated the same way by selecting the sub- domains accordingly. One only has to move the possibly non-homogeneous jump data to the right-hand side functional.

It must be emphasized that we are not requiring our harmonic mappings to be locally univalent. In other words, the Jacobian need not be of constant sign in the domain Ω. The orientation of the mapping may reverse, corresponding to a folding in the associated minimal surface. It is also possible for the minimal surface to exhibit several sheets above a point in the (u, v)–plane. Thus the lifted mapping f e may be locally or globally univalent even when the underlying mapping f is not.

Ahlfors L. V. (1979) Complex Analysis: An Introduction to the Theory of Analytic Functions of one Complex Variable, Third Edition, Mcgraw Hill, New York. Conwey, John B. (1987) Functions of One Complex Variable: Secon Edition Edition, Springer - Verlag, New York. Domínguez P., Contreras A. y Cano L. FCFM, BUAP (2017) Marsden Jerrold E. (1999) Basic Complex Analysis: Third Edition, W.H. Freeman, New York

The concept of bilinear isometry can be naturally extended to the context of spaces of vector-valued continuous functions. Examples of bilinear isometries defined on these spaces can be found, for instance, in [7, Proposition 5.2], where the author provide certain compact spaces X and Banach spaces E for which there exists a bilinear isometry T : C(X, E) × C(X, E) −→ C(Y, E).

This definition was rewritten in [6] considering only annihilation operators prov- ing they are enough to define the Hermite-Sobolev space. In the same work, the authors deal with weighted Sobolev spaces for weights in the Muckenhoupt class A p , defined for 1 < p < ∞ , as the set of weights (non-negative and locally integrable

The  constant  changes  undergone  by  society  lead  us  to  propose  simple,  diverse  spaces  (flexible  and  customi-­‐ zable)  that  can  be  adapted  to  each  person's  needs.  In  conceptual  terms,  tablets  are  a  good  example  of  a  device   that  adapts  to  the  needs  of  a  changing  society.  The  experience  of  uniting  university  and  business  has  led  to  the   evident  break  with  the  traditional  classroom  space  as  the  absolute  protagonist  of  learning  and  requires  expan-­‐ sion  into  other  areas,  transforming  the  new  environments  into  living  mutable  elements,  where  exchanges  and   contributions  of  the  diverse  actors  involved  are  constant.

operators Bojanic and Khan [2] and Taberska [10] estimated the rate of convergence for functions having derivative of B.V. The analogous problem on the convergence rate for the Bernstein polynomials and certain other integral operators were studied in [1],[5], [6] and [8]. Very recently Ispir et al. [9] considered the Kantorovich process of a generalized sequence of linear positive operators and estimated the rate of convergence for absolutely continuous functions having a derivative coinciding a.e., with a function of bounded variation.

It has been pointed out that there is a need for a detailed geometrical characterization of soil pore structure in order to better understand its role in soil functioning, including its contribution to accumulation and protection of soil organic matter, to optimization of soil water and air regimes, and to storage and availability of plant nutrients [von Lutzow et al., 2006]. Mathe- matical morphology [Serra, 1982] provides a set of non-fractal parameters (in the realm of classical non-fractal geometry) that has been used to characterize soil structures that were a priori dierent [San José Martínez et al., 2013]. Fol- lowing the rationale of this theory, a set of fractals parameters beyond V F D and SF D is needed to built an exhaustive fractal description of soil structure. The same rationale suggests the utility of measures of soil pore connectiv- ity. Recent studies of intra-aggregate porosity with 2D images of soil sections seem to suggest that lacunarity could be a suitable fractal parameter choice [Kravchenko et al., 2011; Sukop, 2001; Sukop et al., 2002]. Therefore, our nd- ings could improve knowledge of the fractal geometry of soil pore structure with a comprehensive geometrical description with fractal attributes of soil pore space as a 3D shape. These fractals attributes should probably include, but not restricted to, V F D , SF D and lacunarity.