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Basic Sets of Special Monogenic Polynomials in Fréchet Modules

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Basic Sets of Special Monogenic Polynomials in Fréchet Modules Hindawi Journal of Complex Analysis Volume 2017, Article ID 2075938, 11 pages https://doi.org/10.1155/2017/2075938 Research Article Basic Sets of Special Monogenic Polynomials in Fréchet Modules Gamal Farghaly Hassan,1,2 Lassaad Aloui,3 and Allal Bakali1 1 Department of Mathematics, Faculty of Sciences, Northern Border University, P.O. Box 1321, Arar, Saudi Arabia 2Faculty of Science, University of Assiut, Assiut 71516, Egypt 3Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis-El Manar, Tunis, Tunisia Correspondence should be addressed to Gamal Farghaly Hassan; [email protected] Received 20 June 2016; Revised 31 October 2016; Accepted 18 December 2016; Published 14 February 2017 Academic Editor: Konstantin M. Dyakonov Copyright © 2017 Gamal Farghaly Hassan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is concerned with the study of the theory of basic sets inFrechet´ modules in Clifford analysis. The main aim of this account, which is based on functional analysis consideration, is to formulate criteria of general type for the effectiveness (convergence properties) of basic sets either in the space itself or in a subspace of finer topology. By attributing particular forms for the Frechet´ module of different classes of functions, conditions are derived from the general criteria for the convergence properties in open and closed balls. Our results improve and generalize some known results in complex and Clifford setting concerning the effectiveness of basic sets. 1. Introduction aboutthestudyofbasicsetsofpolynomialsincomplexanal- ysis, we refer to [17–20]. Thetheoryofbasesinfunctionspacesplaysanimportantrole In theory of basic sets of polynomials in hypercomplex in mathematics and its applications, for example, in approx- analysis, Abul-Ez and Constales gave in [21, 22] the extension imation theory, partial differential equations, geometry, and of the theory of bases of polynomials in one complex variable mathematical physics. to the setting of Clifford analysis. This is the natural general- The subject of basic sets of polynomials in one complex ization of complex analysis to Euclidean space of dimension variable, in its classical form, was introduced by Whittaker larger than two, where the holomorphic functions have [1, 2] who laid down the definition of basic sets, basic series, values in Clifford algebra and are null solutions of a linear and effectiveness of basic sets. Many well-known polynomials differential operator. An important subclass of the Clifford such as Laguerre, Legendre, Hermite, Bernoulli, Euler, and holomorphic functions called special monogenic functions is Bessel polynomials form simple bases of polynomials (see considered, for which a Cannon theorem on the effectiveness [3–7]). A significant advance was contributed to the subject inclosedandopenball[21,23]wasestablished.Manyauthors by Cannon [8, 9] who obtained necessary and sufficient studied the basic sets of polynomials in Clifford analysis [24– conditions for the effectiveness of basic sets for classes of 30]. functions with finite radius of regularity and entire functions. In [31], Adepoju laid down a treatment of the subject The theory of basic sets of polynomials can be generalized of basic sets of polynomials of a single complex variable in to higher dimensions in several different ways, for instance, to Banach space which is based on functional analysis consid- several complex variables or to hypercomplex analysis. erations. Also, the authors in [12, 32] studied the basic sets of The theory of basic sets of polynomials in several complex polynomials of several complex variables in Banach space. variables was developed at the end of the 1950s by Mursi We shall lay down in this paper a treatment of the subject and Maker [10] and later by Nassif [11] and was studied in of basic sets based primarily on functional analysis and Clif- more detail afterwards by others (c.f [12–15]). Also, the rep- ford analysis. The aim of this treatment is to construct a crite- resentation of matrix functions by bases of polynomials has rion, of general type, for effectiveness of basic sets in Frechet´ been studied by Makar and Fawzy [16]. For more information modules. By attributing particular forms to these Frechet´ 2 Journal of Complex Analysis modules, we derive, in the remaining articles of the present Remark 2. Notice that becomes a real vector space if R is paper, from the general criterion of effectiveness already identified with R0 = A0 ⊂ A. obtained, particular conditions for effectiveness in the differ- In the following, all A-modules will be right A- ent forms of the regions which are relevant to our subsequent modules. work. Thus, effectiveness in open and closed balls is studied. In addition, we give some applications of the effectiveness of Definition 3 (A-linear operator). Let and be two basic sets of polynomials in approximation theory concerned unitary A-modules. Then a function :→is said to with be an A-linear operator if, for all , ∈ and , ∈ A, (1) the expansion of Clifford valued functions in closed ( + ) =() +() . (1) and open ball by infinite series in a given sequence of basic sets, The set of all A-linear operators from into is denoted (2) the expansion of Clifford valued functions in closed by (,. ) and open ball by infinite series in a given sequence of Cannon sets of special monogenic polynomials. Definition 4 (proper system of seminorms). Let be a unitary A-module. Then a family P of functions :→ These new results extend and generalize the known results in [0, ∞) complex and Clifford setting given in [12, 21, 23, 31, 32]. is said to be a proper system of seminorms on if the following conditions are fulfilled: 2. Notation and Preliminaries [1] There exists a constant 0 ≥1such that, for all ∈P, ∈A and , ∈: In order to introduce our results, we give several notations and assumptions. (i) ( + ) ≤ (). +() Let us denote by {1,...,} the canonical basis of the () ≤ ||() () = ||() ∈ Euclidean vector space R and by A the associated real (ii) 0 ,and if R. Clifford algebra in which one has the multiplication rules + =−2 , , = 1,...,,where denotes the Kronecker symbol. [2] For any finite number 1,2,..., ∈ P, there exist A vector space basis for the Clifford algebra A is given ∈P and >0such that, for all ∈, by the set { : ⊆ {1,...,}} with = ,..., = ⋅⋅⋅ 1≤ < <⋅⋅⋅< ≤ =1 =1ℎ () ≤ () . 1 ℎ , 1 2 ℎ ,and 0 . sup =1,2,..., (2) Every ∈A can be written in the form =∑ with ∈ R. The conjugate element of ∈A is defined by =∑ = ⋅⋅⋅ =− (= 1,...,) [ ] () =0 ∈P =0. ,where ℎ 1 , , 3 If for all then and 0 =0 =1. +1 We denote also by R = spanR{0,1,...,}=R ⊕ Definition 5 (Frechet´ module). A Frechet´ module over R ⊂ A the space of paravectors =00 +∑=1 . A isaHausdorffspacewithacountablepropersystemof In this notation, the paravector will be represented in the seminorms P =()≥0 satisfying form =0 +with Sc() =0 being the scalar part and <⇒() ≤ () ( ∈ ) Vec() = being the vector part of .TheinducedClifford (i) ; , 2 1/2 norm of arbitrary ∈A is given by || =∑ ( || ) . (ii) a subset of is open if, ∀ ∈ , there exist >0 Some care must be taken when using this norm to and ≥0such that estimate product. We will always use the formula || ≤ /2 2 ||||. { ∈ : (−)≤}⊂, ∀≤, (3) One useful approach to generalize complex analysis to higher dimensional spaces is the Cauchy-Riemann approach (iii) is complete with respect to this topology. which is based on the consideration of functions that are in the kernel of the generalized Cauchy-Riemann operator T P +1 We denote by the topology defined by the family of =∑=0 (/) in R (for more details, see [33, 34]). seminorms on . A A Definition 1 (unitary right -module). A unitary right - Definition 6 (convergent sequences in the topology T). The (, +) × module is an abelian group with a mapping sequence ()≥0 of elements of converges in the topology A →(,) → , ∈ A ; such that for all and T to the element of , if and only if, for all ∈ P,we , ∈: have ( + ) = + (i) . ( −)=0. lim (4) (ii) () = (). →∞ ( + ) = + (iii) . We may also equivalently say that the sequence ()≥0 (iv) 0 =. converges in to with respect to T. Journal of Complex Analysis 3 It is a familiar property for the Frechet´ module that a so that is a linear operator on .Also,putting= in seminorm on is T-continuous, if and only if there is a (7), it can be verified that ∈ P seminorm and a positive finite constant such that () =, (13) () ≤ () ; (∀ ∈) . (5) and is orthonormal to ()≥0. Itisalsoknownthatalinearoperator on is continuous We deduce the continuity of from (6) and (8). if and only if there is a seminorm ∈ P and a constant Theorem 11. ( ) such that Let ≥0 be an absolute basis for and let be given by (11). Then the family ()≥0 forms a proper system () ≤ () ; (∀ ∈ ). (6) of continuous seminorms. Moreover, for <, there exists a constant , such that 3. Basis and Absolute Basis () ≤, () (14) Definition 7 (basis for Frechet´ module ). Let be a Frechet´ for all ∈. module over A.Asequence()≥0 of nonzero elements of is called a basis for if, for each element ∈,thereis ( ) ( ()) Proof. Firstly, we prove that the family ≥0 is a proper one and only one sequence ≥0 of the Clifford algebra system of seminorms as follows. A,suchthat [1] We observe, from the linearity of and properties ∞ (i) and (ii) of seminorms, that =∑ () . (7) ∞ =0 ( + ) = ∑ ( + ) () =0 Definition 8 (Cauchy’s inequality). We shall assume that ( ) ∞ Cauchy’s inequality holds for the basis ≥0 in the form that = ∑ () + () ( ) for each ∈ P there is a positive finite constant such that =0 () () ≤ () (8) ∞ (15) ≤ ∑ { () + ()} () for all integers and for all ∈. =0 Also, the nature of the problems considered here neces- = () + () , sitates that whenever <, there is a finite positive constant () ≤ || () , ≥1, ,,suchthat 0 0 ∞ whenever , ∈ and ∈A.
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