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Basic Sets of Special Monogenic Polynomials in Frechet Modules.

1. Introduction

The theory of bases in function spaces plays an important role in mathematics and its applications, for example, in approximation theory, partial differential equations, geometry, and mathematical physics.

The subject of basic sets of polynomials in one complex variable, in its classical form, was introduced by Whittaker [1, 2] who laid down the definition of basic sets, basic series, and effectiveness of basic sets. Many well-known polynomials such as Laguerre, Legendre, Hermite, Bernoulli, Euler, and Bessel polynomials form simple bases of polynomials (see [3-7]). A significant advance was contributed to the subject by Cannon [8, 9] who obtained necessary and sufficient conditions for the effectiveness of basic sets for classes of functions with finite radius of regularity and entire functions.

The theory of basic sets of polynomials can be generalized to higher dimensions in several different ways, for instance, to several complex variables or to hypercomplex analysis.

The theory of basic sets of polynomials in several complex variables was developed at the end of the 1950s by Mursi and Maker [10] and later by Nassif [11] and was studied in more detail afterwards by others (c.f [12-15]). Also, the representation of matrix functions by bases of polynomials has been studied by Makar and Fawzy [16]. For more information about the study of basic sets of polynomials in complex analysis, we refer to [17-20].

In theory of basic sets of polynomials in hypercomplex analysis, Abul-Ez and Constales gave in [21,22] the extension of the theory of bases of polynomials in one complex variable to the setting of Clifford analysis. This is the natural generalization of complex analysis to Euclidean space of dimension larger than two, where the holomorphic functions have values in Clifford algebra and are null solutions of a linear differential operator. An important subclass of the Clifford holomorphic functions called special monogenic functions is considered, for which a Cannon theorem on the effectiveness in closed and open ball [21,23] was established. Many authors studied the basic sets of polynomials in Clifford analysis [24-30].

In [31], Adepoju laid down a treatment of the subject of basic sets of polynomials of a single complex variable in Banach space which is based on functional analysis considerations. Also, the authors in [12, 32] studied the basic sets of polynomials of several complex variables in Banach space.

We shall lay down in this paper a treatment of the subject of basic sets based primarily on functional analysis and Clifford analysis. The aim of this treatment is to construct a criterion, of general type, for effectiveness of basic sets in Frechet modules. By attributing particular forms to these Freechet modules, we derive, in the remaining articles of the present paper, from the general criterion of effectiveness already obtained, particular conditions for effectiveness in the different forms of the regions which are relevant to our subsequent work. Thus, effectiveness in open and closed balls is studied. In addition, we give some applications of the effectiveness of basic sets of polynomials in approximation theory concerned with

(1) the expansion of Clifford valued functions in closed and open ball by infinite series in a given sequence of basic sets,

(2) the expansion of Clifford valued functions in closed and open ball by infinite series in a given sequence of Cannon sets of special monogenic polynomials.

These new results extend and generalize the known results in complex and Clifford setting given in [12,21, 23, 31, 32].

2. Notation and Preliminaries

In order to introduce our results, we give several notations and assumptions.

Let us denote by {[e.sub.1], ..., [e.sub.m]} the canonical basis of the Euclidean vector space [R.sup.m] and by [A.sub.m] the associated real Clifford algebra in which one has the multiplication rules [e.sub.i][e.sub.j] + [e.sub.j][e.sub.i] = -2[[delta].sub.ij], i, j = 1, ..., m, where [[delta].sub.ij] denotes the Kronecker symbol.

A vector space basis for the Clifford algebra [A.sub.m] is given by the set {[e.sub.A] : A [subset or equal to] {1, ..., m}} with [mathematical expression not reproducible], 1 [less than or equal to] [[alpha].sub.1] < [[alpha].sub.2] < ... < [[alpha].sub.h] [less than or equal to] m, and [e.sub.[phi]] = [e.sub.0] = 1. Every a [member of] [A.sub.m] can be written in the form a = [[summation].sub.A][a.sub.A][e.sub.A] with [a.sub.A] [member of] R. The conjugate element of a [member of] [A.sub.m] is defined by [bar.a] = [[summation].sub.A][a.sub.A][[bar.e].sub.A], where [mathematical expression not reproducible], and [[bar.e].sub.0] = [e.sub.0] = 1.

We denote also by [R.sup.m+1] = [span.sub.R][{[e.sub.0], [e.sub.1], ..., [e.sub.m]}.sub.m] = R [direct sum] [R.sup.m] [subset] [A.sub.m] the space of paravectors x = [x.sub.0][e.sub.0] + [[summation].sup.m.sub.j=1][x.sub.j][e.sub.j]. In this notation, the paravector x will be represented in the form x = [x.sub.0] + [x.bar] with Sc(x) = [x.sub.0] being the scalar part and Vec(x) = [x.bar] being the vector part of x. The induced Clifford norm of arbitrary a [member of] [A.sub.m] is given by [absolute value of a] = [([[summation].sub.A][[absolute value of ([a.sub.A])].sup.2]).sup.1/2].

Some care must be taken when using this norm to estimate product. We will always use the formula [absolute value of ab] [less than or equal to] [2.sup.m/2][absolute value of a][absolute value of b].

One useful approach to generalize complex analysis to higher dimensional spaces is the Cauchy-Riemann approach which is based on the consideration of functions that are in the kernel of the generalized Cauchy-Riemann operator D = [[summation].sup.m.sub.i=0][e.sub.i]([partial derivative]/[partial derivative][x.sub.i]) in [R.sup.m+1] (for more details, see [33, 34]).

Definition 1 (unitary right [A.sub.m]-module). A unitary right [A.sub.m]-module X is an abelian group (X, +) with a mapping X x [A.sub.m] [right arrow] X; (f, [lambda]) [right arrow] f[lambda] such that for all [lambda],[mu] [member of] [A.sub.m] and f, g [member of] X:

(i) f([lambda] + [mu]) = f[lambda] + f[mu].

(ii) f([lambda][mu]) = (f[lambda])[mu].

(iii) (f + g)[lambda] = f[lambda] + g[lambda].

(iv) f[e.sub.0] = f.

Remark 2. Notice that X becomes a real vector space if R is identified with R[e.sub.0] = [A.sub.0] [subset] [A.sub.m].

In the following, all [A.sub.m]-modules will be right [A.sub.m]-modules.

Definition 3 ([A.sub.m]-linear operator). Let X and Y be two unitary [A.sub.m]-modules. Then a function T : X[right arrow] Y is said to be an [A.sub.m]-linear operator if, for all f, g [member of] X and [lambda], [mu] [member of] [A.sub.m],

T(f[lambda] + g[mu]) = T(f)[lambda] + T(g)[mu]. (1)

The set of all [A.sub.m]-linear operators from X into Y is denoted by L(X, Y).

Definition 4 (proper system of seminorms). Let X be a unitary [A.sub.m]-module. Then a family P of functions p : X [right arrow] [0, [infinity]) is said to be a proper system of seminorms on X if the following conditions are fulfilled:

[[P.sub.1]] There exists a constant [C.sub.0] [greater than or equal to] 1 such that, for all p [member of] P, [lambda] [member of] [A.sub.m] and f, g [member of] X:

(i) p(f + g) [less than or equal to] p(f) + p(g).

(ii) p(f[lambda]) [less than or equal to] [C.sub.0][absolute value of [lambda]]p(f), and p(f[lambda]) = [absolute value of [lambda]]p(f) if [lambda] [member of] R.

[[P.sub.2]] For any finite number [p.sub.1], [p.sub.2], ..., [p.sub.k] [member of] P, there exist p [member of] P and C > 0 such that, for all f [member of] X,

[mathematical expression not reproducible] (2)

[[P.sub.3]] If p(f) = 0 for all p [member of] P then f = 0.

Definition 5 (Frechet module). A Frechet module E over [A.sub.m] is a Hausdorff space with a countable proper system of seminorms P = [([p.sub.i]).sub.i [greater than or equal to] 0] satisfying

(i) i < j [??] [p.sub.i](f) [less than or equal to] [p.sub.j](f); (f [member of] E),

(ii) a subset U of E is open if, [for all]f [member of] U, there exist [epsilon] > 0 and N [greater than or equal to] 0 such that

{g [member of] E: [p.sub.i](f - g) [less than or equal to] [epsilon]} [subset] U, [for all]i [less than or equal to] N, (3)

(iii) E is complete with respect to this topology.

We denote by [T.sub.E] the topology defined by the family P of seminorms on E.

Definition 6 (convergent sequences in the topology [T.sub.E]). The sequence [([f.sub.k]).sub.k [greater than or equal to] 0] of elements of E converges in the topology [T.sub.E] to the element g of E, if and only if, for all [p.sub.i] [member of] P, we have

[mathematical expression not reproducible] (4)

We may also equivalently say that the sequence [([f.sub.k]).sub.k [greater than or equal to] 0] converges in E to g with respect to [T.sub.E].

It is a familiar property for the Frechet module E that a seminorm [sigma] on E is [T.sub.E]-continuous, if and only if there is a seminorm [p.sub.i] [member of] P and a positive finite constant [C.sub.i] such that

[sigma](f) [less than or equal to] [C.sub.i][p.sub.i](f); ([for all]f [member of] E). (5)

It is also known that a linear operator L on E is continuous if and only if there is a seminorm [p.sub.i] [member of] P and a constant [K.sub.i] such that

[absolute value of (L(f))] [less than or equal to] [K.sub.i][p.sub.i](f); ([for all]f [member of] E). (6)

3. Basis and Absolute Basis

Definition 7 (basis for Frechet module E). Let E be a Frechet module over [A.sub.m]. A sequence [([z.sub.n]).sub.n [greater than or equal to] 0] of nonzero elements of E is called a basis for E if, for each element f [member of] E, there is one and only one sequence [([Z.sub.n](f)).sub.n [greater than or equal to] 0] of the Clifford algebra [A.sub.m], such that

f = [[infinity].summation over (n=0)][z.sub.n][Z.sub.n](f). (7)

Definition 8 (Cauchy's inequality). We shall assume that Cauchy's inequality holds for the basis [([z.sub.n]).sub.n [greater than or equal to] 0] in the form that for each [p.sub.i] [member of] P there is a positive finite constant [M.sub.i] such that

[absolute value of ([Z.sub.n](f))][p.sub.i] ([z.sub.n]) [less than or equal to] [M.sub.i][p.sub.i](f) (8)

for all integers n and for all f [member of] E.

Also, the nature of the problems considered here necessitates that whenever i < j, there is a finite positive constant [M.sub.i,j], such that

[[infinity].summation over (n=0)][[p.sub.i]([z.sub.n])/[p.sub.j]([z.sub.n])] = [M.sub.i,j]. (9)

Definition 9 (absolute basis for Frechet module E). The basis [([z.sub.n]).sub.n [greater than or equal to] 0] is called an absolute basis for E if the series

[[infinity].summation over (n=0)][absolute value of ([Z.sub.n](f))][p.sub.i]([z.sub.n])(10)

is convergent in R for all integers i and for all f [member of] E. Thus, in this case, we can write

[[infinity].summation over (n=0)][absolute value of ([Z.sub.n](f))][p.sub.i]([z.sub.n]) = [Q.sub.i](f) < [infinity]; (i [greater than or equal to] 0, f [member of] E). (11)

We start with the following introductory theorems.

Theorem 10. If [([z.sub.k]).sub.k [greater than or equal to] 0] is a basis for E and if Cauchy's inequality (8) is satisfied, then [Z.sub.n] is a continuous linear operator on E, orthonormal to [([z.sub.k]).sub.k [greater than or equal to] 0].

Proof. It easily follows from the uniqueness of representation (7) that if f, g [member of] E and [lambda], [mu] [member of] [A.sub.m], then

[Z.sub.n](f[lambda] + g[mu]) = [Z.sub.n](f)[lambda] + [Z.sub.n](g)[mu], (12)

so that [Z.sub.n] is a linear operator on E. Also, putting f = [z.sub.k] in (7), it can be verified that

[Z.sub.n]([z.sub.k]) = [[delta].sup.k.sub.n], (13)

and [Z.sub.n] is orthonormal to [([z.sub.k]).sub.k [greater than or equal to] 0].

We deduce the continuity of [Z.sub.n] from (6) and (8).

Theorem 11. Let [([z.sub.n]).sub.n [greater than or equal to] 0] be an absolute basis for E and let [Q.sub.i] be given by (11). Then the family [([Q.sub.i]).sub.i [greater than or equal to] 0] forms a proper system of continuous seminorms. Moreover, for i < j, there exists a constant [N.sub.i,j] such that

[Q.sub.i](f) [less than or equal to] [N.sub.i,j][p.sub.j](f) (14)

for all f [member of] E.

Proof. Firstly, we prove that the family [([Q.sub.i]).sub.i [greater than or equal to] 0] is a proper system of seminorms as follows.

[[P.sub.1]] We observe, from the linearity of [Z.sub.n] and properties (i) and (ii) of seminorms, that

[mathematical expression not reproducible] (15)

whenever f, g [member of] E and [lambda] [member of] [A.sub.m].

[[P.sub.2]] Let [Q.sub.1], [Q.sub.2], ..., [Q.sub.k] be defined by (11). Since P is a proper system of seminorms, then there exist p [member of] P and C > 0 such that, for all f [member of] X,

[mathematical expression not reproducible] (16)

[[P.sub.3]] Suppose that [Q.sub.i](f) = 0 [for all]i; then [absolute value of ([Z.sub.n](f))][p.sub.i]([z.sub.n]) = 0 [for all]i, n. Since P is a proper system of seminorms and [([z.sub.k]).sub.k [greater than or equal to] 0] is a basis, for each n [member of] N there exists j such that [p.sub.j]([z.sub.n]) [not equal to] 0. So [Z.sub.n](f) = 0 for all n [member of] N and hence (7) implies that f = 0.

Finally, when i < j, inequality (14) can be obtained from (8,9), and (11) as follows:

[mathematical expression not reproducible] (17)

It follows from condition (5) that the seminorm [Q.sub.i] is continuous on E. Therefore the family [([Q.sub.i]).sub.i [greater than or equal to] 0] forms a proper system of continuous seminorms, as required.

4. Basic Sets

In this section, we lay down the definition of basic sets, basic coefficients, and basic series and show (in Theorem 13) that when the basic series converges, it will converge to the element with which it is associated.

Let [([P.sub.k]).sub.k [greater than or equal to] 0] be a sequence of nonzero elements of E, and suppose that [([[pi].sub.k]([z.sub.n])).sub.n,k[greater than or equal to]0] is a matrix of coefficients in the Clifford algebra [A.sub.m] such that, for each n [greater than or equal to] 0, we have the unique representation

[z.sub.n] = [[infinity].summation over (k=0)][P.sub.k][[pi].sub.k]([z.sub.n]). (18)

In this case, we shall call the sequence [([P.sub.k]).sub.k [greater than or equal to] 0] a basic set on E.

Let f be any element of E and substitute (18) in (7) to obtain the formal series

f ~ [[infinity].summation over (k=0)][P.sub.k][[PI].sub.k](f), (19)

where

[[PI].sub.k](f) = [[infinity].summation over (n=0)][[pi].sub.k]([z.sub.n])[Z.sub.n](f). (20)

When series (20) converges in [A.sub.m], [[PI].sub.k](f) exists and is called the kth basic coefficient of f relative to the set [([P.sub.k]).sub.k [greater than or equal to] 0]. When the basic coefficient [[PI].sub.k](f) exists for all k, series (19) is called the basic series associated with f.

The following theorem is concerned with the basic coefficients [([[PI].sub.k](f)).sub.k [greater than or equal to] 0].

Theorem 12. If [[PI].sub.k](f) is defined for all f in E, the map [[PI].sub.k]: E [right arrow] [A.sub.m] is a continuous linear operator on E.

Proof. Let

[T.sub.N](f) = [N.summation over (n=0)][[pi].sub.k]([z.sub.n])[Z.sub.n](f). (21)

It is clear that [T.sub.N] is a continuous linear operator on E as a finite sum of continuous linear functional [Z.sub.n].

Now, if [[PI].sub.k](f) is defined for all f in E, the sequence [([T.sub.N](f)).sub.N [greater than or equal to] 0] converges pointwise to [[PI].sub.k](f) in E. Therefore, by the Banach-Steinhaus theorem for Frechet space [35], we deduce that [[PI].sub.k] is equally a continuous linear operator on E, and the theorem is established.

We now write

[B.sub.n](f) = [n.summation over (k=0)][P.sub.k][[pi].sub.k](f) (22)

for the nth partial sum of basic series (19). The following theorem establishes the required conformity of the limit of [B.sub.n](f) with the space E.

Theorem 13. If, for every f [member of] E, [B.sub.n](f) is defined for all n and if the sequence [([B.sub.n](f)).sub.n [greater than or equal to] 0] converges in E to some limit B(f), then B(f) = f for all elements f [member of] E.

Proof. We prove that B is a continuous linear operator on E as a limit of finite sum of continuous linear operators as in Theorem 12.

Now, it can be proven, from (13) and (20), that

[[PI].sub.k]([z.sub.n]) = [[pi].sub.k]([z.sub.n]); (n, k [greater than or equal to] 0). (23)

Hence, (18) and (22) together yield

B([z.sub.n]) = [[infinity].summation over (k=0)][P.sub.k][[pi].sub.k]([z.sub.n]) = (n [greater than or equal to] 0). (24)

Let f be any element of E and write

[g.sub.k] = [k.summation over (j=0)][z.sub.j][Z.sub.j](f). (25)

Then, in view of (24) and (25), we have

B([g.sub.k]) = [g.sub.k], (26)

and, by continuity of B, we deduce that B(f) = f and Theorem 13 is therefore established.

5. Effectiveness of Basic Sets

We have seen that when [([B.sub.n](f)).sub.n [greater than or equal to] 0] converges for each f of E, [B.sub.n](f) converges to f. This means that basic series (19) associated with the element f converges to f itself for all f [member of] E. In this case, we say that the set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E. To find a necessary and sufficient condition for the effectiveness of a basic set [([P.sub.k]).sub.k [greater than or equal to] 0] for the space E, we consider, for each seminorm [p.sub.i] [member of] P, the mapping [q.sub.i] : E [right arrow] R defined by

[mathematical expression not reproducible] (27)

Suppose that [q.sub.i](f) is finite for all f [member of] E. We first show that [q.sub.i] is a seminorm on E.

Let f, g [member of] E and [lambda] [member of] [A.sub.m]; it follows from (27) and the linearity of [[PI].sub.k] that

[q.sub.i](f + g) [less than or equal to] [q.sub.i](f) + [q.sub.i](g),

[q.sub.i](f[lambda]) [less than or equal to] [C.sub.0][absolute value of [lambda]][q.sub.i](f). (28)

The first result concerning the effectiveness of basic set [([P.sub.k]).sub.k [greater than or equal to] 0] is the following theorem.

Theorem 14. For [([P.sub.k]).sub.k [greater than or equal to] 0] to be effective for E, it is necessary and sufficient that, for each [p.sub.i] [member of] P, the seminorm [q.sub.i] exist and be continuous.

Proof.

Necessity. When the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E, basic series (19) associated with each element f [member of] E converges to f, and it follows that [B.sub.n] is a continuous linear operator on E.

Therefore, if we write

[C.sub.v,[mu]](f) = [B.sub.v](f) - [B.sub.[mu]-1](f); (v [greater than or equal to] [mu] > 0), (29)

then from (27) we shall have

[mathematical expression not reproducible] (30)

If [mu] [right arrow] [infinity], then [C.sub.v,[mu]](f) [right arrow] 0. Then for every [epsilon] > 0 there exists a natural number m, such that if [mu] [greater than or equal to] m, then [p.sub.i]([C.sub.v,[mu]](f)) < [epsilon]. Hence, we have

[mathematical expression not reproducible] (31)

If v [right arrow] [infinity] with fixed [mu], then [C.sub.v,[mu]](f) [right arrow] f - [B.sub.[mu]-1](f). Hence, for some [epsilon] > 0, there is N such that, for all v > N and [mu] [less than or equal to] m,

[mathematical expression not reproducible] (32)

This shows that the seminorm qi exists.

Now, to prove the continuity of [q.sub.i], let [([f.sub.n]).sub.n [greater than or equal to] 0] be a sequence in E which converges to an element f [member of] E. By this hypothesis, if [epsilon] > 0, there exists a natural number M such that if n > M, then [p.sub.i] ([f.sub.n] - f) < [epsilon].

Hence, if K([epsilon]) = sup{M, N, m}, it follows that if n [greater than or equal to] K([epsilon]), then

[mathematical expression not reproducible] (33)

Thus [lim.sub.n[right arrow][infinity]][q.sub.i]([f.sub.n]) = [q.sub.i](f) and [q.sub.i] is a continuous seminorm on E.

Sufficiency. We observe from (22) and (27) that

[p.sub.i]{[B.sub.n](f)} [less than or equal to] [q.sub.i](f); (n [greater than or equal to] 0). (34)

Since [q.sub.i] is continuous on E, we deduce that the sequence [([B.sub.n]).sub.n [greater than or equal to] 0] is an equicontinuous sequence on E.

Now define the subspace Y of E by

Y := {g [member of] Y, the sequence [([B.sub.n](g)).sub.n [greater than or equal to] 0] is a Cauchy sequence}. (35)

It follows from the equicontinuity of [([B.sub.n]).sub.n [greater than or equal to] 0] that the set Y is closed. Hence, the set Y [subset] E is everywhere dense and is closed, so that Y = [bar.Y] = E. Therefore [([B.sub.n](f)).sub.n [greater than or equal to] 0] is a Cauchy sequence on E and since E is complete, the sequence [([B.sub.n](f)).sub.n [greater than or equal to] 0] converges for all f [member of] E and hence it converges to f in E. Thus, the set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E and Theorem 14 is established.

Theorem 15. Suppose that [([z.sub.n]).sub.n [greater than or equal to] 0] is an absolute basis for E. Then the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] will be effective for E if and only if, for any continuous seminorm [p.sub.i] [member of] P, there is a continuous seminorm [p.sub.j] [member of] P and a positive finite number [K.sub.i,j] such that

[q.sub.i]([z.sub.n]) [less than or equal to] [K.sub.i,j][p.sub.j]([z.sub.n]); [for all]n [member of] N. (36)

Proof.

Necessity. If the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E, then, by Theorem 14, the application [q.sub.i] is a continuous seminorm on E. Hence, by (5), there is a seminorm [p.sub.j] [member of] P and a positive number [K.sub.i,j] such that

[q.sub.i](f) [less than or equal to] [K.sub.i,j][p.sub.j](f); [for all]f [member of] E. (37)

Putting f = [z.sub.n], n [greater than or equal to] 0, we obtain (36).

Sufficiency. Multiplying the basic coefficient [[PI].sub.k](f) of (20) by [P.sub.k] and using (11), (27), and (36), we obtain

[mathematical expression not reproducible] (38)

where [Q.sub.j](f) is defined by (11). According to inequality (14), there is a seminorm [p.sub.l] [member of] P and a positive number [M.sub.jl] such that

[mathematical expression not reproducible] (39)

It follows from condition (5) that [q.sub.i] is continuous on E. Then, by Theorem 14, we deduce that the set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E and the proof of Theorem 15 is therefore terminated.

6. Alternative Treatment of the Problem

In this treatment, we consider the Frechet module E as a subspace of a Banach module F with a continuous norm [sigma] such that

[sigma](f) [less than or equal to] [p.sub.i](f), [for all]i [member of] N, [for all]f [member of] E, (40)

where P = [([p.sub.i]).sub.i [greater than or equal to] 0] is the family of seminorms defined, as before, in the space E. Thus, the topology induced in E by the topology [T.sub.F] of F determined by the norm [sigma] is coarser than the topology [T.sub.E] defined on E by the family P of seminorms.

Let [([z.sub.k]).sub.k [greater than or equal to] 0] be a basis for E and let [([P.sub.k]).sub.k [greater than or equal to] 0] be a sequence of nonzero elements of F. We suppose that [([[pi].sub.k]([z.sub.n])).sub.n,k [greater than or equal to] 0] is a matrix of [A.sub.m] such that, for each n [greater than or equal to] 0, we have the unique representation

[z.sub.n] = [[infinity].summation over (k=0)][P.sub.k][[pi].sub.k]([z.sub.n]), (41)

and the convergence is in F. In this case, we call the sequence [([P.sub.k]).sub.k [greater than or equal to] 0] a basic set on F. Let f be an element of E and substitute (41) in (7) to obtain the formal series

f ~ [[infinity].summation over (k=0)][P.sub.k][[PI].sub.k](f), (42)

where

[[PI].sub.k](f) = [[infinity].summation over (n=0)][[pi].sub.k]([z.sub.n])[Z.sub.n](f). (43)

When series (43) converges in [A.sub.m], we call [[PI].sub.k](f) the basic coefficient of f, and when [[PI].sub.k](f) exists for each k, series (42) is called the basic series of f. Recall that the partial sum [B.sub.n] : E [right arrow] F is defined (see (22)) by

[B.sub.n](f) = [n.summation over (k=0)][P.sub.k][[PI].sub.k](f). (44)

Theorem 12 remains unchanged, while the alternative form of Theorem 13 is the following.

Theorem 16. If, for every f [member of] E, [B.sub.n](f) is defined for all n and if the sequence [([B.sub.n](f)).sub.n [greater than or equal to] 0] converges in F to some element B(f), then B(f) = f for all elements f [member of] E.

Proof. We prove, as before, that, for each integer n, [B.sub.n] is a linear operator from E to F. We show now that [B.sub.n] is continuous. In fact, if [([f.sub.m]).sub.m [greater than or equal to] 0] is a sequence of elements of E converging to an element g of E, it can be deduced from (40) and (44) and Theorem 12 that

[mathematical expression not reproducible] (45)

Hence [B.sub.n]([f.sub.m]) converges to [B.sub.n](g) in F and hence [B.sub.n] : E [right arrow] F is a continuous linear operator. Proceeding exactly as in the proof of Theorem 13, we can deduce that B : E [right arrow] F is a continuous linear operator.

Now, set [g.sub.k] = [[summation].sup.k.sub.j=0][z.sub.j][Z.sub.j](f). It is clear that [g.sub.k] [right arrow] f in E and B([g.sub.k]) = [g.sub.k]; hence B(f) = f, as required.

We see that when [([B.sub.n](f)).sub.n [greater than or equal to] 0] converges in F, for every element f [member of] E, [B.sub.n](f) converges in F to f. This means that basic series (42) associated with the element f converges in F to the element f, for all f [member of] E. In this case, we say that the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E in F.

The necessary and sufficient condition for effectiveness of [([P.sub.k]).sub.k [greater than or equal to] 0] for E in F is obtained through the expression

[mathematical expression not reproducible] (46)

It can be proven in exactly the same way as before that [q.sup.[sigma]] is a seminorm on E. The revised version of Theorem 14 is as follows.

Theorem 17. For the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] to be effective for E in F, it is necessary and sufficient that [q.sup.[sigma]] exist and be continuous on E.

Proof.

Necessity. Since [B.sub.n] is continuous linear operator from E to F, we apply the same method as in the proof of Theorem 14.

Sufficiency. We see here, from (22) and (46), that

[sigma]([B.sub.n](f)) [less than or equal to] [q.sup.[sigma]](f); [for all]f [member of] E. (47)

Since [q.sup.[sigma]] is continuous on E, then the sequence [([B.sub.n]).sub.n [greater than or equal to] 0] will be equicontinuous from E to F. The proof is then completed in exactly the same way as in the proof of Theorem 14.

Now, if [([z.sub.n]).sub.n [greater than or equal to] 0] is an absolute basis for E, the effectiveness of the set [([P.sub.k]).sub.k [greater than or equal to] 0] for E in F will be estimated through the expression [q.sup.[sigma]]([z.sub.n]) as it is seen from the following Theorem which is the alternative form of Theorem 15.

Theorem 18. Suppose that [([z.sub.n]).sub.n [greater than or equal to] 0] is an absolute basis for E. Then the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] will be effective for E in F if and only if there is a seminorm [p.sub.i] [member of] P and a constant [K.sub.i] such that

[q.sup.[sigma]]([z.sub.n]) [less than or equal to] [K.sub.i][p.sub.i]([z.sub.n]); [for all]n [greater than or equal to] 0. (48)

Proof.

Necessity. If [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E in F, then, by Theorem 17, [q.sup.[sigma]] is a continuous norm on E. Hence, by (5), there is a seminorm [p.sub.i] [member of] P and a constant [K.sub.i] such that

[q.sup.[sigma]](f) [less than or equal to] [K.sub.i][p.sub.i](f); [for all]f [member of] E. (49)

Putting f = [z.sub.n], we obtain (48).

Sufficiency. As in the proof of Theorem 15, we deduce from (11), (43), (46), and (48) that

[mathematical expression not reproducible] (50)

Applying inequality (14), we obtain

[q.sup.[sigma]](f) [less than or equal to] [C.sub.0][K.sub.i][M.sub.i,j][p.sub.j](f) = [C.sub.0][L.sub.i,j][p.sub.j](f), [for all]f [member of] E. (51)

So we deduce that [q.sup.[sigma]] is continuous on E and hence Theorem 17 implies that [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for E in F. Theorem 18 is therefore established.

7. Applications

We need to mention some definitions and notations in Clifford analysis [21, 22, 33, 34].

Definition 19 (monogenic function). Let [OMEGA] [subset] [R.sup.m+1] be an open set; then an [A.sub.m]-valued function f is called left (resp. right) monogenic in [OMEGA] if it satisfies Df = 0 (resp. fD = 0) in [OMEGA]. Here, D = [[summation].sup.m.sub.i=0][e.sub.i]([partial derivative]/[partial derivative][x.sub.i]), defined in Section 2, is the generalized Cauchy-Riemann operator.

Definition 20 (special monogenic polynomial). A polynomial P(x) is special monogenic if and only if DP(x) = 0 (so P(x) is monogenic) and there exists [a.sub.i,j] [member of] [A.sub.m] for which P(x) = [[summation].sup.finite.sub.i,j][[bar.x].sup.i][x.sup.j][a.sub.i,j].

Definition 21 (special monogenic function). Let [OMEGA] be a connected open subset of [R.sup.m+1] containing 0 and let f be monogenic in [OMEGA]. Then f is called special monogenic in [OMEGA] if and only if its Taylor series near zero (which exists) has the form f(x) = [[summation].sup.[infinity].sub.n=0]0 [P.sub.n](x) for certain special monogenic polynomials [P.sub.n](x).

The fundamental references for special monogenic function are [36, 37].

Remark 22. Note that if [P.sub.n](x) is a homogeneous special monogenic polynomial of degree n, then (see [21,29]) [P.sub.n](x) = [P.sub.n](x)[alpha]: [alpha] is some constant in [A.sub.m] and

[P.sub.n](x) = [n!/(m)] [summation over (k+l=n)][[((m - 1)/2).sub.k]/k!] [[((m + 1)/2).sub.l]/l!] [[bar.x].sup.k][x.sup.l] x [member of] [R.sup.m+1], (52)

where, for b [member of] R, [(b).sub.l] = b(b + 1) ... (b + l - 1).

It is well known that [([P.sub.n](x)).sub.n [greater than or equal to] 0] is an Appell sequence with respect to [partial derivative]/[partial derivative][x.sub.0] or (1/2)D (which represent the same operator for monogenic functions): (1/2)[bar.D][P.sub.n](x) = n[P.sub.n-1](x) and [bar.D] = [[summation].sup.m.sub.i=0][[bar.e].sub.i]([partial derivative]/[partial derivative][x.sub.i]) in [R.sup.m+1] (see [38-40]).

The maximum value of [absolute value of ([P.sub.n](x))] in [absolute value of x] = R is given by (see [21])

[mathematical expression not reproducible] (53)

An open ball is usually denoted by B(r), and closed ball is denoted by [bar.B](r), where

B(r) = {x [member of] [R.sup.m+1], [absolute value of x] < r},

[bar.B](r) = {x [member of] [R.sup.m+1], [absolute value of x] [less than or equal to] r}. (54)

Also, the class of special monogenic functions in an open ball B(r) is written as H(r) and [bar.H](r) denotes the class of special monogenic functions in closed ball [bar.B](r).

The first application of the above theory is to the effectiveness in an open ball.

7.1. Effectiveness in Open Balls. We propose to derive in the present section, from the results of Section 5, conditions for effectiveness of basic sets in open balls. For this case, we take the Frechet module E to be the class H(r), r > 0, of special monogenic functions in the open ball B(r).

Let [[rho].sub.0] be a certain positive number less than r and construct the sequence [([[rho].sub.n]).sub.n [greater than or equal to] 0] as follows:

[mathematical expression not reproducible] (55)

So

[[rho].sub.0] < [[rho].sub.1] < [[rho].sub.2] < ... < [[rho].sub.n] < ... < r. (56)

The countable family P of seminorms [([p.sub.i]).sub.i [greater than or equal to] 0] on the Frechet module H(r) is defined as follows.

For f [member of] H(r), we set

[mathematical expression not reproducible] (57)

Thus, when i < j, [[rho].sub.i] < [[rho].sub.j]. So

[p.sub.i](f) [less than or equal to] [p.sub.j](f); [for all]f [member of] H(r), (58)

and, therefore, condition (i) of Definition 5 is satisfied.

The topology on H(r) defined by the family [([p.sub.i]).sub.i [greater than or equal to] 0] is the topology of normal convergence over the compact sets [bar.B]([[rho].sub.i]), i [greater than or equal to] 0. It is easy to show that the [A.sub.m]-module H(r) is complete for this topology; that is to say, H(r) is a Frechet module.

We shall take a basis for H(r), the Appell sequence [([P.sub.n](x)).sub.n [greater than or equal to] 0]. In fact every function f [member of] H(r) has the unique expansion

f(x) = [[infinity].summation over (n=0)][P.sub.n](x)[a.sub.n](f). (59)

Thus (7) is true. In this case, Cauchy's inequality (8) takes the form (see [22])

[mathematical expression not reproducible] (60)

It can be verified also that [([P.sub.n](x)).sub.n [greater than or equal to] 0] is an absolute basis for H(r) in the sense that series (11), which is rewritten here as

[[infinity].summation over (n=0)][absolute value of ([a.sub.n](f))][[rho].sup.n.sub.i], (61)

is convergent [for all]i [greater than or equal to] 0.

Finally, when i < j, [[rho].sub.i] < [[rho].sub.j], so

[[infinity].summation over (n=0)][[p.sub.i]([P.sub.n](x))/[p.sub.j]([P.sub.n](x))] = [[infinity].summation over (n=0)][[[rho].sup.n.sub.i]/[[rho].sup.n.sub.j]] < [infinity] (62)

and then relation (9) holds.

Now, let [([P.sub.k]).sub.k [greater than or equal to] 0] = [([P.sub.k](x)).sub.k [greater than or equal to] 0] be a basic set. Expression (18) is the unique representation

[P.sub.n](x) = [[infinity].summation over (k=0)][P.sub.k](x)[[pi].sub.n,k] (63)

and if f [member of] H(r), then by substituting (63) in (59) we obtain the basic series of f:

f(x) ~ [[infinity].summation over (k=0)][P.sub.k](x)[[PI].sub.k] (64)

where

[[PI].sub.k](f) = [[infinity].summation over (n=0)][[pi].sub.n,k][a.sub.n](f); (k [greater than or equal to] 0) (65)

is the basic coefficient of f.

In this case, the expression [q.sub.i]([P.sub.n](x)) is called the Cannon sum for the set ([P.sub.k]) and is denoted by [F.sub.n]([[rho].sub.i]):

[mathematical expression not reproducible] (66)

The Cannon function for the same set in [bar.B]([rho]) is

[mathematical expression not reproducible] (67)

It should be observed that (63), (66), and (67) together yield

K([rho]) [greater than or equal to] [rho] (68)

and it is easily seen that K([rho]) is a monotonic increasing function of [rho].

The fundamental theorem for effectiveness for H(r) is deducible from Theorem 15. It is stated in the following form.

Theorem 23. The necessary and sufficient condition for the basic set [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective for H(r) is that

K([rho]) < r, [for all][rho] < r. (69)

Proof. Suppose that [rho] is any positive number less than r; then there exists a number [[rho].sub.i] such that

[rho] [less than or equal to] [[rho].sub.i] < r. (70)

If the set [([P.sub.k](x)).sub.k [greater than or equal to] 0] is effective for H(r), then, by Theorem 15, there exist [[rho].sub.j] < r and a constant [K.sub.i,j] such that

[q.sub.i]([P.sub.n](x)) [less than or equal to] [K.sub.i,j][p.sub.j]([P.sub.n](x)); [for all]n [member of] N. (71)

Hence, (66) gives

[F.sub.n]([[rho].sub.i]) [less than or equal to] [K.sub.i,j][[rho].sup.n.sub.j]. (72)

Using (67) and (70), we can deduce that

K([rho]) [less than or equal to] K ([[rho].sub.i]) [less than or equal to] [[rho].sub.j] < r (73)

and condition (69) follows. Thus (69) is necessary.

On the other hand, suppose that condition (69) is satisfied and let [[rho].sub.i] be any element of sequence (56). So we have

K([[rho].sub.i]) < r. (74)

Since the sequence [([[rho].sub.i]).sub.i [greater than or equal to] 0] converges to r as i tends to infinity, then there exists an integer j > i such that

K([[rho].sub.i]) < [[rho].sub.j] < r. (75)

Then, by definition (67) of K([rho]), there exists [C.sub.i,j] such that

[F.sub.n]([[rho].sub.i]) [less than or equal to] [C.sub.i,j][[rho].sup.n.sub.j]. (76)

Applying (53), (57), and (66), it follows that

[q.sub.i]([P.sub.n](x)) [less than or equal to] [C.sub.i,j][p.sub.j] ([P.sub.n](x)); [for all]n [member of] N. (77)

Hence, by Theorem 15, the set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for H(r), and the theorem is satisfied.

7.2. Effectiveness in Closed Balls. Let R be any fixed positive number and take the number r to be any finite number greater than R. The [A.sub.m]-module F of Section 6 will be taken as the class [bar.H](R) of special monogenic functions in the closed ball [bar.B](R), with the norm [sigma] defined by

[mathematical expression not reproducible] (78)

Thus, the topology [T.sub.[bar.H](R)] determined by the norm [sigma] is the topology of normal convergence on [bar.B](R). It is well known that [bar.H](R) is complete for this topology; that is to say, [bar.H](R) is a Banach module. The subspace E of [bar.H](R) will be taken as the Frechet module H(r) (see Section 7.1) which will be equipped with the family of seminorms [([p.sub.i]).sub.k [greater than or equal to] 0] defined by

[p.sub.i](f) = M (f, [[rho].sub.i]); i [member of] N, f [member of] H(r), (79)

where

[mathematical expression not reproducible] (80)

It is clear that

[sigma](f) [less than or equal to] [p.sub.i](f): i [member of] N, f [member of] H(r). (81)

So condition (40) is satisfied.

The basis for H(r) is taken, as before, to be the Appell sequence [([P.sub.n](x)).sub.n [greater than or equal to] 0] which accords to Cauchy's inequality and it is also an absolute basis. Moreover, condition (9) is satisfied.

Now, let [([P.sub.k](x)).sub.k [greater than or equal to] 0] be a basic set of [bar.H](R) and suppose that [P.sub.n](x) admits the representation

[P.sub.n](x) = [[infinity].summation over (k=0)][P.sub.k](x)[[pi].sub.n,k], (82)

where the convergence is in [bar.H](R), so that

[mathematical expression not reproducible] (83)

Write the basic coefficient and the basic series as in (64) and (65). We are concerned with the convergence in [bar.H](R); that is to say, basic series (64) converges to f in [bar.H](R) if

[mathematical expression not reproducible] (84)

The basic set [([P.sub.k]).sub.k [greater than or equal to] 0] will be effective for H(r) in [bar.H](R) if the basic series of each special monogenic function in B(r) converges to f normally in [bar.B](R). In this case, we say that the basic set [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for H(r) in [bar.B](R), R < r.

The theorem about such effectiveness is deducible from Theorem 18.

As in (66), we can see, by using (78), that

[q.sup.[sigma]]([P.sub.n](x)) = [F.sub.n](R). (85)

We shall establish the following Theorem.

Theorem 24. The necessary and sufficient condition for the basic set [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective for H(r) in [bar.B](R), r > R > 0, is that

K(R) < r. (86)

Proof. Suppose that [([P.sub.k]).sub.k [greater than or equal to] 0] is effective for H(r) in [bar.B](R). Hence, according to Theorem 18, there exist a norm [p.sub.i] [member of] P and a constant [K.sub.i] such that

[q.sup.[sigma]] ([P.sub.n](x)) [less than or equal to] [K.sub.i][p.sub.i]([P.sub.n](x)), n [greater than or equal to] 0. (87)

Hence, (67), (79), and (85) together yield

K(R) [less than or equal to] [[rho].sub.i] < r (88)

and condition (86) is necessary.

On the other hand, suppose that condition (86) is satisfied. Since [[rho].sub.n] [right arrow] r as n [right arrow] [infinity], there is a number [[rho].sub.i] such that K(R) < [[rho].sub.i] < r. Then, by definition (67) of K([rho]), there exists [K.sub.i] such that

[F.sub.n](R) [less than or equal to] [K.sub.i][[rho].sup.n.sub.i], n [greater than or equal to] 0. (89)

Hence, from (53), (57), and, (85) it follows that

[q.sup.[sigma]] ([P.sub.n](x)) [less than or equal to] [K.sub.i][p.sub.i]([P.sub.n](x)), n [greater than or equal to] 0. (90)

By Theorem 18, it follows that the basic set [([P.sub.k](x)).sub.k [greater than or equal to] 0] is effective for H(r) in [bar.B](R) as required.

Taking r sufficiently near to R, Theorem 24 leads to the following corollary.

Corollary 25. The necessary and sufficient condition for the basic set [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective in [bar.B](R) is that

K(R) = R. (91)

7.3. Cannon Sets. When [([P.sub.k]).sub.k [greater than or equal to] 0] is a basic set of special monogenic polynomials, representation (63) is finite. Thus, if [N.sub.n] is the number of nonzero coefficients in (63), then [N.sub.n] is finite. If this number accords further to the restriction that

[N.sup.1/n.sub.n] [right arrow] 1, as n [right arrow] [infinity], (92)

the corresponding basic set [([P.sub.k](x)).sub.k [greater than or equal to] 0] of polynomials is called Cannon set of special monogenic polynomials (see [21]). Write

[mathematical expression not reproducible], (93) [mathematical expression not reproducible]. (94) Then, in view of (66), we shall have

[mathematical expression not reproducible]. (95)

Hence, if we put

[mathematical expression not reproducible], (96)

then, for Cannon sets, relation (67) implies that

[lambda]([rho]) = K([rho]). (97)

Therefore, for Cannon sets, the Cannon sum and the Cannon function [[omega].sub.n]([rho]) and [lambda]([rho]) are given by (93) and (96), respectively.

For Cannon sets, the Cannon function [lambda]([rho]) will replace K([rho]) in all the concerned relations: (69), (86), and (91). Hence, concerning the effectiveness of Cannon sets of special monogenic polynomials (see [21, 23]), we have the following results which are special cases of our results.

Corollary 26. The necessary and sufficient condition for the Cannon sets of special monogenic polynomials [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective for H(r) is that

[lambda]([rho]) < r, [for all][rho] < r. (98)

Corollary 27. The necessary and sufficient condition for the Cannon sets of special monogenic polynomials [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective for H(r) in [bar.B](R), r > R > 0, is that

[lambda](R) < r. (99)

Corollary 28. The necessary and sufficient condition for the Cannon sets of special monogenic polynomials [([P.sub.k](x)).sub.k [greater than or equal to] 0] to be effective for in [bar.B](R) is that

[lambda](R) = R. (100)

Disclosure

The current address of Gamal Farghaly Hassan is "Department of Mathematics, Faculty of Sciences, Northern Border University, P.O. Box 1321, Arar, Saudi Arabia". http://dx.doi.org/10.1155/2017/2075938

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors wish to acknowledge the approval and the support of this research study from the Deanship of Scientific Research in Northern Border University, Arar, Saudi Arabia (Grant no. 5-7-1436-5).

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Gamal Farghaly Hassan, (1,2) Lassaad Aloui, (3) and Allal Bakali (1)

(1) Department of Mathematics, Faculty of Sciences, Northern Border University, P. O. Box 1321, Arar, Saudi Arabia

(2)Faculty of Science, University of Assiut, Assiut 71516, Egypt

(3) Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis-El Manar, Tunis, Tunisia

Correspondence should be addressed to Gamal Farghaly Hassan; gamal6new@yahoo.com

Received 20 June 2016; Revised 31 October 2016; Accepted 18 December 2016; Published 14 February 2017

Academic Editor: Konstantin M. Dyakonov
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Title Annotation:Research Article
Author:Hassan, Gamal Farghaly; Aloui, Lassaad; Bakali, Allal
Publication:Journal of Complex Analysis
Article Type:Report
Date:Jan 1, 2017
Words:9632
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