Product Theorems for Certain Summability Methods in Non-archimedean Fields

ANNALES MATHÉMATIQUES

BLAISE PASCAL

P.N. Natarajan

Product Theorems for Certain Summability Methods in Non-archimedean Fields

Volume 10, n^o2 (2003), p. 261-267.

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Product Theorems for Certain Summability Methods in Non-archimedean Fields

P.N. Natarajan

Abstract

In this paper, K denotes a complete, non-trivially valued, non- archimedean field. Sequences and infinite matrices have entries inK.

The main purpose of this paper is to prove some product theorems involving the methodsM and(N, p_n)in such fieldsK.

AMS subject classification: 40, 46.

Keywords and phrases: regular summability methods,M,(N, p_n)meth- ods, product theorems, consistency, analytic functions.

Throughout the present paper, K denotes a complete, non-trivially val- ued, non-archimedean field. Sequences and infinite matrices have entries in K.

Given an infinite matrix A = (a_nk), n, k = 0,1,2,· · · and a sequence x = {x_k}, k = 0,1,2,· · ·, by the A-transform of x = {x_k}, we mean the sequenceA(x) ={(Ax)_n}, where

(Ax)_n=

∞

X

k=0

a_nkx_k, n= 0,1,2,· · ·,

it being assumed that the series on the right converge. If lim

n→∞(Ax)_n =l, we say thatx={x_k}is A-summable to l. If lim

n→∞(Ax)_n =lwhenever lim

k→∞x_k = l, we say that the matrix method A is regular. Necessary and sufficient conditions forAto be regular in terms of its entries are well-known (see [2]).

The (N, p_n) methods (or Nörlund methods) were introduced in K and some of their properties were studied earlier by Srinivasan (see [6]). A more detailed study of the(N, p_n) methods was taken up by the author later and published in a series of articles (for instance, see [4], [5]).

P.N. Natarajan

The(N, p_n) method is defined by the matrixA= (a_nk),where a_nk = p_n−k

P_n , k≤n;

= 0, k > n, where p₀ 6= 0,|p_j| < |p₀|, j = 1,2,· · · and P_n =

n

X

k=0

p_k, n= 0,1,2,· · ·. The following result is known ([4], Theorem 1).

Theorem 1.1: The (N, p_n) method is regular if and only if

n→∞lim p_n= 0.

Let{λ_n}be a sequence inK such that lim

n→∞λ_n= 0.LetM = (b_nk),where b_nk = λ_n−k, k≤n;

= 0, k > n.

In this context, we note that theM method reduces to theY method of Srinivasan (see [6]), whenK =Q_p, thep-adic field for a primep, λ₀ =λ₁=

1

2, λ_n= 0, n≥2.

We need the following definition in the sequel.

Definition: Two matrix methods A= (a_nk), B= (b_nk)are said to be consis- tent, if whenever x={x_k} isA-summable to s and B-summable to t, then s=t.

It is clear that the relation “matricesAandBare consistent" is an equiv- alence relation.

We now recall that a product theorem means the following: given regular methods A, B, does x = {x_k} ∈ (A) imply B(x) ∈ (A), limits being the same, where (A) is the convergence field of A? i.e., does “A(x) converges"

imply “A(B(x)) converges to the same limit"?

The main purpose of this paper is to prove some product theorems involv- ing theM,(N, p_n)methods inK.In the sequel, we suppose that the(N, p_n) methods are regular.

Theorem 1.2: If (N, p_n)(x) converges to l, then (N, p_n)(M(x)) converges to l

∞

X

n=0

λ_n

! .

Proof: Let

τ_n = p₀x_n+p₁x_n−1+· · ·+p_nx₀

P_n ,

t_n = λ_nx₀+λ_n−1x₁+· · ·+λ₀x_n, n= 0,1,2,· · · . By hypothesis, lim

n→∞τ_n=l.Since lim

n→∞p_n= 0andp₀6= 0, lim

n→∞P_n=P, P 6= 0.

Now,

τ_n⁰ = (N, p_n) ({t_n})

= p₀t_n+p₁t_n−1+· · ·+p_nt₀ P_n

= 1

P_n[p₀(λ_nx₀+λ_n−1x₁+· · ·+λ₀x_n) +p₁(λ_n−1x₀+λ_n−2x₁+· · ·+λ₀x_n−1)

+· · ·+p_n(λ₀x₀)]

= 1

P_n[λ₀(p₀x_n+p₁xn−1+· · ·+p_nx₀) +λ₁(p₀xn−1+p₁xn−2+· · ·+pn−1x₀)

+· · ·+λ_n(p₀x₀)]

= 1

P_n[λ₀P_nτ_n+λ₁P_n−1τ_n−1+· · ·+λ_nP₀τ₀]

= 1

P_n[{λ₀P_n(τ_n−l) +λ₁P_n−1(τ_n−1−l) +· · ·

+λ_nP₀(τ₀−l)}+l{λ₀P_n+λ₁P_n−1+· · ·+λ_nP₀}]

= 1

P_n[{λ₀P_n(τ_n−l) +λ₁P_n−1(τ_n−1−l) +· · ·+λ_nP₀(τ₀−l)}

+l{λ₀(P_n−P) +λ₁(P_n−1−P) +· · ·+λ_n(P₀−P)}

+lP{λ₀+λ₁+· · ·+λ_n}].

Using Theorem 1 of [3] and the fact that|P_n|=|P₀|, n= 0,1,2,· · ·,it follows that

n→∞lim 1

P_n[λ₀P_n(τ_n−l) +λ₁P_n−1(τ_n−1−l) +· · ·+λ_nP₀(τ₀−l)] = 0

P.N. Natarajan

and

n→∞lim 1

P_n[λ₀(P_n−P) +λ₁(Pn−1−P) +· · ·+λ_n(P₀−P)] = 0, so that

n→∞lim τ_n⁰ =l

∞

X

n=0

λ_n,

i.e., (N, p_n)(M(x)) converges to l

∞

X

n=0

λ_n, completing the proof of the theorem.

Corollary 1.3: If we want to get the same limit l, we have to choose the sequence{λ_n} such that

∞

X

n=0

λ_n= 1, an example being theY method of Srini- vasan.

Corollary 1.4: TheY and(N, p_n)methods are consistent.

We make use of well-known properties of analytic elements (a general reference in this direction is [1]) to prove our next result.

Theorem 1.5: Let |λ_n| ≤ |λ₀|, n= 0,1,2,· · ·. If M({a_n}) converges to l, then M((N, p_n)({a_n})) converges tol too.

Proof: Let F be a complete, algebraically closed extension of K; let U be the disk |x| ≤1in F;let H(U) be theF-algebra of analytic elements in U,which is known as the set of restricted power series with coefficients inF.

LetAbe the algebra of analytic functions in the diskD ofF :|x|<1.

Let φ(x) =

∞

X

n=0

λ_nxⁿ. We note that φ ∈ H(U) and φ in invertible in A, since |λ_n| ≤ |λ₀|, n= 0,1,2,· · ·.

LetMcbe the linear mapping defined by M in the space of power series:

ifM({a_n}) ={c_n},then

M(f) =c

∞

X

n=0

c_nxⁿ.

It is easily seen that

Mc(f) =φ(x)f(x).

SinceM({a_n}) has limitl,

φ(x)f(x) =

∞

X

n=0

(l+_n)xⁿ

= l

1−x+, where =

∞

X

n=0

_nxⁿ∈H(U), since lim

n→∞_n= 0. Sinceφ is invertible inA, we have

f(x) = l

(1−x)φ+ φ.

But l

(1−x)φ +

φ ∈ A. Thus f ∈ A and therefore is bounded because so are l

1−x,1

φ, and so l

(1−x)φ +

φ. Consequently the sequence{a_n}is also bounded (see, for instance, [1]).

Let π(x) =

∞

X

n=0

p_nxⁿ. π∈H(U),since lim

n→∞p_n= 0.

Let (N, p[_n)(f) =

∞

X

n=0

c_nxⁿ. Since {P_n} converges to a limit P 6= 0 and since{a_n} is bounded,

∞

X

n=0

c_nxⁿ= 1

P{π(x)f(x) +θ(x)},

whereθ(x) =

∞

X

n=0

θ_nxⁿ, θ_n=

1 Pn − ¹

P

Xⁿ

k=0

p_ka_n−k, n= 0,1,2,· · ·.

Noting that lim

n→∞θ_n= 0, we haveθ ∈H(U).

Thus, we have,

PMc((N, p[_n)(f)) = φ(x)(π(x)f(x) +θ(x))

= lπ(x)

1−x+(x)π(x) +φ(x)θ(x)

= lπ(1)

1−x+ l(π(x)−π(1))

1−x +(x)π(x) +φ(x)θ(x).

P.N. Natarajan

It is well-known (see [1]) thatx−1dividesπ(x)−π(1) inH(U)and so PMc((N, p[_n)(f)) = lπ(1)

1−x+τ(x), where τ(x)∈H(U). Sinceπ(1) =P,

Mc((N, p[_n)(f)) = l 1−x+ 1

Pτ(x),

which proves thatM((N, p_n)({a_n})) has limitl. This completes the proof of the theorem.

Acknowledgement. I thank the referee for his constructive suggestions, which enabled me to present the material in a much better form.

References

[1] A. Escassut. Analytic elements inp-adic Analysis. World Scientific Pub- lishing Co., 1995.

[2] A.F. Monna. Sur le théorème de Banach-Steinhaus. Indag. Math., 25: 121–131, 1963.

[3] P.N. Natarajan. Multiplication of series with terms in a non-archimedean field. Simon Stevin, 52: 157–160, 1978.

[4] P.N. Natarajan. On Nörlund method of summability in non-archimedean fields. J.Analysis, 2: 97–102, 1994.

[5] P.N. Natarajan and V.Srinivasan. Silvermann-Toeplitz theorem for dou- ble sequences and series and its application to Nörlund means in non- archimedean fields. Ann.Math. Blaise Pascal, 9: 85–100, 2002.

[6] V.K. Srinivasan. On certain summation processes in the p-adic field.

Indag. Math., 27: 368–374, 1965.

P.N. Natarajan

Ramakrishna Mission Vivekananda College

Department of Mathematics Mylapore

Chennai 600 004 INDIA

[email protected]