P.N. NATARAJAN

V. SRINIVASAN

Convolution of Nörlund methods in non-archimedean fields

*Annales mathématiques Blaise Pascal*, tome 4, n° 2 (1997), p. 41-47

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# Convolution of Nörlund methods in non-archimedean fields

P.N. Natarajan

Department of Mathematics

Ramakrishna Mission Vivekananda College

Chennai - 600 004

India

V. Srinivasan

Department of Mathematics

V. Ramakrishna Polytechnic

Chennai - 600 019

India

## Abstract

In the present paper we obtain a few inclusion theorems for the convolution of Nörlund methods in the form $(N, r_n) \subseteq (N, p_n) * (N, q_n)$ in complete, non-trivially valued, non-archimedean fields.

Throughout the present paper $K$ denotes a complete, non-trivially valued, non-archimedean field. Infinite matrices and sequences, which are considered in the sequel, have entries in $K$. If $A = (a_{nk})$, $a_{nk} \in K$, $n, k = 0, 1, 2, \dots$ is an infinite matrix, the $A$-transform $Ax = \{(Ax)_n\}$ of the sequence $x = \{x_k\}$, $x_k \in K$, $k = 0, 1, 2, \dots$ is defined by

$$ (Ax)_n = \sum_{k=0}^{\infty} a_{nk} x_k, \quad n = 0, 1, 2, \dots, $$

where it is assumed that the series on the right converge. If $\lim_{n \to \infty} (Ax)_n = s$, we say that $\{x_k\}$ is $A$-summable to $s$, written as $x_k \to s(A)$ or $A$-lim $x_k = s$. If $\lim_{k \to \infty} (Ax)_n = s$ whenever $\lim_{k \to \infty} x_k = s$, we say that $A$ is regular. The following result is well-known (see [2], [4]).
**Theorem 1** $A = (a_{nk})$ is regular if and only if

$$ \sup_{n,k} |a_{nk}| < \infty; \qquad (1) $$

$$ \lim_{n \to \infty} a_{nk} = 0, \quad \text{for every fixed } k; \qquad (2) $$

and

$$ \lim_{n \to \infty} \sum_{k=0}^{\infty} a_{nk} = 1. \qquad (3) $$

Any matrix $A$ for which (1) holds is called a $K_r$-matrix. If $A$ and $B$ are two infinite matrices such that $x_k \to s(A)$ implies $x_k \to s(B)$, we say that $A$ is included in $B$, written as $A \subseteq B$. $A$ is said to be row-finite if for $n = 0, 1, 2, \dots$, there exists a positive integer $k_n$ such that $a_{nk} = 0$, $k > k_n$.

Given two infinite matrices $A$ and $B$, their convolution is defined as the matrix $C = (c_{nk})$, where

$$ c_{nk} = \sum_{i=0}^{k} a_{ni} b_{n,k-i}, \quad n, k = 0, 1, 2, \dots \qquad (4) $$

In such a case we write $C = A * B$.

The following properties of the convolution can be easily proved.

1. If $A$ and $B$ are both row-finite or both $K_r$, then their convolution $C$ is row-finite or $K_r$ respectively and their row sums satisfy

$$ \sum_{k=0}^{\infty} c_{nk} = \left( \sum_{k=0}^{\infty} a_{nk} \right) \left( \sum_{k=0}^{\infty} b_{nk} \right), \quad n = 0, 1, 2, \dots \qquad (5) $$

2. If $A, B$ are both regular, then $C$ is regular too.

The Nörlund method of summability i.e., $(N, p_n)$ method in $K$ is defined as follows (see [5]): $(N, p_n)$ is defined by the infinite matrix $(a_{nk})$ where

$$ \begin{aligned} a_{nk} &= \frac{p_{n-k}}{P_n}, \quad k \le n; \\ &= 0, \quad k > n, \end{aligned} $$

where $p_0 \ne 0$, $|p_0| > |p_j|$, $j = 1, 2, \dots$ and $P_n = \sum_{k=0}^{n} p_k$, $n = 0, 1, 2, \dots$. It is to be noted that $|P_n| = |p_0| \ne 0$ so that $P_n \ne 0$, $n = 0, 1, 2, \dots$. The following result is very useful in the sequel.

**Theorem 2** (See [5], Theorem 1.) $(N, p_n)$ is regular if and only if

$$ p_n \to 0, n \to \infty. \qquad (6) $$
The purpose of the present paper is to prove a few inclusion theorems for the convolution of Nörlund methods in the form $(N, r_n) \subseteq (N, p_n) * (N, q_n)$.

We need to define $\{\bar{p}_n\}$ by

$$p_0\bar{p}_0 = 1, \quad p_0\bar{p}_n + p_1\bar{p}_{n-1} + \cdots + p_n\bar{p}_0 = 0, \quad n \ge 1 \qquad (7)$$

$$\text{i.e., } \bar{p}(x) = \sum_{n=0}^{\infty} \bar{p}_n x^n = \frac{1}{\sum_{n=0}^{\infty} p_n x^n} = \frac{1}{p(x)}, \text{ assuming that these series converge.}$$

The following result is an easy consequence of Kojima-Schur theorem (see [2], [4]).

**Lemma 1** Let $A = (a_{nk})$ be a row-finite matrix and $(N, p_n)$ be a regular Nörlund method. Then $A$-lim $x_k$ exists whenever $(N, p_n)$-lim $x_k$ exists if and only if

$$\sup_{0 \le \gamma \le k_n} |P_\gamma \sum_{k=\gamma}^{k_n} a_{nk} \bar{p}_{k-\gamma}| = O(1), \quad n \to \infty; \qquad (8)$$

$$\lim_{n \to \infty} \sum_{k=\gamma}^{k_n} a_{nk} \bar{p}_{k-\gamma} = \delta_{\gamma}, \quad \text{for every fixed } \gamma; \qquad (9)$$

and

$$\lim_{n \to \infty} \sum_{k=0}^{k_n} a_{nk} = \delta. \qquad (10)$$

**Corollary 1** $(N, p_n) \subseteq A$ if and only if (8), (9) and (10) hold with $\delta_\gamma = 0$, $\gamma = 0, 1, 2, \dots$ and $\delta = 1$.

**Corollary 2** If $(N, p_n)$ and $(N, q_n)$ are regular Nörlund methods, then $(N, p_n) \subseteq (N, q_n)$ if and only if $h_n \to 0$, $n \to \infty$ where

$$h(x) = \sum_{n=0}^{\infty} h_n x^n = \frac{\sum_{n=0}^{\infty} q_n x^n}{\sum_{n=0}^{\infty} p_n x^n} = \frac{q(x)}{p(x)}$$

(see [5]).

Let $(N, p_n)$, $(N, q_n)$, $(N, r_n)$ be regular Nörlund methods. Let $p_n(x) = \sum_{i=n}^{\infty} p_i x^i$, $p_0(x) = p(x)$ with similar expressions for $q_n(x)$, $r_n(x)$. Let

$$\begin{align}
\frac{p(x)q(x)}{r(x)} &= \sum_{\gamma=0}^{\infty} \theta_{\gamma} x^{\gamma}; \notag \\
\frac{p_{n+1}(x)q(x)}{r(x)} &= \sum_{\gamma=0}^{\infty} \alpha_{n\gamma} x^{\gamma}; \tag{11} \\
\frac{p(x)q_{n+1}(x)}{r(x)} &= \sum_{\gamma=0}^{\infty} \beta_{n\gamma} x^{\gamma}, \notag
\end{align}$$
and

$$ \frac{1}{r(x)} \{p(x)q(x) - p_{n+1}(x)q(x) - p(x)q_{n+1}(x)\} = \sum_{\gamma=0}^{\infty} \varphi_{n\gamma}x^{\gamma}. \quad (12) $$

It now follows that

$$ \varphi_{n\gamma} = \theta_{\gamma} - \alpha_{n\gamma} - \beta_{n\gamma} \quad (13) $$

and

$$ \alpha_{n\gamma} = \beta_{n\gamma} = 0, \quad 0 \le \gamma \le n. \quad (14) $$

Taking $C = (N, p_n) * (N, q_n)$, C is a row-finite matrix with

$$ c_{nk} = \frac{1}{P_n Q_n} \sum_{i=\max(0, k-n)}^{\min(k,n)} p_{n-i} q_{n-k+i} \quad \text{with } k_n = 2n. \quad (15) $$

We write

$$ f_{n\gamma} = \sum_{k=\max(0, 2n-\gamma)}^{2n} c_{nk} \bar{r}_{k+\gamma-2n}, \quad n, \gamma \ge 0. \quad (16) $$

**Lemma 2**

$$ P_n Q_n f_{n\gamma} = \varphi_{n\gamma}, \quad 0 \le \gamma \le 2n+1. \quad (17) $$

**Proof.** The result follows as in [6].

**Theorem 3** $(N, p_n) * (N, q_n)$-lim $x_k$ exists whenever $(N, r_n)$-lim $x_k$ exists if and only if

$$ \sup_{0 \le \gamma \le 2n} |R_{2n-\gamma} \varphi_{n\gamma}| = O(1), \quad n \to \infty; \quad (18) $$

and

$$ \lim_{n \to \infty} \frac{\varphi_{n, 2n-\gamma}}{P_n Q_n} = \delta_{\gamma}, \quad \text{for every fixed } \gamma. \quad (19) $$

**Proof.** Let $(N, p_n) * (N, q_n)$-lim $x_k$ exist whenever $(N, r_n)$-lim $x_k$ exists. Applying Lemma 1 with $(N, p_n) = (N, r_n)$ and $A = (N, p_n) * (N, q_n) = (c_{nk})$, we have,

$$ \sup_{0 \le \gamma \le 2n} |R_{\gamma} \sum_{k=\gamma}^{2n} c_{nk} \bar{r}_{k-\gamma}| = O(1), \quad n \to \infty \quad (20) $$

and

$$ \lim_{n \to \infty} \sum_{k=\gamma}^{2n} c_{nk} \bar{r}_{k-\gamma} = \delta_{\gamma}, \quad \text{for every fixed } \gamma. \quad (21) $$

Using (16) and (20), we get

$$ \sup_{0 \le \gamma \le 2n} |R_{2n-\gamma} f_{n\gamma}| = O(1), \quad n \to \infty. \quad (22) $$
Using (17), we note that $|f_{n\gamma}| = \frac{|\varphi_{n\gamma}|}{|p_0| |q_0|}$ since $|P_n| = |p_0|$ and $|Q_n| = |q_0|$. Consequently, in view of (22), we get

$$ \sup_{0 \le \gamma \le 2n} |R_{2n-\gamma}\varphi_{n\gamma}| = O(1), \quad n \to \infty. $$

In view of (16) and (21), we have

$$ \lim_{n \to \infty} f_{n,2n-\gamma} = \delta_{\gamma}, \quad \text{for every fixed } \gamma. $$

Now, using (17), we get

$$ \lim_{n \to \infty} \frac{\varphi_{n,2n-\gamma}}{P_n Q_n} = \delta_{\gamma}, \quad \text{for every fixed } \gamma. $$

Thus (18) and (19) hold. Conversely (18) and (19) imply (20) and (21) respectively. Using (5), we have, $\lim_{n \to \infty} \sum_{k=0}^{2n} c_{nk} = 1$. Using Lemma 1, the result follows, completing the proof of the theorem.

**Corollary 3** $(N, r_n) \subseteq (N, p_n) \ast (N, q_n)$ if and only if (18) and (19) hold with $\delta_\gamma = 0$.

**Corollary 4** If $\lim_{n \to \infty} \bar{r}_n = 0$, then $(N, r_n) \subseteq (N, p_n) \ast (N, q_n)$ if and only if (18) holds.

**Proof.** The result follows using (9) and the fact that $(N, p_n) \ast (N, q_n)$ is regular.

**Theorem 4** If

$$ \varphi_{n,2n-\gamma} = o(1), \quad n \to \infty, \quad \text{for every fixed } \gamma, \qquad (23) $$

and either

$$ \varphi_{n\gamma} = O(1), \quad n, \gamma \to \infty, \qquad (24) $$

or

$$ \theta_{\gamma}, \alpha_{n\gamma}, \beta_{n\gamma} = O(1), \quad n, \gamma \to \infty, \qquad (25) $$

then

$$ (N, r_n) \subseteq (N, p_n) \ast (N, q_n). $$

**Proof.** Using (23), (19) follows with $\delta_\gamma = 0$ since $|P_n| = |p_0|$ and $|Q_n| = |q_0|$. Because of (13) and (25), (24) holds. So if (24) or (25) holds, (18) holds since $R_n = O(1)$, $n \to \infty$, $(N, r_n)$ being a regular method. The result now follows from Corollary 3.

We shall now take up an application of Theorem 4.
**Theorem 5** Let $\overline{p}_n, \overline{q}_n \to 0$, $n \to \infty$ and $t_n = p_0q_n + p_1q_{n-1} + \cdots + p_nq_0$, $n = 0, 1, 2, \dots$ Then

$$ (N, t_n) \subseteq (N, p_n) * (N, q_n) $$

and

$$ (N, p_n) \subseteq (N, t_n), (N, q_n) \subseteq (N, t_n). $$

**Proof.** We apply Theorem 4 with $r_n = t_n$. With the usual notation we have $t(x) = p(x)q(x)$ and $\bar{t}(x) = \overline{p}(x)\overline{q}(x)$. Since $\overline{p}_n, \overline{q}_n \to 0$, $n \to \infty$, $\bar{t}_n \to 0$, $n \to \infty$ (see [3], Theorem 1). Consequently (23) follows using (9). In view of (11), we have,

$$ \sum_{\gamma=0}^{\infty} \theta_{\gamma} x^{\gamma} = \frac{p(x)q(x)}{t(x)} = 1, $$

so that

$$ \theta_0 = 1 \text{ and } \theta_{\gamma} = 0, \gamma \ge 1; $$

$$ \sum_{\gamma=0}^{\infty} \alpha_{n\gamma} x^{\gamma} = \frac{p_{n+1}(x)q(x)}{t(x)} = p_{n+1}(x)\bar{p}(x), $$

so that

$$ \alpha_{n\gamma} = \sum_{\lambda=0}^{\gamma-(n+1)} \bar{p}_{\lambda} p_{\gamma-\lambda}, \quad \gamma \ge n+1; \\ = 0, \quad 0 \le \gamma \le n. $$

Consequently $\alpha_{n\gamma} = O(1)$, $n, \gamma \to \infty$. Similarly $\beta_{n\gamma} = O(1)$, $n, \gamma \to \infty$. In view of Theorem 4, $(N, t_n) \subseteq (N, p_n) * (N, q_n)$. Now $\frac{t(x)}{p(x)} = q(x)$ and $q_n \to 0$, $n \to \infty$, $(N, q_n)$ being regular, by (6). So by Corollary 2, $(N, p_n) \subseteq (N, t_n)$. Similarly $(N, q_n) \subseteq (N, t_n)$. The proof of the theorem is now complete.

## References

[1] G. Bachman, *Introduction to p-adic numbers and valuation theory*, Academic Press, 1964.

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

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

[4] P.N. Natarajan, *Criterion for regular matrices in non-archimedean fields*, *J. Ramanujan Math. Soc.* 6 (1991), 185-195.
[5] P.N. Natarajan, On Nörlund method of summability in non-archimedean fields, *J. Analysis* 2 (1994), 97-102.

[6] D.C. Russell, Convolution of Nörlund summability methods, *Proc. London Math. Soc.* 9 (1959), 1-20.

Manuscrit reçu en Septembre 1997