MATHEMATICS

A. A. MELENTSOV and E. B. MURAEV

ON THE THEORY OF SUMMATION OF DOUBLE SERIES BY BOREL METHODS

(Presented by Academician A. N. Kolmogorov on 2 XI 1959)

In the works of V. G. Chelidze and V. A. Berekashvili \((^{1,2})\), the Borel summation of double series is considered, defined as follows:

Let a double series be given

\[ \sum_{i,k=0}^{\infty} a_{ik} \tag{1} \]

and its partial sums

\[ A_{mn}=\sum_{i=0}^{m}\sum_{k=0}^{n} a_{ik}. \]

Suppose that the double power series

\[ A(x,y)=\sum_{i,k=0}^{\infty} A_{ik}\frac{x^i y^k}{i!k!} \]

converges for all values \(x \ge 0\) and \(y \ge 0\).

The series (1) is called \(B_\lambda\)-summable to the sum \(S\) if

\[ \lim_{(x,y)_\lambda \to \infty} e^{-(x+y)}A(x,y)=S. \]

By the symbol \((x,y)_\lambda \to \infty\) we understand tending to infinity inside the sector defined by the inequalities

\[ \lambda \le \frac{y}{x} \le \frac{1}{\lambda}, \qquad 0<\lambda<1. \]

We call the series (1) \(B\)-summable to the sum \(S\) if

\[ \lim_{x\to\infty,\ y\to\infty} e^{-(x+y)}A(x,y)=S. \]

The latter means that for every \(\varepsilon>0\) there exists a number \(T(\varepsilon)\) such that

\[ \left|e^{-(x+y)}A(x,y)-S\right|<\varepsilon \]

as soon as \(x>T\) and \(y>T\).

V. G. Chelidze \((^1)\) proved the following proposition:

If the double series (1) converges and has sum \(S\), and the partial sums \(A_{mn}\) of this series satisfy the condition

\[ |A_{mn}| \le M(m+1)^\alpha (n+1)^\beta, \tag{2} \]

where \(M,\alpha,\beta\) are positive numbers independent of \(m\) and \(n\), then the double series (1) is \(B_\lambda\)-summable to the sum \(S\) for all values of the number \(\lambda\) \((0<\lambda<1)\).

V. A. Berekashvili showed \((^2)\) that condition (2) can be replaced by the weaker one, namely

\[ |A_{mn}| \leqslant M a^{o(n)+o(n)}, \tag{3} \]

where \(M\) and \(a\) are positive numbers independent of \(m\) and \(n\).

We have succeeded in proving the regularity of the \(B_\lambda\)-method under more general conditions than those of V. G. Chelidze and V. A. Berekashvili.

Theorem 1. Let the double series (1) converge, have sum \(S\), and moreover satisfy two conditions:

\[ \left| \sum_{k=0}^{\infty} A_{ik}\frac{y^k}{k!} \right| \leqslant M_i e^{(1+\lambda')y}, \tag{4} \]

\[ \left| \sum_{i=0}^{\infty} A_{ik}\frac{x^i}{i!} \right| \leqslant N_k e^{(1+\lambda')x}, \tag{5} \]

where \(M_i, N_k, \lambda' < 1\) are positive numbers independent of \(x\) and \(y\). Then the series (1) is \(B_\lambda\)-summable to the sum \(S\) for \(\lambda>\lambda'\).

Let the double power series

\[ a(x,y)=\sum_{i,k=0}^{\infty} a_{ik}\frac{x^i y^k}{i!k!} \]

converge for all values of \(x\) and \(y\) (i.e., \(a(x,y)\) is an entire function). Put

\[ \Phi(x,y)=\int_0^x\int_0^y e^{-(t+\tau)}a(t,\tau)\,dt\,d\tau. \]

The series (1) is called \(B'_\lambda\)-summable to the sum \(S\) if

\[ \lim_{(x,y)_\lambda\to\infty} \Phi(x,y)=S. \]

If

\[ \lim_{x\to\infty,\;y\to\infty} \Phi(x,y)=S, \]

then we call the series (1) \(B'\)-summable to the sum \(S\).

The methods \(B\) and \(B'\) are limiting cases of the methods \(B_\lambda\) and \(B'_\lambda\) as \(\lambda\to 0\). Restricting ourselves to the consideration of such double series (1) for which the functions \(a(x,y)\) and \(A(x,y)\) are simultaneously entire functions, we prove the following theorems:

Theorem 2. The methods \(B_\lambda\) and \(B'_\lambda\) are equivalent if and only if the relation

\[ \lim_{(x,y)_\lambda\to\infty} \left[ \Phi''_{xy}(x,y)+\Phi'_x(x,y)+\Phi'_y(x,y) \right]=0 \]

holds.

An analogous assertion holds for the methods \(B\) and \(B'\).

This theorem is an analogue of a well-known theorem in the theory of Borel summation of simple series (\((^3)\), p. 229, Theorem 123).

Theorem 3. If the function \(\Phi(x,y)\), defined in the domain \(x\geqslant 0,\ y\geqslant 0\), has continuous first partial derivatives and a continuous mixed derivative of second order and satisfies the conditions:

\[ 1)\quad \Phi(x,0)=O(e^{\lambda x})\quad \text{as } x\to\infty; \]

\[ \Phi(0,y)=O(e^{\lambda y})\quad \text{as } y\to\infty,\quad 0<\lambda<1; \]

2)
\[ \left|\Phi''_{xy}(x,y)+\Phi'_x(x,y)+\Phi'_y(x,y)+\Phi(x,y)\right|\le Me^{\lambda(x+y)} \]
for all \(x\ge 0,\ y\ge 0\);

3)
\[ \lim_{x\to\infty,\ y\to\infty} \left[\Phi''_{xy}(x,y)+\Phi'_x(x,y)+\Phi'_y(x,y)+\Phi(x,y)\right]=S, \]
then
\[ \lim_{(x,y)_{\lambda+\delta}\to\infty}\Phi(x,y)=S \quad\text{for }0<\delta<1-\lambda. \]

Proof. Without loss of generality, one may assume that \(S=0\). Putting
\[ \Phi''_{xy}(x,y)+\Phi'_x(x,y)+\Phi'_y(x,y)+\Phi(x,y)=\varepsilon(x,y), \]
we obtain
\[ \frac{\partial^2}{\partial x\,\partial y}\left[e^{x+y}\Phi(x,y)\right] = e^{x+y}\varepsilon(x,y); \]
therefore,
\[ \Phi(x,y)=e^{-y}\Phi(x,0)+e^{-x}\Phi(0,y)-e^{-(x+y)}\Phi(0,0)+ e^{-(x+y)}\int_0^x\int_0^y e^{t+\tau}\varepsilon(t,\tau)\,dt\,d\tau. \]

Using the hypotheses of the theorem and the last relation, it is easy to show that
\[ \lim_{(x,y)_{\lambda+\delta}\to\infty}\Phi(x,y)=0. \]

This implies the validity of our theorem.

This theorem, in a simpler formulation, is known for functions of one variable ((\(^{3}\), p. 138, Theorem 53). As in the case of one variable ((\(^{3}\), p. 230, Theorem 124), it finds application in the theory of Borel summation.

Theorem 4. Let \(0<\lambda'<\lambda<1\). If the double series (1) is summable by the method \(B\) to the value \(S\) and
\[ |A(x,y)|\le Me^{(1+\lambda')\lambda(x+y)}, \]
where \(M\) is a positive number independent of \(x\) and \(y\), then the series (1) is summable by the method \(B_{\lambda}'\) to the value \(S\).

For the proof it suffices to establish the relation
\[ e^{-(x+y)}A(x,y)=\Phi''_{xy}(x,y)+\Phi'_x(x,y)+\Phi'_y(x,y)+\Phi(x,y), \]
and then to use the preceding theorem.

Remark. With respect to double series satisfying the condition of Theorem 4, the method \(B_{\lambda}'\) is stronger than the method \(B\). Bearing in mind an example of a single series summable by the method \(B'\) and not summable by the method \(B\) ((\(^{3}\), p. 230), it is easy to construct a double series summable by the method \(B_{\lambda}'\) and not summable by the method \(B\).

Ural State University
named after A. M. Gorky

Received
6 X 1959

REFERENCES

\(^{1}\) V. G. Chelidze, Reports of the Academy of Sciences of the Georgian SSR, 8, No. 8 (1947).
\(^{2}\) V. A. Berekashvili, Reports of the Academy of Sciences of the Georgian SSR, 14, No. 4, 193 (1953).
\(^{3}\) G. Hardy, Divergent Series, Moscow, 1951.