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MATHEMATICS
S. P. GEISBERG
ON SOME PROPERTIES OF SUMMATION METHODS
(Presented by Academician V. I. Smirnov, 15 X 1960)
Notation. \(A=(a_{nk})\) is a complex matrix; \(A^\varphi=(a_{n\varphi_k})\) is a submatrix of the matrix \(A\), in which the second index runs through the values of a subsequence \(\varphi=\{\varphi_k\}\) of the natural numbers. If \(A\) is the matrix \((a_{nk})\), then by \(A_1, A_2, A_3\) we denote the sets of sequences \(\xi=\{\xi_k\}\) of complex numbers for which, respectively*:
\[ \sum_n \left|\sum_k a_{nk}\xi_k\right|<\infty; \]
\[ \sup_n \left|\sum_k a_{nk}\xi_k\right|<\infty; \]
the sequence \(\eta=\{\eta_n\}\), \(\eta_n=\sum_k a_{nk}\xi_k\), converges.
Further, \(Q=\{q_k(x)\}\) is a sequence of functions continuous on \([0,\infty)\); \(Q^\varphi=\{q_{\varphi_k}(x)\}\) is its subsequence; \(Q_1, Q_2, Q_3\) are the sets of sequences \(\xi=\{\xi_k\}\) for which, respectively,
\[ \int_0^\infty \left|\sum q_k(x)\xi_k\right|\,dx<\infty; \]
\[ \sup_{0<\alpha<\beta<\infty} \left| \int_\alpha^\beta \left(\sum q_k(x)\xi_k\right)\,dx \right|<\infty; \]
there exists
\[ \int_0^\infty \left(\sum q_k(x)\xi_k\right)\,dx. \]
The matrix \(A\) (the sequence \(Q\)) is called a summation method; the sets \(A_3, A_2, A_1\) \((Q_3, Q_2, Q_1)\) are called, respectively, the fields of summability, boundedness, and absolute summability.
By \(c, c_0, m\) we denote, respectively, the sets of convergent sequences, sequences converging to zero, and bounded sequences; \(e\) is the set of sequences \(\xi=\{\xi_k\}\) for which \(\sum |\xi_k|<\infty\).
For a matrix \(A\) and a sequence \(\xi=\{\xi_k\}\) we put
\[ \|\xi\|_{A_1}^p = \sum_n \left| \sum_{k=p}^{\infty} a_{nk}\xi_k \right|; \qquad \|\xi\|_{A_2}^p = \|\xi\|_{A_3}^p = \sup_n \left| \sum_{k=p}^{\infty} a_{nk}\xi_k \right|; \]
\[ \|\xi\|_{A_1}^p = \sum_{k=p}^{\infty}|\xi_k|; \qquad \|\xi\|_{2}^p = \|\xi\|_{3}^p = \sup_{k\ge p}|\xi_k|. \]
Similarly, for sequences \(Q\) and \(\xi=\{\xi_k\}\) we put
\[ \|\xi\|_{Q_1}^p = \int_0^\infty \left| \sum q_k(x)\xi_k \right|\,dx; \qquad \|\xi\|_{Q_2}^p = \|\xi\|_{Q_3}^p = \sup_{0<\alpha<\beta<\infty} \left| \int_\alpha^\beta \left(\sum q_k(x)\xi_k\right)\,dx \right|. \]
In § 1 of the present note we obtain some relations between concrete summation methods; § 2 is devoted to the study of the interrelations of the fields of different summation methods. In § 3 we give several properties of operators in the space \(c_0\), analogous to properties of matrix summation methods.
\[ \text{* If the limits of summation are not indicated, summation is carried out over all integer values of the index from }0\text{ to }\infty. \]
§ 1. We shall consider the following methods:
- The Cesàro method of order \(a\), or the method \((C,a)\) \((a>0)\):
\[ a_{nk}= \begin{cases} 1, & n=k=0,\\[6pt] \dfrac{k\binom{\,n-k+a-1\,}{a-1}}{n\binom{\,n+a\,}{a}}, & n\geqslant 1;\ n\geqslant k\geqslant 0,\\[10pt] 0, & k>n. \end{cases} \]
- The Euler method of order \(a\), or the method \((E,a)\) \((a>0)\):
\[ a_{nk}= \begin{cases} \dfrac{a^{\,n-k}}{(a+1)^{n+1}}\binom{n}{k}, & n\geqslant k\geqslant 0,\\[8pt] 0, & k>0. \end{cases} \]
- Abel’s method, or method \(A\):
\[ q_k(x)=ke^{-kx}. \]
- The integral Borel method, or method \(B\):
\[ q_k(x)=e^{-x}\frac{x^k}{k!}. \]
Theorem 1. In order that: a) \((C,a)_1^\varphi \subset e\); b) \(A_1^\varphi \subset e\), it is necessary and sufficient that there exist an index \(s\) and a number \(r>1\) such that, for \(k>s\),
\[ \varphi_{k+1}/\varphi_k>r. \tag{1} \]
Theorem 2. In order that \((E,a)_1^\varphi \subset e\), it is necessary and sufficient that the following conditions be fulfilled:
1) If \(\xi=\{\xi_k\}\in (E,a)_1^\varphi\), then \(\xi_k=O(\gamma^k)\) for some \(\gamma>0\).
2) There exist an index \(p\) and a number \(q>0\) such that, for all \(k>p\),
\[ \varphi_{k+1}-\varphi_k>q\sqrt{\varphi_k}. \tag{2} \]
Theorem 3. In order that \(B_1^\varphi \subset e\), it is necessary and sufficient that the following conditions be fulfilled:
1) If \(\xi=\{\xi_k\}\in B_1^\varphi\), then \(\xi_k=O(\gamma^k)\) for some \(\gamma>0\).
2) There exist an index \(p\) and a number \(q>0\) such that, for all \(k>p\),
\[ \varphi_{k+1}-\varphi_k>q\sqrt{\varphi_k}. \tag{3} \]
The sufficiency of condition (1) in case a) of Theorem 1 was proved in paper (1). The necessity of conditions (1), (2), (3) is proved with the aid of the following two theorems, which are of independent interest.
Theorem 4. Let the method \(A\) be given by the matrix \((a_{nk})\), and suppose that: 1) \(e\in A_1\); 2) \((m-e)\cap A_1=\varnothing\); 3) \(\sum_k |a_{nk}|<\infty\) \((n=0,1,\ldots)\). Then there exist an index \(p\) and a number \(q>0\) such that, for all \(\xi\in e\), we have
\[ \|\xi\|_{A_1}^{p}>q\|\xi\|_1^{p}. \]
Theorem 5. Let the method \(Q\) be given by a sequence \(\{q_k(x)\}\) of functions continuous on \([0,\infty)\), and suppose that: 1) \(e\in Q_1\); 2) \((m-e)\cap Q_1=\varnothing\); 3) \(\sum_k |q_k(x)|<\infty\) \((x\in[0,\infty))\). Then there exist an index \(p\) and a number \(q>0\) such that, for all \(\xi\in e\), we have
\[ \|\xi\|_{Q_1}^{p}>q\|\xi\|_1^{p}. \]
Theorems analogous to Theorems 1 and 2 for summability fields are proved in (2); however, the necessity of conditions (1), (2), (3) is established there only for condition (1) in the case of the method \((C,\alpha)\), \(0<\alpha\leqslant 1\). For the remaining cases, necessity can be established by the same device as in the proof of Theorems 1 and 2, with the aid of Theorem 9 established below.
§ 2. Let \(A\) and \(B\) be two summability methods given by matrices \((a_{nk})\) and \((b_{nk})\), respectively, with
\[ \sum_k \left(|a_{nk}|+|b_{nk}|\right)<\infty,\qquad n=0,1,\ldots . \]
Theorem 6. If for some \(i\) and \(j\), \(1 \leqslant i,j \leqslant 3\), we have \(A_i \subset B_j\), then for every \(\delta>0\) there exist an index \(p(\delta)\) and a number \(q(\delta)>0\) such that for all \(\xi \in A_i \cap m\) we have
\[ \|\xi\|^{(\delta)}_{A_i}+\delta q(\delta)\|\xi\|^{p(\delta)}_{2} >q(\delta)\|\xi\|^{p(\delta)}_{B_j}. \]
Remark. For finite-row matrices one may put \(\delta=0\) \((p(0)<\infty,\ q(0)>0)\).
For methods \(P\) and \(Q\), given by the sequences \(\{p_k(x)\}\) and \(\{q_k(x)\}\), with
\[ \sum_k (|p_k(x)|+|q_k(x)|)<\infty, \]
analogous relations hold if everywhere \(A_i\) is replaced by \(P_i\) and \(B_i\) by \(Q_i\).
As consequences of Theorem 6 we obtain:
Theorem 7. If \(A_1 \subset e\), then there exist an index \(p\) and a number \(q>0\) such that, for all \(\xi \in A_1 \cap m\),
\[ \|\xi\|^p_{A_1}>q\|\xi\|_1^p. \]
Theorem 8. If \(A_1 \subset m\), then there exist an index \(p\) and a number \(q>0\) such that, for all \(\xi \in A_1 \cap m\),
\[ \|\xi\|^p_{A_1}>q\|\xi\|_2^p. \]
Theorem 9. If \(A_2 \subset m\) \((A_3 \subset m)\), then there exist an index \(p\) and a number \(q>0\) such that for all \(\xi \in A_2 \cap m\) \((\xi \in A_3 \cap m)\) we have
\[ \|\xi\|^p_{A_2}>q\|\xi\|_2^p. \]
Theorem 9 for the case \(A_3 \subset m\), with the matrix \(A\) the matrix of Toeplitz, is asserted in \((^3)\). With the help of Theorem 9 the following result, due to Zeller \((^4)\), is easily proved:
Theorem 10. If \(c \subset A_3 \subset m\), then \(c=A_3\).
Theorem 8 leads to an analogue of Theorem 10 for fields of absolute summability.
Theorem 11. If \(e \subset A_1 \subset m\) and, for every \(p\), the bounded solutions of the system of equations
\[ \sum_n \tau_n a_{nk}=0 \qquad (k=p,\ p+1,\ldots) \]
are contained in \(e\), then \(A_1 \subset c\).
Applying Theorem 8 to methods 1 and 2 and using the results of § 1, we obtain:
Theorem 12. If \((C,\alpha)^{\varphi}_{1}\subset m\), then \((C,\alpha)^{\varphi}_{1}=e\).
Theorem 13. If \((E,\alpha)^{\varphi}_{1}\subset m\), then \((E,\alpha)^{\varphi}_{1}=e\).
§ 3. Introduce on \(c_0\) the norm \(\|\xi\|=\sup_k|\xi_k|\). Then \(c_0\) will be a \(B\)-space. Let \(A\) be a linear continuous operator mapping \(c_0\) into itself. Denote by \(c_n\) the \(n\)-th adjoint \(c_0\)-space and by \(A_n\) the \(n\)-th adjoint operator to \(A\).
Theorem 14. If from \((A_4,x)\in c_0\) it follows that \(x\in c_2\), then from \((A_4,x)\in c_0\) it follows that \(x\in c_0\); moreover, there exists \(q>0\) such that for all \(x\in c_0\)
\[ \|(A_4,x)\|>q\|x\|. \]
Further, it is not difficult to see that the operator \(A_1\) can be represented in the form
\[ \eta=(A_1,\xi),\qquad \eta=\{\eta_n\},\qquad \xi=\{\xi_k\},\qquad \eta_n=\sum_k a_{nk}\xi_k, \]
where the matrix \((a_{nk})\) depends only on the operator \(A\).
Theorem 15. If from \((A_4,x)\in c_1\) it follows that \(x\in c_2\), and for each \(p\) the bounded solutions of the system of equations
\[ \sum_n \tau_n a_{nk}=0 \qquad (k=p,\ p+1,\ldots) \]
are contained in \(c_1\), then from \((A_4,x)\in c_1\) it follows that \(x\in c_0\).
Theorems 14 and 15 are, obviously, in a certain sense analogues of Theorems 10 and 11.
Tartu State University
Received
7 IX 1960
CITED LITERATURE
- N. Obreshkov, Dokl. Bulgarsk. Akad. Nauk, 8, 1 (1955).
- K. Zeller, W. Meyer-König, Math. Zs., 66, No. 3 (1956).
- A. L. Brudno, DAN, 91, No. 2 (1953).
- K. Zeller, Math. Zs., 53, 5 (1951).