MATHEMATICS

O. A. ZIZA

ON THE SUMMATION OF ORTHOGONAL SERIES BY EULER METHODS

(Presented by Academician A. N. Kolmogorov, 9 XII 1961)

Consider an orthogonal series from \(L^2\), i.e., a series of the form

\[ \sum_{n=0}^{\infty} a_n \varphi_n(x), \qquad x \in [a,b], \tag{1} \]

where \(\{\varphi_n(x)\}\), \(n=0,1,\ldots\), is an orthonormal \((ON)\) system of functions on the interval \([a,b]\), and

\[ \sum_{n=0}^{\infty} a_n^2 < \infty . \tag{2} \]

We introduce two definitions.

Definition 1*. Two summation methods \(T_1\) and \(T_2\) are called equivalent in the space \(L^2\) if, from the summability of every orthogonal series from \(L^2\) (1) by one of these methods on any set \(D\) of positive measure, there follows the summability of this series by the other method almost everywhere on \(D\).

Definition 2**. Let \(\{n_m\}\), \(m=0,1,\ldots\), be an increasing sequence of natural numbers. We shall say that the numerical series

\[ \sum_{n=0}^{\infty} u_n \tag{3} \]

with partial sums \(U_n\) is summed by the method \(T[n_m]\) to the sum \(S\), if \(\lim\limits_{m\to\infty} U_{n_m}=S\).

It is known that all Cesàro methods \((C,\alpha)\), \(\alpha>0\), are equivalent in the space \(L^2\) to the method \(T[2^m]\). For \(\alpha=1\) this theorem was proved by S. Kaczmarz; for any \(\alpha>0\), by A. Zygmund \((^2)\), pp. 219 and 222).

An analogous theorem was also proved by A. Zygmund for methods of normal Riesz means \((^3)\). Recall that the series (3) is summed to the sum \(S\) by the method of normal Riesz means \((R,P_n,\delta)\), where \(0<P_n<P_{n+1}\to\infty\), if

\[ \lim_{\omega\to\infty} \sum_{P_n<\omega} \left(1-\frac{P_n}{\omega}\right)^\delta u_n = S \]

(see \((^4)\), p. 115).

A. Zygmund’s result is as follows: the Riesz methods \((R,P_n,\delta)\) for all \(\delta>0\) are equivalent in the space \(L^2\) to the method \(T[n_m]\), where the numbers \(n_m\) are determined from the condition

\[ P_{n_m} \leqslant 2^m < P_{n_m+1}. \]

D. E. Menshov showed \((^1)\) that for an arbitrary summation method a theorem of this type does not hold: there exists a matrix perfectly regular—

* This definition differs somewhat from Definition 1 of paper \((^1)\), p. 353.
** See \((^1)\), Definition 2, p. 353.

method \(T\), not equivalent in the space \(L^2\), in the sense of Definition 1 given above, to any method of the form \(T[n_m]\)*.

In the present paper we consider the summation of orthogonal series from \(L^2\) by the Euler and Borel methods.

The series (3) is summable to the sum \(S\) by the Euler method \((E,q)\), \(0<q<\infty\), if
\[ \lim_{n\to\infty} E_n^q=S, \]
where
\[ E_n^q=\frac{1}{(1+q)^n}\sum_{k=0}^{n}\binom{n}{k}q^{\,n-k}U_k . \]

The series (3) is summable to \(S\) by the exponential Borel method \(B\), if
\[ \lim_{x\to\infty} B(x)=S, \]
where
\[ B(x)=e^{-x}\sum_{n=0}^{\infty}\frac{x^n}{n!}U_n, \]
moreover the latter series converges for all \(x\geq 0\) ((4), pp. 224 and 229)**.

Let us formulate the main theorem.

Theorem 1. In the space \(L^2\), all Euler methods \((E,q)\), \(q>0\), and the Borel method are equivalent to one another and equivalent to the method \(T[m^2]\).

We note that summability of orthogonal series by Euler methods was considered by Yu. Meder \((5,6)\). In particular, he obtained the following results: 1) from the summability almost everywhere on \([a,b]\) of the series (1) by any Euler method \((E,q)\) there follows the summability almost everywhere of this series by the method \(T[2^m]\); 2) there exists an orthogonal series from \(L^2\), summable almost everywhere on \([a,b]\) by the method \(T[2^m]\) and almost nowhere summable by the method \((E,q)\) for any \(q>0\).

Let us outline the proof of Theorem 1. Denote the \((E,q)\)-mean of the series (1) by \(E_n^q(x)\).

Lemma 1. The series
\[ \sum_{n=1}^{\infty}\sqrt n\,\bigl[E_n^q(x)-E_{n-1}^q(x)\bigr]^2 \]
converges almost everywhere on \([a,b]\) for every \(q>0\).

From this lemma follows the equivalence in \(L^2\) of all methods \((E,q)\) and the method \(B\). Indeed***, if the series (1) is summable on a set \(D\), \(mD>0\), by the method \(B\), then, by Szasz’s theorem ((7), § 4), the method \(B\) also sums on \(D\) the \((E,q)\)-means of this series. From the latter fact and Lemma 1, using the Tauberian theorem of P. Schmidt (see, for example, (4), p. 385), we obtain convergence almost everywhere on \(D\) of the sequence \(\{E_n^q(x)\}\), i.e. summability of the series (1) by the method \((E,q)\). Since the method \(B\) is stronger than all methods \((E,q)\) (see, for example, (4), p. 230), it follows that all methods \((E,q)\) and the method \(B\) are equivalent in \(L^2\).

Let us now outline the proof of the equivalence of the methods \((E,1)\) and \(T[m^2]\). For this we shall need the following modification of the method \((R,P_n,\delta)\). We shall say that the series (3) is summable to \(S\) by the method \((\bar R,p_n)\), if \(p_0>0\),
\[ p_n\geq 0,\qquad P_n=\sum_{k=0}^{n}p_k\to\infty \]
and
\[ \lim_{n\to\infty}\frac{1}{P_n}\sum_{k=0}^{n}p_kU_k=S \]
((4), p. 79).

* This follows from the following. The method constructed in (1) is such that for any sequence \(\{n_m\}\) either there is a series of the form (1), summable by this method almost everywhere but not summable almost nowhere by the method \(T[n_m]\), or there is a series of the form (1), summable almost everywhere by the method \(T[n_m]\) but not summable almost nowhere by the constructed method ((1), pp. 405 and 418).

* In what follows, only the exponential* Borel method is considered.

*** This argument is analogous to the proof of equivalence in \(L^2\) of the methods \((C,\alpha)\), \(\alpha\geq 1\), and the Abel method ((2), p. 220).

The theorem of A. Zygmund cited above, as well as certain estimates for the \((R, \overline{P}_n, \delta)\)-means of series (1) from paper (³), are also valid for the corresponding methods \((\overline{R}, p_n)\). (In particular, for \(p_n = \frac{1}{n}\), see also (⁸).)

Lemma 2. There exists a Riesz method \((\overline{R}, q_n)\) which is weaker* than the method \((E, 1)\) and which is equivalent in the space \(L^2\) to the method \(T[m^2]\).

Lemma 3. There exists a Riesz method \((\overline{R}, t_n)\), equivalent in the space \(L^2\) to the method \(T[m^2]\), and such that if the method \((E, 1)\) sums series (3), then it also sums the \((\overline{R}, t_n)\)-means of this series.

It follows at once from Lemma 2 that from the summability of series (1) on a set of positive measure by the method \(T[m^2]\), there follows the summability almost everywhere on this set of series (1) by the method \((E, 1)\). The converse assertion is obtained from Lemma 3 by arguments analogous to those given above (after Lemma 1). In doing so one uses the estimates (somewhat modified) for the Riesz means of series (1) given in paper (³).

We shall give one more theorem, which is a consequence of Theorem 1.

Definition 3. A function \(\Omega(n) > 0\) is called an exact Weyl multiplier for the summability method \(T\) if: 1) from the condition \(\sum c_n^2 \Omega(n) < \infty\) there follows the summability almost everywhere by the method \(T\) of the series \(\sum c_n \varphi_n(x)\), where \(\{\varphi_n(x)\}\) is an arbitrary ON-system on \([a,b]\); 2) for every function \(0 < \omega(n) = o(\Omega(n))\) there exists an orthogonal series \(\sum b_n \psi_n(x)\), not summable by the method \(T\) at any point of the interval \([a,b]\), and such that \(\sum b_n^2 \omega(n) < \infty\).

As is known, an exact Weyl multiplier for convergence is the function \(\ln^2 n\). These are the theorems of D. E. Menshov—H. Rademacher and D. E. Menshov (see, for example, (²), pp. 190 and 195).

A. Zygmund in paper (³) found exact Weyl multipliers for the methods \((R, P_n, \delta)\). (A correction to Zygmund’s formulation was indicated by Chen Jiang-gong ((⁹), p. 14).)

Theorem 2. An exact Weyl multiplier for the Euler methods \((E, q)\) and for the Borel method is the function \(\ln^2 n\).

This theorem follows from Theorem 1 and the following lemma.

Lemma 4. If the method \(T\) is equivalent in the space \(L^2\) to some method \(T[n_m]\), then an exact Weyl multiplier for the method \(T\) is the function

\[ \Omega(n) = \ln^2 m \quad \text{for } n_m \leq n < n_{m+1}, \qquad m = 1,2,\ldots . \]

(For \(0 \leq n < n_1\) the values of \(\Omega(n)\) may be defined arbitrarily.)

Lemma 4 follows easily from the theorems on exact Weyl multipliers formulated above.

I express my deep gratitude to D. E. Menshov for his constant attention to and guidance of my work.

All-Union Correspondence
Power Engineering Institute

Received
9 XII 1961

REFERENCES

¹ D. E. Menshov, Tr. Mosk. matem. obshch., 10, 351 (1961).
² S. Kaczmarz, H. Steinhaus, Theory of Orthogonal Series, M., 1958.
³ A. Zygmund, Bull. Acad. Polon. Sci. Lettres (Cracovie), sér. A, 6, 293 (1927).
⁴ G. Hardy, Divergent Series, M., 1951.
⁵ J. Meder, Ann. Polon. math., 5, No. 2 (1958).
⁶ J. Meder, Bull. Acad. Polon. Sci., 7, No. 10 (1959).
⁷ O. Szasz, Roczn. Polskiego towarz. mat., 25, 75 (1952).
⁸ J. Meder, Rozprawy Matematyczne, Polska Acad. Nauk, 17, 1 (1959).
⁹ Chen Kien-Kwong, Summation of the Fourier Series of Orthogonal Function, Peking, 1957.

* The case of equivalence is not excluded here.