MATHEMATICS

A. A. TALALYAN

ON THE EXISTENCE OF A TRIGONOMETRIC SERIES UNIVERSAL WITH RESPECT TO SUBSERIES

(Presented by Academician M. V. Keldysh, January 3, 1961)

In 1941 D. E. Men'shov (^1) proved the following fundamental theorem on the representation of measurable functions by trigonometric series:

For every almost everywhere finite measurable function \(f(x)\), defined on \([-\pi,\pi]\), there exists a trigonometric series

\[ \frac{a_0}{2}+\sum_{n=1}^{\infty} a_n\cos nx+b_n\sin nx, \]

which converges to \(f(x)\) almost everywhere on \([-\pi,\pi]\).

In proving this theorem, D. E. Men'shov used the following theorem, proved by him in paper (^2):

For every almost everywhere finite measurable function \(f(x)\), defined on \([-\pi,\pi]\), and for every positive number \(\sigma\), one can determine a function \(G(x)\), continuous on \([-\pi,\pi]\), and a measurable set \(E\), possessing the following properties:

\[ \alpha)\ E\subset[-\pi,\pi],\quad \operatorname{mes} E>2\pi-\sigma; \]

\[ \beta)\ G(x)=f(x)\quad (x\in E); \]

\[ \gamma)\ \text{the Fourier series of the function }G(x)\text{ converges almost everywhere on }[-\pi,\pi]. \]

Using Men'shov's lemmas proved in paper (^2), and applying a method different from the method used in (^1), one can prove the following theorem, which is a strengthening of Men'shov's theorem on the representation of measurable functions by trigonometric series:

Theorem 1. There exists a trigonometric series

\[ \frac{a_0}{2}+\sum_{n=1}^{\infty} a_n\cos nx+b_n\sin nx \tag{1} \]

such that for every almost everywhere finite measurable function \(f(x)\), defined on \([-\pi,\pi]\), there exists a subseries of the series (1)

\[ \sum_{k=1}^{\infty} a_{n_k}\cos n_k x+b_{n_k}\sin n_k x \quad (n_1<n_2<\cdots<n_k<\cdots), \tag{2} \]

which converges to \(f(x)\) almost everywhere on \([-\pi,\pi]\).

This theorem is a positive solution of the question posed in the survey article (^3) (see (^3), § 10, problem 3).

We shall outline the proof of Theorem 1 in general terms.

We construct a series (1) having the following properties (we formulate these properties as lemmas, because they may be regarded as properties of trigonometric polynomials in general).

Lemma 1. For any almost everywhere finite measurable function \(\psi(x)\), given on some set \(E\), \(\operatorname{mes} E>0\), for any natural number \(N\) and positive number \(\delta\), from the terms of the series (1) one can form a trigonometric polynomial
\[ \sum_{j=1}^{q}\left(a_{p_j}\cos p_jx+b_{p_j}\sin p_jx\right) \]
and determine measurable sets \(P\) and \(Q\) such that the following conditions are satisfied:
\[ N<p_1<p_2<\cdots<p_q; \tag{3} \]
\[ P\subset E,\qquad \operatorname{mes} P>\operatorname{mes} E-\delta; \tag{4} \]
\[ Q\subset CE,\qquad \operatorname{mes} Q>\operatorname{mes} CE-\delta \qquad (CE=(-\pi,\pi)-E); \tag{5} \]
\[ \left|\sum_{j=1}^{q} a_{p_j}\cos p_jx+b_{p_j}\sin p_jx-\psi(x)\right|<\delta \tag{6} \]
for \(x\in P\);
\[ \left|\sum_{j=1}^{i} a_{p_j}\cos p_jx+b_{p_j}\sin p_jx\right|<\delta \tag{7} \]
for \(x\in Q\) and for all \(i\), where \(1\le i\le q\).

Lemma 2. For any positive number \(\varepsilon\) one can determine a positive number \(\delta\) such that, whatever the measurable function \(f(x)\) defined on some measurable set \(E\subset(-\pi,\pi)\), \(\operatorname{mes} E>0\), and satisfying the inequality
\[ |f(x)|\le \delta,\qquad x\in E, \tag{8} \]
and whatever the natural number \(N\) and positive number \(\eta\), there exist a measurable set \(F\) and a polynomial
\[ \sum_{j=1}^{r} a_{k_j}\cos k_jx+b_{k_j}\sin k_jx, \]
chosen from the series (1), which satisfy the following conditions:
\[ F\subset E,\qquad \operatorname{mes} F>\operatorname{mes} E-\varepsilon; \tag{9} \]
\[ N<k_1<k_2<\cdots<k_r; \tag{10} \]
\[ \left|\sum_{j=1}^{\rho} a_{k_j}\cos k_jx+b_{k_j}\sin k_jx\right|\le \varepsilon \tag{11} \]
for any \(x\in F\) and for all \(\rho\), where \(1\le \rho\le r\);
\[ \left|\sum_{j=1}^{r} a_{k_j}\cos k_jx+b_{k_j}\sin k_jx-f(x)\right|>\eta \tag{12} \]
for any \(x\in F\).

It is further proved that the series (1), constructed in such a way that Lemmas 1 and 2 hold for it, satisfies the conditions of Theorem 1.

In constructing the series (1) possessing the indicated properties, the following lemma of D. E. Menshov is used (2):

Lemma 3 (D. E. Menshov). Let \(ax+b\) be a linear function which does not change sign on some interval \([c,d]\).

Then, for any positive number \(\eta\) and integer \(\nu > 8\), one can define a function \(\psi(x)\) and a measurable set \(E\) that have the following properties:

1) \(\psi(x)\) is continuous on \([c,d]\) and is linear on each of a certain finite number of intervals obtained by partitioning the interval \([c,d]\);

2) \(\psi(c)=ac+b,\ \psi(d)=ad+b\);

3) \(E \subset [c,d],\ \operatorname{mes} E > (d-c)\left(1-\dfrac{8}{\nu}\right)\);

4) \(|\psi(x)| \leqslant 2\varkappa \nu\) \((c \leqslant x \leqslant d)\), where \(\varkappa = \max\limits_{c \leqslant x \leqslant d}|ax+b|\);

5)
\[ \left|\int_{c'}^{d'} \psi(t)\,dt\right| < \eta \]
for all \(c'\) and \(d'\), where \(c \leqslant c' < d' \leqslant d\);

6)
\[ \left|\int_c^d \psi(t)\frac{\sin n(t-x)}{t-x}\,dt\right| \leqslant B\varkappa \nu \]
\((x \in E,\ n=1,2,\ldots)\), where \(B\) is an absolute constant;

7) \(\psi(x)=ax+b\) for \(x \in E\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
27 XII 1960

REFERENCES

1. D. E. Men’shov, Matem. sborn., 9, no. 3, 667 (1941).
2. D. E. Men’shov, Matem. sborn., 8, no. 3, 493 (1940).
3. A. A. Talalyan, UMN, 15, no. 5, 77 (1960).