UDC 517.552.3+517.512.5

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR D. MENSHOV

LIMITS OF UNCERTAINTY IN THE SENSE OF \(T\)-MEANS FOR TRIGONOMETRIC SERIES

Let us take an infinite matrix

\[ T=\|a_{mk}\| \qquad (m,k=0,1,2,\ldots) \tag{1} \]

and some infinite series

\[ \sum_{n=0}^{\infty} u_n \tag{2} \]

with partial sums \(S_m\) \((m=0,1,2,\ldots)\). We shall call the quantities

\[ A_m(T)=\sum_{k=0}^{\infty} a_{mk}S_k \qquad (m=0;1,2,\ldots) \tag{3} \]

the \(T\)-means of the series (2), determined by the matrix (1).

It is said that the series (2) is summable by the method \(T\) to the value \(S\), if the series on the right-hand side of equality (3) converge for every \(m=0,1,2,\ldots\) and the quantities \(A_m(T)\) tend to the limit \(S\) as \(m\to\infty\). The summation method \(T\) is called regular if every series (2) converging to a finite value \(S\) is summable by the method \(T\) to the same value \(S\).

The summation method \(T\) is called a method with finite rows if the matrix (1) defining this method satisfies the condition

\[ a_{mk}=0 \qquad (k>\nu_m,\ m=0,1,2,\ldots), \tag{4} \]

where \(\nu_m\) are natural numbers, in general depending on \(m\).

Theorem 1. Let an arbitrary regular summation method \(T\) with finite rows be given, determined by the matrix (1). Then, for any two measurable functions \(F(x)\) and \(G(x)\) satisfying the inequality

\[ G(x)\leq F(x) \tag{5} \]

almost everywhere on the interval \([-\pi,\pi]\), one can define a trigonometric series

\[ \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx), \tag{6} \]

which has the following properties:

1) If

\[ A_m(x,T) \qquad (m=0,1,2,\ldots) \tag{7} \]

are the \(T\)-means of the series (6), determined by the matrix (1), then the upper limit in measure on \([-\pi,\pi]\) of the sequence (7) is equal to \(F(x)\), and the lower limit in measure on \([-\pi,\pi]\) of the same sequence is equal to \(G(x)\).

2)

\[ \lim_{n\to\infty} a_n=0,\qquad \lim_{n\to\infty} b_n=0. \tag{8} \]

* The functions \(F(x)\) and \(G(x)\) may be equal to \(+\infty\) or \(-\infty\) on sets of positive measure.

** The definition of upper and lower limits in measure of a sequence of functions is given in (1), p. 4.

If in Theorem 1 we assume that \(F(x)=G(x)\) almost everywhere on \([-\pi,\pi]\), then we obtain the following theorem.

Theorem 2. For any measurable function \(F(x)\), defined almost everywhere on the segment \([-\pi,\pi]\), one can define a trigonometric series (6), satisfying condition (8), which converges in measure on \([-\pi,\pi]\) to the function \(F(x)\).

In the proof of Theorem 1 the following is used.

Lemma A. Let there be given: an arbitrary regular summability method \(T\) with finite rows, defined by the matrix (1); an arbitrary function \(\varphi(x)\), continuous on the segment \([-\pi,\pi]\); an arbitrary positive number \(\sigma<1\), and arbitrary natural numbers \(L\) and \(L_0\).

Then one can define a natural number \(L'>L\), a trigonometric polynomial

\[ H(x)=\sum_{j=L+1}^{L'}(a_j\cos jx+b_j\sin jx) \]

and sets \(E, G_n\) \((L<n\le L')\), which satisfy the following conditions:

1) \(\operatorname{mes} E<\sigma,\ E\subset[-\pi,\pi]\);

2) \(\operatorname{mes} G_n<\sigma,\ G_n\subset[-\pi,\pi]\) \((L<n\le L')\);

3) if \(\tau_k(x)\) and \(B_n(x,T)\) are determined from the equalities

\[ \tau_k(x)= \begin{cases} 0 & (0\le k\le L),\\[4pt] \displaystyle\sum_{j=L+1}^{k}(a_j\cos jx+b_j\sin jx) & (L<k\le L'),\\[8pt] H(x) & (k>L'), \end{cases} \]

\[ B_n(x,T)=\sum_{k=0}^{\infty} a_{nk}\tau_k(x)\qquad (n=0,1,2,\ldots), \]

then

\[ |B_n(x,T)-\varphi(x)|<\sigma \qquad (x\in[-\pi,\pi]-E,\ n\ge L'); \]

4) \(B_n(x,T)=\theta_n(x)\varphi(x)+\eta_n(x)\) \((x\in[-\pi,\pi]-G_n,\ L<n\le L')\),

where \(|\theta_n(x)|<K,\ |\eta_n(x)|<\sigma\) \((x\in[-\pi,\pi]-G_n,\ L<n\le L')\), and \(K\) is a constant depending only on the summability method \(T\);

5) \(B_n(x,T)=0\) \((0\le n\le L)\);

6) \(L_0<L',\ \sqrt{a_n^2+b_n^2}<\sigma\) \((L<n\le L')\).

Definition of a trigonometric series (6) satisfying the conditions of Theorem 1. Let us take an arbitrary summability method \(T\) and arbitrary functions \(F(x), G(x)\) satisfying the conditions of Theorem 1, and put

\[ f_{2\nu}(x)=F(x),\qquad f_{2\nu+1}(x)=G(x) \qquad (\nu=0,1,2,\ldots). \]

Take a sequence of functions \(h_m(x)\) \((m=0,1,2,\ldots)\), continuous on \([-\pi,\pi]\), which is an almost uniformly convergent sequence to the sequence of functions \(f_m(x)\) \((m=0,1,2,\ldots)\) almost everywhere on \([-\pi,\pi]\). Put, further,

\[ u_m(x)=\max [h_m(x),h_{m+1}(x)],\qquad v_m(x)=\min [h_m(x),h_{m+1}(x)] \]

\[ (m=0,1,2,\ldots), \]

\[ w_{2\nu}(x)=v_\nu(x),\qquad w_{2\nu+1}(x)=u_\nu(x), \qquad (\nu=0,1,2,\ldots), \]

\[ \mu_m=[2^m\Omega_m]=1 \qquad (m=1,2,\ldots), \]

* If the function \(F(x)\) is finite almost everywhere on \([-\pi,\pi]\), then Theorem 2 is easily obtained from previously known theorems.

** The definition of almost uniformly convergent sequences is given in (1), p. 27. The existence of a sequence of functions \(h_m(x)\) with the stated properties follows from Lemma 3.3 in (1), p. 27.

where \([a]\) denotes the integer part of the number \(a\), and

\[ \Omega_m=\max_{x\in[-\pi,\pi]} |w_m(x)-w_{m-1}(x)| \qquad (m=1,2,\ldots), \]

\[ r_0=0,\qquad r_m=\sum_{s=1}^{m}\mu_s \qquad (m=1,2,\ldots), \]

\[ Q_0(x)=0,\qquad Q_t(x)=w_{m-1}(x)+(t-r_{m-1})\frac{w_m(x)-w_{m-1}(x)}{\mu_m} \]

\[ (r_{m-1}<t\le r_m,\ m=1,2,\ldots). \]

It is easy to see that the functions \(Q_t(x)\) are continuous on \([-\pi,\pi]\) and are defined for all \(t=0,1,2,\ldots\). We now define an increasing sequence of natural numbers \(M_t\) \((t=0,1,2,\ldots)\) and the trigonometric series (6) in the following way.

Put \(M_0=1,\ a_0=a_1=b_1=0\). Suppose next that the natural number \(M_{t-1}\) and the numbers \(a_j,b_j\) for \(j=1,2,\ldots,M_{t-1}\), where \(t\) is some natural number, have already been defined. Put

\[ S_{t-1,0}(x)=0,\qquad S_{t-1,k}(x)= \begin{cases} \displaystyle \sum_{j=1}^{k}(a_j\cos jx+b_j\sin jx), & (1\le k\le M_{t-1}),\\[1.2em] \displaystyle \sum_{j=1}^{M_{t-1}}(a_j\cos jx+b_j\sin jx), & (M_{t-1}<k), \end{cases} \]

\[ D_{t-1,n}(x,T)=\sum_{k=0}^{\infty} a_{nk}S_{t-1,k}(x). \]

It is easy to see that, for a given \(t\), the sequence of functions \(D_{t-1,n}(x,T)\) \((n=0,1,2,\ldots)\) converges uniformly on \([-\pi,\pi]\) to the function

\[ S_{t-1,M_{t-1}}(x)=\sum_{j=1}^{M_{t-1}}(a_j\cos jx+b_j\sin jx), \]

as \(n\to\infty\). In this case we can define a natural number \(M_t'\) satisfying the conditions

\[ M_t'>M_{t-1},\qquad \gamma_t<1/2^t, \]

where

\[ \gamma_t=\sup_{n,n'>M_t'}\max_{x\in[-\pi,\pi]} |D_{t-1,n}(x,T)-D_{t-1,n'}(x,T)|. \]

We now apply Lemma A, in which we put

\[ \varphi(x)=Q_t(x)-D_{t-1,M_{t-1}}(x,T), \tag{9} \]

\[ \sigma=1/2^t,\qquad L=M_{t-1},\qquad L_0=M_t'. \tag{10} \]

Then, on the basis of this lemma, we can define a natural number \(M_t>M_{t-1}\), a trigonometric polynomial

\[ H_t(x)=\sum_{j=M_{t-1}+1}^{M_t}(a_j\cos jx+b_j\sin jx) \tag{11} \]

and the sets

\[ E_t,\qquad G_{t,n}\qquad (M_{t-1}<n\le M_t), \]

which satisfy all the conditions of Lemma A, if they are taken correspondingly-

respectively, instead of \(L'\), \(H(x)\), \(E\), and \(G_n\), and define \(\varphi(x)\), \(\sigma\), \(L\), and \(L_0\) from equalities (9) and (10).

Thus we define trigonometric polynomials \(H_t(x)\) for all \(t=1,2,\ldots\), where the natural numbers \(M_t\) \((t=0,1,2,\ldots)\) form an increasing sequence. Since we have set \(M_0=1\), \(a_0=a_1=b_1=0\), all terms of the trigonometric series (6) will be determined. It can be shown that the trigonometric series thus obtained satisfies all the conditions of Theorem 1.

Moscow State University
named after M. V. Lomonosov

Received
23 VI 1967

REFERENCES

1. D. Menshov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 32, 1 (1950).