UDC 517.522.3+517.512.5

MATHEMATICS

L. V. ZHIZHIASHVILI

ON THE DIVERGENCE OF FOURIER SERIES

(Presented by Academician I. N. Vekua on 18 III 1970)

1. It is well known (see (¹) or (²), p. 599) that there exists a \(2\pi\)-periodic function \(f(x)\in L(-\pi,\pi)\), whose Fourier series \(\sigma[f]\) does not converge in the metric \(L(-\pi,\pi)\). On the other hand (see (³), p. 424), if \(f(x)\in L\log^{+}L\), then the series \(\sigma[f]\) converges in the metric \(L(-\pi,\pi)\). Naturally the question arises: what can be said about the convergence of the series \(\sigma[f]\) in the sense of the space \(L(-\pi,\pi)\) in the case when \(f(x)\in L(\log^{+}L)^\alpha\) for all \(\alpha\in(0,1)\)?

Further, it is known (see (⁴), p. 258) that if \(f(x)\in L(-\pi,\pi)\) and its integral modulus of continuity \(\omega(\delta,f)_L\) satisfies the condition

\[ \omega(\delta,f)_L=O\left\{\left(\log\frac{1}{\delta}\right)^{-1-\varepsilon}\right\},\quad \varepsilon>0\quad(\delta\to0), \tag{1} \]

then the series \(\sigma[f]\) converges both in the metric \(L(-\pi,\pi)\) and almost everywhere. Whether condition (1) with \(\varepsilon=0\) guarantees the convergence of the series \(\sigma[f]\) almost everywhere or in the sense of the metric \(L(-\pi,\pi)\) is unknown. However (see (³), p. 288, or (⁵), p. 100), if

\[ \omega(\delta,f)_L=o\left\{\left(\log\frac{1}{\delta}\right)^{-1}\right\}\quad(\delta\to0), \tag{2} \]

then the series \(\sigma[f]\) converges in the metric \(L(-\pi,\pi)\). It is unknown whether, in relation (2), the symbol \(o\) can be replaced by \(O\).

Below we shall give assertions that answer the questions posed (with respect to convergence in the metric \(L(-\pi,\pi)\)); results will also be given that are generalizations of the assertions obtained by us to the case of multiple trigonometric Fourier series.

1. Let \(f(x)\in L(-\pi,\pi)\). As usual, by \(\bar f(x)\) we denote the conjugate function to \(f(x)\), and by the symbol \(\bar\sigma[f]\) the conjugate trigonometric series to \(\sigma[f]\).

Theorem 1. There exists a function \(f_0(x)\in L(-\pi,\pi)\) such that:

a) \[ \omega(\delta,f_0)_L=O\left\{\left(\log\frac{1}{\delta}\right)^{-1}\right\},\quad \omega(\delta,\bar f_0)_L=O\left\{\left(\log\frac{1}{\delta}\right)^{-1}\right\}\quad(\delta\to0); \]

b) the series \(\sigma[f_0]\) and \(\bar\sigma[f_0]\) do not converge in the metric \(L(-\pi,\pi)\).

Theorem 1 shows that condition (2) is essential for the convergence of the series \(\sigma[f]\) in the sense of the metric \(L(-\pi,\pi)\); consequently, condition (1) with \(\varepsilon=0\), generally speaking, does not guarantee the convergence of the series \(\sigma[f]\) in the metric \(L(-\pi,\pi)\) (cf. the corresponding assertion from the book (⁴), p. 258).

We also note that in item b) of Theorem 1 one may restrict oneself to considering only one of the series \(\sigma[f_0]\) and \(\bar\sigma[f_0]\), since the functions \(f_0(x)\) and \(\bar f_0(x)\) are summable, and in this case (see (²), p. 602) the divergence of one series in the metric \(L(-\pi,\pi)\) entails the divergence of the other series in the sense of the same metric.

Using the results of P. L. Ul’yanov (⁶), according to Theorem 1 we obtain

Corollary. There exists a function \(f_0(x)\) such that \((f_0(x),\bar f_0(x))\in L(\log^{+}L)^\alpha\) for all \(\alpha\in[0,1)\), but the series \(\sigma[f_0]\) and \(\sigma[\bar f_0]\) do not converge in the metric \(L(-\pi,\pi)\).

Theorem 1 shows that for the convergence of \(\sigma[f]\) in the metric \(L(-\pi,\pi)\) the condition \(f(x)\in L\log^{+}L\) is essential.

1. Let now \(R=[-\pi,\pi;-\pi,\pi;\ldots;-\pi,\pi]\), and consider a function \(f(x_1,x_2,\ldots,x_n)\in L(R)\) which is \(2\pi\)-periodic with respect to each of the variables. By the symbols \(\omega(\delta_1,\delta_2,\ldots,\delta_n,f)_L\), \(\omega(\delta_1,\delta_2,\ldots,\delta_{n-1},f)_L,\ldots,\omega(\delta_1,\delta_2,f)_L,\ldots,\omega(\delta_1,f)_L,\ldots,\omega(\delta_n,f)_L\) we shall denote, respectively, the complete and partial integral moduli of continuity of the function \(f(x_1,x_2,\ldots,x_n)\), where it is assumed that \(\delta_k\) corresponds to \(x_k\) \((k=1,2,\ldots,n)\). By \(\sigma_n[f]\) we denote the \(n\)-fold (see (5), p. 256) trigonometric Fourier series of the function \(f(x_1,x_2,\ldots,x_n)\), and by the expressions \(\overline{\sigma}_n[f,x_1]\), \(\overline{\sigma}_n[f,x_2],\ldots,\overline{\sigma}_n[f,x_1,x_2],\ldots,\overline{\sigma}_n[f,x_{n-1},x_n],\ldots,\overline{\sigma}_n[f,x_1,x_2,\ldots,x_n]\) we shall denote the \(n\)-fold (see (5), p. 256) conjugate trigonometric series. As for (see (5), p. 257) the conjugate functions of the variables, we denote them by the symbols \(\overline{f}_1(x_1,x_2,\ldots,x_n), \overline{f}_2(x_1,x_2,\ldots,x_n),\ldots,\overline{f}_{1,2,\ldots,n}(x_1,x_2,\ldots,x_n)\).

Since, in the study of multiple series, one may consider different types of convergence, we shall restrict ourselves to considering convergence in the sense of Pringsheim.

Theorem 2. Let \(f(x_1,x_2,\ldots,x_n)\in L(R)\) and \(1\le i\le m\) \((m=1,2,\ldots,n)\). If

\[ \omega(\delta_i,\delta_{i+1},\ldots,\delta_m,f)_L = o\left\{\prod_{k=i}^{m}\left(\log\frac{1}{\delta_k}\right)^{-1}\right\} \quad (\delta_k\to 0,\ k=1,2,\ldots,n), \tag{3} \]

then the series \(\sigma_n[f]\) converges in the metric \(L(R)\).

The theorem just given is, in a certain sense, final, since the following is true.

Theorem 3. There exists a function \(f_1(x_1,x_2,\ldots,x_n)\in L(R)\) for which

\[ \text{a) }\quad \omega(\delta_i,\delta_{i+1},\ldots,\delta_m,f_1)_L = O\left\{\prod_{k=i}^{m}\left(\log\frac{1}{\delta_k}\right)^{-1}\right\} \quad (\delta_k\to 0); \tag{4} \]

b) the integral moduli of continuity of all conjugate functions for \(f_1(x_1,x_2,\ldots,x_n)\) also satisfy relations (4);

c) the series \(\sigma_n[f_1]\) does not converge in the metric \(L(R)\).

We note that even the fulfillment of condition (3) in the sense that, for some \(m_0\) and \(i_0\) (which may be equal), \(o\) is replaced by \(O\), does not guarantee convergence in the metric \(L(R)\) of the series \(\sigma_n[f]\).

We also note that some series among the \(n\)-fold conjugate trigonometric series may be convergent in the metric \(L(R)\), something that does not occur in the one-dimensional case.

Scientific Research Institute of Applied Mathematics
of Tbilisi State University

Received
7 I 1970

References

1. A. Zygmund, Proc. Lond. Math. Soc., 34, 392 (1932).
2. N. K. Bari, Trigonometric Series, Moscow, 1961.
3. A. Zygmund, Trigonometric Series, 1, Moscow, 1965.
4. A. Zygmund, Trigonometric Series, 2, Moscow, 1965.
5. L. V. Zhizhiashvili, Conjugate functions and trigonometric series, Tbilisi, 1969.
6. P. L. Ul’yanov, Izv. Acad. Sci. USSR, Ser. Math., 32, 649 (1968).