Reports of the Academy of Sciences of the USSR

1. Volume 137, No. 4

MATHEMATICS

L. V. TAIKOV

ON THE DIVERGENCE OF FOURIER SERIES

(Presented by Academician A. N. Kolmogorov on 4 XI 1960)

1. N. N. Luzin \((^1)\) constructed a trigonometric series

\[ \frac{a_0}{2}+\sum_{k=1}^{\infty} a_k \cos kx+ \]

\[ +\, b_k \sin kx \]

with coefficients tending to zero, which diverges unboundedly almost everywhere on \([0,2\pi)\).* Subsequently, the analogous problem was considered for trigonometric series on which more and more restrictions were imposed. A. N. Kolmogorov \((^3)\) gave an example of a Fourier–Lebesgue series diverging almost everywhere, and somewhat later \((^4)\) an example of a Fourier series diverging at every point. In A. N. Kolmogorov’s examples the conjugate series

\[ \sum_{k=1}^{\infty} a_k \sin kx - b_k \cos kx \]

is not a Fourier series (see, for example, \((^5)\)). The next step was made by Hardy and Rogosinski \((^6)\): they constructed a Fourier series diverging almost everywhere, whose conjugate is also a Fourier series.

In parallel, the study of sets of convergence and divergence of trigonometric series was carried out. As is known \((^7)\), the sets of points of unbounded divergence of a sequence of continuous functions on a closed set are of type \(G_\delta\). Herzog and Piranian \((^8)\) constructed, for a given set \(E\) of type \(G_\delta\) on the unit circle, a Taylor series which diverges unboundedly on \(E\) and converges on the complementary set; then S. B. Stechkin \((^9)\) constructed a trigonometric series which converges on a prescribed set \(E \subset [0,2\pi)\) of type \(F_\sigma\) and diverges unboundedly on \(E_1=[0,2\pi)-E\); finally, Zeller \((^{10})\) constructed a Fourier series with the properties indicated above.

In the present note a theorem is proved that is more precise than Zeller’s theorem. From it, in particular, follows the existence of a pair of conjugate Fourier series diverging unboundedly at every point.

A formulation of a somewhat weaker theorem was published by me in \((^{11})\).

2. Lemma 1. Let \([a,b]\subset[0,2\pi)\), \([e,d]\subset(a,b)\), and \(\varepsilon>0\). Then there exists a trigonometric polynomial

\[ T(x)=\frac{a_0}{2}+\sum_{k=1}^{N} a_k \cos kx + b_k \sin kx, \]

having the following properties:

\[ 1)\quad \|T\|_L=\int_{0}^{2\pi} |T(x)|\,dx < \varepsilon; \]

* As S. B. Stechkin \((^2)\) showed, the trigonometric series constructed by N. N. Luzin in fact diverges at every point.

2) for all \(x\in [0,2\pi)\setminus [a,b]\),

\[ |S_k(x,T)|=\left|\frac{a_0}{2}+\sum_{j=1}^{k} a_j\cos jx+b_j\sin jx\right|<\varepsilon \]

\[ |\widetilde S_k(x,T)|=\left|\sum_{j=1}^{k} a_j\sin jx-b_j\cos jx\right|<\varepsilon,\qquad k=1,2,\ldots; \]

3) for every \(x\in[c,d]\) there is an index \(k_x\) such that \(|S_{k_x}(x,T)|>1/\varepsilon\).

This lemma is a slight modification of one result of A. N. Kolmogorov (see \((^{12})\), § 8.404) and is proved analogously.

For each natural \(n\) we construct the points \(x_l=\dfrac{2\pi l}{2n+1}\), \(l=0,1,\ldots,2n\), and choose \(n_0\) so that

\[ \frac{2\pi}{2n_0+1}<\min\left\{\frac{c-a}{4},\,\frac{d-b}{4}\right\}=\delta. \]

For each \(n\ge n_0\), among the division points \(x_l\) we choose the point \(x_{l_0}\) nearest from the left to the point \(c\), and the point \(x_{l_1}\) nearest from the right to the point \(d\). From the Fejér polynomial \(K_{2n(n+1)}(x)\) of order \(2n(n+1)\) we form the polynomial

\[ \varphi(x)=\frac{1}{m'}\sum_{l=l_0}^{l_1} K_{2n(n+1)}(x-x_l),\qquad \text{where } m'=l_1-l_0+1. \]

Choose \(\eta=\eta(n)\) so that for \(x_l-\eta\le x\le x_l+\eta\) the inequality \(K_{(2n+1)*}(x-x_l)\ge n(n+1)/2\) holds and the Dirichlet sum of order \(2n(n+1)\) is positive, i.e. \(D_{(2n+1)*}(x-x_l)>0\).

Put \(h=(a+c)/2\), \(e=(d+b)/2\). Let \(x_{\nu_0}\) be the point nearest from the left to the point \(h\), and \(x_{2\nu_1}\) the point nearest from the right to the point \(e\) among all points \(x_{2\nu}\), \(\nu=0,1,\ldots,n\). Consider the trigonometric polynomial

\[ \psi(x)=\frac{1}{m''}\sum_{\nu=\nu_0}^{\nu_1} K_{m_\nu}(x-x_{2\nu}),\qquad m''=\nu_1-\nu_0+1. \]

Choose the numbers \(m_{\nu_0},\ldots,m_{\nu_1}\) so that the following conditions are satisfied: \(m_{\nu_0}>(2n+1)^2\); to each \(x\in(x_j+\eta,x_{j+1}-\eta)\), \(2\nu_0\le j\le 2\nu_1\), there corresponds a natural \(k=k_x\) such that \(m_j\le k<m_{j+1}\), \(\sin(k+1/2)(x_{2j}-x)\ge 1/3\), and \(2k+1\) is divisible by \(2n+1\). The possibility of such a choice of \(m_\nu\) was proved by A. N. Kolmogorov (see \((^{12})\)).

Put

\[ T(x)=\frac{\varphi(x)+\psi(x)}{(\ln n)^{1/2}}. \]

For sufficiently large \(n\) the polynomial \(T(x)\) will satisfy all the conditions of the lemma.

Lemma 2. Let \(M>0\). Then for every trigonometric polynomial

\[ T(x)=\frac{a_0}{2}+\sum_{k=1}^{N} a_k\cos kx+b_k\sin kx \]

there exists a trigonometric polynomial of the form

\[ t(x)=\sum_{k=Q}^{R} c_k\cos kx+d_k\sin kx, \]

where \(Q>M\), such that for every \(x\in[0,2\pi)\)

\[ |t(x)|\le |T(x)|,\qquad |\widetilde t(x)|\le |\widetilde T(x)|, \tag{1} \]

\[ \frac14\sup_k |S_k(x,T)|\le \sup_n |S_n(x,t)|<\sup_k\{\,|S_k(x,t)|+|S_k(x,\widetilde T)|\,\}. \tag{2} \]

Proof. Put

\[ t(x)=\frac12(\cos Px+\sin 3Px)\,T(x), \]

where \(P=M+N+1\). Then

\[ \widetilde t(x)=\frac12(\sin Px-\cos 3Px)\,T(x), \]

and property (1) is obvious.

Let us prove property (2). The following identities hold: for \(k\leq P\),

\[ S_k(x,t)=\frac14\{\cos Px\,[T(x)-S_{P-k-1}(x,T)]+\sin Px\,[\widetilde T(x)-S_{P-k-1}(x,\widetilde T)]\}; \tag{3} \]

for \(P\leq k\leq P+N\),

\[ S_k(x,t)=\frac12\{\cos Px\cdot S_{k-P}(x,T)+S_{2P-k-1}(x,t)\}; \tag{4} \]

for \(P+N\leq k\),

\[ S_k(x,t)=\frac12\{\cos Px\cdot T(x)+S_k(x,T\sin 3Px)\}. \tag{5} \]

The second term in equality (5) is expanded analogously to equalities (3) and (4).

Estimating (3), (4), and (5) in absolute value, we obtain the right-hand side of inequality (2).

Let us prove the left-hand side of inequality (2). Choose \(k_x\) so that

\[ \sup_k |S_k(x,T)|=S_{k_x}(x,T). \]

Then

\[ \cos Px\cdot S_{k_x}(x,T)=S_{P+k_x}(x,t)-S_{P-k_x-1}(x,t). \]

If \(|\cos Px|\geq \frac12\), then

\[ |S_{P+k_x}(x,t)-S_{P-k_x-1}(x,t)|\geq \frac12 |S_{k_x}(x,T)| =|\sup_k |S_k(x,T)||. \]

Consequently, one of the terms is not less than \(\frac14\sup_k |S_k(x,T)|\). If, however, \(|\cos Px|\leq \frac12\), then \(|\sin 3Px|\geq \frac12\), and we may use the polynomial \(\sin 3PxT(x)\) for the estimate. The conjugate polynomial \(\widetilde t(x)\) is considered analogously. The lemma is completely proved.

3. Theorem. Let \(E\subset [0,2\pi)\) be an arbitrary set of type \(F_\sigma\). Then there exists a pair of conjugate Fourier series converging on \(E\) and diverging unboundedly on \(E_1=[0,2\pi)-E\).

Proof. For intervals satisfying the conditions

\[ (c_k,d_k)\subset [a_k,b_k]\subset [0,2\pi), \]

and numbers \(\varepsilon_k=2^{-k}\), \(k=1,2,\ldots\), we can, relying on Lemmas 1 and 2, construct by induction a sequence of polynomials

\[ T_k(x)=\sum_{n=p_k}^{q_k} a_n\cos nx+b_n\sin nx, \]

satisfying the conditions:

1) \(p_k>q_{k-1}\);

2) \(\|T_k\|_L<2^{-k},\quad \|\widetilde T\|_L<2^{-k}\);

3) for all \(m\) and \(x\in [0,2\pi)-[a_k,b_k]\),

\[ |S_m(x,T_k)|<2^{-k},\qquad |\widetilde S_m(x,T_k)|<2^{-k}; \]

4) for every \(x \in [c_k, d_k]\) there is a natural number \(p_x^{(k)} \geqslant p_x\) such that

\[ \left| S_{p_x^{(k)}}(x, T_k) \right| > 2^k,\qquad \left| \widetilde S_{p_x^{(k)}}(x, T_k) \right| > 2^k . \]

Using the Herzog—Piranian—Zeller construction (see, for example, \((^5)\)), we can construct a Fourier series that satisfies the theorem.

The consequence of this theorem noted in item 1 could also have been obtained from A. N. Kolmogorov’s polynomials (see, for example, \((^{12})\)) and Lemma 2.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
30 X 1960

CITED LITERATURE

\(^1\) N. Lusin, Rend. Circ. Mat. Palermo, 32, 386 (1911).
\(^2\) S. B. Stechkin, Izv. AN SSSR, ser. matem., 21, 711 (1957).
\(^3\) A. Kolmogoroff, Fund. Math., 4, 324 (1923).
\(^4\) A. Kolmogoroff, C. R., 183, 1327 (1926).
\(^5\) P. L. Ul’yanov, UMN, 12, issue 3 (75) (1957).
\(^6\) G. H. Hardy, V. V. Rogozinskii, Fourier Series, Moscow, 1959.
\(^7\) F. Hausdorff, Set Theory, Moscow—Leningrad, 1937.
\(^8\) F. Herzog, G. Piranian, Duke Math. J., 16, No. 3, 529 (1949).
\(^9\) S. B. Stechkin, UMN, 6, No. 2, 148 (1951).
\(^10\) K. Zeller, Arch. d. Math., 6, No. 4, 335 (1955).
\(^11\) L. V. Taĭkov, All-Union Conference on the Theory of Functions, Yerevan, 1960, p. 103.
\(^12\) A. Zygmund, Trigonometric Series, Moscow—Leningrad, 1939, pp. 175–180.