MATHEMATICS

I. N. PAK

ON THE PROPERTIES OF SUMS OF CERTAIN TRIGONOMETRIC SERIES

(Presented by Academician V. I. Smirnov on 31 I 1963)

In the present paper we investigate certain properties of sums of cosine and sine series. Many authors have dealt with this problem. The works of L. Fejér (¹), who posed and solved a number of problems on finding properties of sums of the indicated series, deserve special attention. However, many questions of this broad problem still remain unsolved.

Lemma 1. For a sequence \(\{c_n\}\) with bounded variation and \(c_n \to 0\), there exists a sequence \(\{d_n\}\) such that
\[ c_n=d_n+d_{n+1},\qquad d_n\to 0. \tag{1} \]

The proof follows from Lemma 1 (⁵), if one takes into account that \(c_n=p_n-q_n\), where \(p_n\downarrow 0\) and \(q_n\downarrow 0\) \((d_n=\Delta c_n+\Delta c_{n+2}+\Delta c_{n+4}+\ldots)\).

Lemma 2. For \(\{c_n\}\) with bounded variation and \(c_n\to 0\), a necessary and sufficient condition for the \(k\)-fold monotonicity of \(\{d_n\}\), where \(c_n=d_n+d_{n+1}\), is
\[ \Delta^{k+1}c_n+\Delta^{k+1}c_{n+1}+\Delta^{k+1}c_{n+4}+\ldots \geq 0 \qquad (n=1,2,\ldots). \tag{2} \]

Condition (2) is intermediate between the conditions \(\Delta^k c_n\geq 0\) and \(\Delta^{k+1}c_n\geq 0\).

In (¹) Fejér proved that if \(\{b_n\}\) is convex (twice monotone) and \(b_n\to 0\), then
\[ \sum_{n=1}^{\infty} b_n\sin nx>0,\qquad 0<x<\pi. \]

This assertion is also valid under somewhat weaker restrictions. It is obtained from the following theorem, in whose proof Lemmas 1 and 2 play an essential role.

Theorem 1. If \(\Delta^2 b_n+\Delta^2 b_{n+2}+\Delta^2 b_{n+4}+\ldots \geq 0\) \((n=2,3,\ldots)\) and \(b_n\to 0\), then the following two relations hold:
\[ \sum_{n=1}^{\infty} b_n\sin nx\geq \beta_1\sin x,\qquad 0\leq x\leq \pi; \tag{3} \]
\[ \sum_{n=1}^{\infty} b_n\sin nx\geq \Delta b_1\sin x+2\beta_2\cos\frac{x}{2}\sin\frac{3}{2}x,\qquad 0\leq x\leq \pi; \tag{4} \]
where \(\beta_1=\Delta^2 b_1+\Delta^2 b_3+\Delta^2 b_5+\ldots\) and \(\beta_2=\Delta^2 b_2+\Delta^2 b_4+\ldots\).

The theorem is proved on the basis of the transformation
\[ \sum_{n=1}^{\infty} b_n\sin nx = \sum_{n=1}^{\infty}(\bar b_n+\bar b_{n+1})\sin nx = \]
\[ =\bar b_1\sin x+2\cos\frac{x}{2}\sum_{n=1}^{\infty}\bar b_{n+1}\sin\left(n+\frac12\right)x \]

and the relations

\[ \sum_{n=1}^{\infty} \bar b_{n+1}\sin\left(n+\frac12\right)x \geqslant -\bar b_2\sin\frac{x}{2}, \]

\[ \sum_{n=1}^{\infty} \bar b_{n+1}\sin\left(n+\frac12\right)x \geqslant \Delta \bar b_2\sin\frac{3}{2}x-\bar b_3\sin\frac{x}{2}, \qquad 0\leqslant x\leqslant 2\pi. \]

Remark 1. Suppose \(\Delta^2 b_n+\Delta^2 b_{n+2}+\cdots\geqslant 0\) also for \(n=1\). Then, by Lemma 2, \(\Delta b_1\geqslant \beta_2\geqslant 0\). Consequently,

\[ \sum_{n=1}^{\infty} b_n\sin nx \geqslant \Delta b_1\sin x+2\beta_2\cos\frac{x}{2}\sin\frac{3}{2}x \geqslant 4\beta_2\sin x\cos^2\left(\frac{x}{2}\right), \qquad 0\leqslant x\leqslant \pi. \tag{5} \]

From (3) and (5) it is clear that if \(\Delta^2 b_n+\Delta^2 b_{n+2}+\cdots\geqslant 0\) \((n=1,2,\ldots)\) and \(b_n\to 0\), then the sum of the sine series under consideration is nonnegative on \((0,\pi)\). We note that the condition \(\Delta^2 b_n+\Delta^2 b_{n+2}+\cdots\geqslant 0\) cannot be replaced by the condition \(\Delta b_n\geqslant 0\).

Remark 2. Suppose, in particular, that \(\{b_n\}\) \((n=1,2,\ldots)\) is convex, \(b_n\to 0\), and not all \(b_n=0\) (this is so if \(b_1\ne 0\)). In this case it is obvious that \(\beta_1\) and \(\beta_2\) cannot both be equal to zero; then the right-hand side in at least one of the relations (3) or (5) is positive on \((0,\pi)\), whence Fejér’s assertion follows. Moreover, in the case under consideration the relations (3) or (4), (5) give a positive lower bound on \((0,\pi)\).

Remark 3. The numbers \(\beta_1\) and \(\beta_2\) can be positive also in the case when \(\{b_n\}\) is not convex.

Next, as Fejér showed (1), the sum of the series \(\sum_{n=1}^{\infty} a_n\cos nx\) decreases monotonically on \((0,\pi)\) if \(a_n\to 0\) and \(\Delta^4 a_n\geqslant 0\) \((n=1,2,\ldots)\). Consequently, under these conditions

\[ \sum_{n=1}^{\infty} a_n\cos nx \geqslant \sum_{n=1}^{\infty} a_n\cos n\pi. \]

The last inequality is valid also under weaker restrictions; namely, Theorems 2 and 3 hold.

Theorem 2. If \(\Delta^4 a_n+\Delta^4 a_{n+2}+\cdots\geqslant 0\) \((n=1,2,\ldots)\) and \(a_n\to 0\), then

\[ \sum_{n=1}^{\infty} a_n\cos nx \geqslant \sum_{n=1}^{\infty} a_n\cos n\pi + 2\alpha\cos^2\left(\frac{x}{2}\right), \]

where

\[ \alpha=\sum_{k=1}^{\infty} k\Delta^4 a_{2k-1}\geqslant 0. \]

This theorem is proved with the aid of Lemmas 1 and 2, the transformation

\[ \sum_{n=1}^{\infty} a_n\cos nx = -\bar a_1+\operatorname{ctg}\frac{x}{2}\sum_{n=1}^{\infty}\Delta \bar a_n\sin nx \qquad (a_n=\bar a_n+\bar a_{n+1}) \]

and Theorem 1.

Remark. The condition \(\Delta^4 a_n+\Delta^4 a_{n+2}+\cdots\geqslant 0\) cannot be replaced by the condition \(\Delta^3 a_n\geqslant 0\).

Theorem 3. If \(\Delta^2 c_n+\Delta^2 c_{n+2}+\cdots\geqslant 0\) \((n=1,2,\ldots)\) and \(c_n\to 0\), then

\[ \sum_{n=1}^{\infty}\frac{c_n}{n}\cos nx \geqslant \sum_{n=1}^{\infty}\frac{c_n}{n}\cos n\pi + 2\beta\cos^2\left(\frac{x}{2}\right), \]

where

\[ \beta=\Delta^2 c_1+\Delta^2 c_3+\Delta^2 c_5+\cdots. \]

Remark. The sequence \(\{a_n\}=\{c_n/n\}\), where \(\{c_n\}\) satisfies the conditions of Theorem 3, need not satisfy the conditions of Theorem 2. Conversely, the existence of a sequence satisfying the conditions of Theorem 2 but not satisfying the conditions of Theorem 3 is obvious, since in Theorem 3 \(a_n=c_n/n=O(n^{-1})\), whereas in Theorem 2 this is not assumed. Thus, Theorems 2 and 3 are incomparable.

Let us now consider cosine and sine series with a convex sequence of coefficients tending to zero. For them Theorems 4–7 hold; in the proofs of these theorems one mainly uses the Young–Kolmogorov theorem \((^{2,3})\) stating that

\[ \frac{a_0}{2}+\sum_{n=1}^{\infty} a_n \cos nx \geqslant 0, \tag{6} \]

if \(\{a_n\}\) \((n=0,1,2,\ldots)\) is convex and \(a_n \to 0\), and Theorem 1.

Theorem 4. If \(\{a_n\}\) \((n=1,2,\ldots)\) is convex and \(a_n \to 0\), then

\[ \sum_{n=1}^{\infty} a_n \cos nx \leqslant a_1 \cos x+\cdots \]

\[ \cdots+a_{N-1}\cos (N-1)x+\frac{a_N}{2}\cos Nx,\qquad \frac{\pi}{2N}\leqslant x\leqslant \frac{\pi}{N}. \]

In particular, putting here \(N=1\), we obtain

\[ \sum_{n=1}^{\infty} a_n \cos nx \leqslant \frac{a_1}{2}\cos x,\qquad \frac{\pi}{2}\leqslant x\leqslant \pi. \]

Theorem 5. If \(\{a_n\}\) \((n=1,2,\ldots)\) is convex and \(a_n \to 0\), then

\[ \sum_{n=1}^{\infty} a_n \cos nx \geqslant a_1\cos x+\cdots+a_{N-1}\cos (N-1)x+\frac{a_N}{2}\cos Nx, \]

\[ \frac{3\pi}{2N}\leqslant x\leqslant \frac{2\pi}{N},\qquad N\geqslant 2. \tag{7} \]

The segments \([3\pi/2N,\,2\pi/N]\) cover the whole interval \((0,\pi]\) with the exception of the interval \((2\pi/3,\,3\pi/4)\). A lower bound in the wider interval \(\pi/2\leqslant x\leqslant 3\pi/4\) is given by the following formula:

\[ \sum_{n=1}^{\infty} a_n \cos nx \geqslant a_1\cos x+\frac{a_2}{2}\cos^2 x-\left(a_2+\frac{a_3}{2}\cos x\right)\sin^2 x. \]

Theorem 6. If \(\{b_n\}\) \((n=1,2,\ldots)\) is convex and \(b_n \to 0\), then

\[ \sum_{n=1}^{\infty} b_n \sin nx \leqslant b_1\sin x+\cdots+b_{N-1}\sin (N-1)x+\frac{b_N}{2}\sin Nx, \]

\[ \frac{\pi}{N}\leqslant x\leqslant \frac{3\pi}{2N},\qquad N\geqslant 2, \]

\[ \sum_{n=1}^{\infty} b_n \sin nx \leqslant b_1\sin x+\frac{b_2}{2}\sin 2x,\qquad \frac{\pi}{2}\leqslant x\leqslant \pi. \]

Theorem 7. If \(\{b_n\}\) \((n=1,2,\ldots)\) is convex and \(b_n \to 0\), then

\[ \sum_{n=1}^{\infty} b_n \sin nx \geqslant \frac{b_1}{2}\sin x,\qquad 0<x\leqslant \frac{\pi}{2}, \]

\[ \sum_{n=1}^{\infty} b_n \sin nx \geqslant \left(\frac{b_1}{2}+b_2\cos x+\frac{b_3}{2}\cos^2 x\right)\sin x,\qquad \frac{\pi}{2}\leqslant x\leqslant \frac{2}{3}\pi, \]

\[ \sum_{n=1}^{\infty} b_n \sin nx \leqslant b_1\sin x+b_2\sin 2x+\frac{b_3}{2}\sin 3x,\qquad \frac{2}{3}\pi\leqslant x\leqslant \frac{5}{6}\pi, \tag{8} \]

\[ \sum_{n=1}^{\infty} b_n \sin nx \geqslant b_1 \sin x + b_2 \sin 2x + \frac{b_3}{2}\cos x \sin 2x, \qquad \frac{3}{4}\pi \leqslant x \leqslant \pi. \]

Remark 1. By the Young–Kolmogorov theorem (see (6))

\[ \sum_{n=1}^{\infty} a_n \cos nx \geqslant -\frac{a_0}{2}, \]

if \(\{a_n\}\) \((n=0,1,2,\ldots)\) is convex and \(a_n \to 0\). The least value of \(a_0/2\) that can be found from the Young–Kolmogorov theorem is equal to
\(a_0/2 = a_1 - a_2/2\) \((\Delta^2 a_0=0)\) and, as we see, does not depend on \(x\). In Theorem 5 the lower bound of the cosine series depends on \(x\).

For example, putting \(N=2\) in (7), we obtain

\[ \sum_{n=1}^{\infty} a_n \cos nx \geqslant a_1 \cos x + \frac{a_2}{2}\cos 2x, \qquad \frac{3}{4}\pi \leqslant x \leqslant \pi; \]

\[ a_1 \cos x + \frac{a_2}{2}\cos 2x > -a_1 + \frac{a_2}{2} \quad \text{for} \quad \frac{3}{4}\pi \leqslant x < \pi, \]

\[ a_1 \cos \pi + \frac{a_2}{2}\cos 2\pi = -a_1 + \frac{a_2}{2}. \]

Remark 2. The right-hand sides in (8) are all positive in the corresponding intervals.

Theorem 8. If \(a_n \geqslant 0\), \(\{a_n\}\) is of bounded variation and \(a_n \to 0\), then the function \(C(x)\), where

\[ C(x)=\sum_{n=1}^{\infty} a_n \cos nx, \]

has a root in \((0,\pi)\).

In the proof of this theorem the \(A\)-integrability of \(C(x)\) is used ((4), p. 659). In the particular case when the series for \(C(x)\) is a Fourier–Lebesgue series, the theorem follows, for example, from the equality

\[ \int_{0}^{\pi} C(x)\,dx=0. \]

Theorem 9. For any segment \([x,\pi]\), \(0<x<\pi\), there exists an infinite set of cosine series with convex coefficient sequences tending to zero whose sums \(C(x)\) are negative on \([x,\pi]\).

This theorem is proved on the basis of the fact that \(C(x)<0\) for \(x\) satisfying the inequality

\[ a_2 \cos^2\left(\frac{x}{2}\right)-a_1 \cos x > 0 \qquad (0<x\leqslant \pi). \]

Corollary of Theorems 8 and 9. There exist cosine series

\[ \sum_{n=1}^{\infty} a_n \cos nx \]

with a convex coefficient sequence tending to zero, whose sums have a root arbitrarily close to zero.

Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich

Received
27 I 1963

References

1. L. Fejer, Trans. Am. Math. Soc., 39, 18 (1936).
2. W. H. Young, Proc. London Math. Soc., 12, 49 (1913).
3. A. N. Kolmogoroff, Bull. Intern. Acad. Polon., ser. A, 84 (1923).
4. N. K. Bari, Trigonometric Series, Moscow, 1961.
5. I. N. Pak, Tr. ucheb. inst. svyazi, no. 12 (1962).