Mathematics

I. N. Pak

On the Properties of Sums of Certain Sine and Cosine Series

(Presented by Academician V. I. Smirnov, 21 I 1965)

There exists a sine series \(S(x)=\sum_{n=1}^{\infty} b_n \sin nx\) with \(b_n>b_{n+1}>0\) and \(b_n\to 0\) such that \(S(x)<0\) in \((\pi-l,\pi)\), \(l>0\). On the other hand, by Fejér’s theorem \((^1)\), \(S(x)>0\) on \((0,\pi)\) if \(\Delta^2 b_n \geq 0\) \((n=1,2,\ldots)\) and \(b_n\to 0\) \((b_1\ne 0)\). In \((^4)\) it was established that \(S(x)\geq 0\) on \((0,\pi)\) if \(b_n\downarrow 0\) and \(\beta_n=\Delta^2 b_n+\Delta^2 b_{n+2}+\cdots\geq 0\) \((n=1,2,\ldots)\); if at least one of the numbers \(\beta_1\) and \(\beta_2\) is positive, then \(S(x)>0\) on \((0,\pi)\), and a positive lower bound for \(S(x)\) on \((0,\pi)\) is given. These results are generalized in the following theorem.

Theorem 1. Let \(b_n\downarrow 0\) and
\[ \beta_n=\Delta^2 b_n+\Delta^2 b_{n+2}+\Delta^2 b_{n+4}+\cdots \geq 0 \]
\((n=1,2,\ldots)\). Denote by \(\{\beta_{n_k}\}\) \((k=1,2,\ldots)\) the subsequence formed from all positive elements of the sequence \(\{\beta_n\}\), preserving the former values of the indices \(n\). Then the sum
\[ S(x)=\sum_{n=1}^{\infty} b_n \sin nx \geq 0 \]
and vanishes on \((0,\pi)\) at exactly those points which are common zeros of the system of functions
\[ \left\{\sin \frac{n_k}{2}x\right\}\quad (k=1,2,\ldots). \tag{1} \]

The numbers \(\beta_n\) cannot all be equal to zero if \(b_1\ne 0\). Clearly, the system (1) has no common zeros on \((0,\pi)\) if at least one of the numbers \(\beta_1\) or \(\beta_2\) is positive, since in this case, respectively, \(\sin x/2\) or \(\sin x\) has no zeros on \((0,\pi)\). The numbers \(\beta_1\) and \(\beta_2\) cannot simultaneously be equal to zero if \(b_n\to 0\), \(\Delta^2 b_n\geq 0\), and \(S(x)\ne 0\).

We note that there exists a sine series whose coefficients satisfy the conditions of Theorem 1, and whose sum vanishes at a point lying inside the interval \((0,\pi)\).

The proof of Theorem 1 is based on the representation
\[ S(x)=2\operatorname{ctg}\left(\frac{x}{2}\right)\sum_{k=1}^{\infty}\beta_n \sin^2\left(\frac{n_k}{2}x\right), \tag{2} \]
which is derived from formula (5) \((^5)\).

If \(\beta_n\geq 0\) for \(n\geq N+1\), then for any \(m\geq N\) and \(0<x<\pi\) it is true that
\[ S(x)\geq 2\operatorname{ctg}\left(\frac{x}{2}\right)\sum_{n=1}^{m}\beta_n \sin^2\left(\frac{n}{2}x\right). \tag{3} \]

Let, for example, \(b_n=1/n^2\). Then all \(\beta_n>0\). For definiteness, putting \(m=2\) in (3), we obtain:
\[ \sum_{n=1}^{\infty}\frac{\sin nx}{n^2} > \left(\frac{\pi^2}{6}-1\right)\sin x+ \left(7-\frac{2}{3}\pi^2\right)\sin x\cos^2\left(\frac{x}{2}\right), \quad 0<x<\pi. \]

Consider the cosine series \(C_1(x)=\sum_{n=1}^{\infty} a_n \cos(2n-1)x\) (it may diverge at \(x=0\)). The function \(C_1(x)\) is even; moreover, \(C_1(x)=-C_1(\pi-x)\). Therefore it suffices to study \(C_1(x)\) for \(0<x<\pi/2\). If \(\Delta^3 a_n\geqslant 0\) \((n=1,2,\ldots)\) and \(a_n\to 0\), then it is known \((^1)\) that \(C_1(x)>0\) on \((0,\pi/2)\). (Moreover, \(C_1(x)\) decreases monotonically on \((0,\pi/2)\).) On the other hand, there exists a cosine series \(\sum_{n=1}^{\infty} a_n\cos(2n-1)x\) with \(\Delta^2 a_n\geqslant 0\) and \(a_n\to 0\), whose sum \(C_1(x)<0\) on \((\pi/2-\varepsilon,\pi)\), \(\varepsilon>0\). In \((^6)\) it was established that \(C_1(x)\geqslant 0\) on \((0,\pi/2)\), if \(a_n\downarrow 0\) and
\(\Delta^3a_n+\Delta^3a_{n+2}+\Delta^3a_{n+4}+\cdots\geqslant 0\) \((n=1,2,\ldots)\).

In Theorem 2 these results are generalized.

Theorem 2. Let \(a_n\downarrow 0\) and
\[ \alpha_n=\Delta^3a_n+\Delta^3a_{n+2}+\Delta^3a_{n+4}+\cdots\geqslant 0 \]
\((n=1,2,\ldots)\). Denote by \(\{a_{n_k}\}\) \((k=1,2,\ldots)\) the subsequence consisting of all positive elements of \(\{a_n\}\), with the former values of the indices \(n\) preserved. Then the sum \(C_1(x)=\sum_{n=1}^{\infty} a_n\cos(2n-\)
\[ -1)x\geqslant 0 \]
and vanishes on \((0,\pi/2)\) at those and only those points which are common zeros of the system
\[ \{\sin n_kx\}\qquad (k=1,2,\ldots). \tag{4} \]

In particular, if at least one of the numbers \(\alpha_1\) or \(\alpha_2\) is positive, then the system (4) has no common zeros on \((0,\pi/2)\), since neither \(\sin x\) nor \(\sin 2x\) has zeros on \((0,\pi/2)\). The numbers \(\alpha_1\) and \(\alpha_2\) cannot vanish simultaneously if \(\Delta^3a_n\geqslant 0\) \((n=1,2,\ldots)\), \(a_n\to 0\), and \(C_1(x)\ne 0\).

Theorem 2 is proved on the basis of the transformation
\[ C_1(x)=\cos x\sum_{k=1}^{\infty}\alpha_{n_k}\frac{\sin^2(n_kx)}{\sin^2 x}. \tag{5} \]

Theorem 3. If \(a_n\to 0\), \(\Delta^2a_n\geqslant 0\),
\[ p_n=\sum_{k=1}^{\infty}(k+1)\Delta^4a_{n+2k}\geqslant 0 \]
for \(n\geqslant N+1\), then for any integer \(m\geqslant N\)
\[ C(x)\geqslant C(\pi)+2\operatorname{ctg}^2\left(\frac{x}{2}\right)\sum_{n=1}^{m}p_n\sin^2\left(\frac{n}{2}x\right), \tag{6} \]
where
\[ C(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos nx. \]
If \(m=\infty\), then (6) becomes an equality.

The proof of Theorem 3 is based on the transformation
\[ C(x)=C(\pi)+\operatorname{ctg}\left(\frac{x}{2}\right)\sum_{n=1}^{\infty}\Delta d_n\sin nx, \]
where \(\Delta d_n=\Delta^2a_n+\Delta^2a_{n+2}+\cdots\) \((^5)\), with the use of (3).

The conditions of Theorem 3 are certainly fulfilled if \(a_n\downarrow 0\) and
\[ \Delta^4a_n+\Delta^4a_{n+2}+\Delta^4a_{n+4}+\cdots\geqslant 0 \]
\((n\geqslant N+1)\).

Let \(N=0\). Then \(p_n\geqslant 0\) \((n=1,2,\ldots)\). If \(C(x)\ne\mathrm{const}\), then not all \(p_n\) can be equal to zero. In this case \(C(x)\) attains its least value at the points \(x=\pi\pmod{2\pi}\). Consequently, \(C(x)\geqslant C(\pi)\). This result is known \((^1)\), but under stronger restrictions: \(\Delta^4a_n\geqslant 0\) \((n=1,2,\ldots)\) and \(a_n\to 0\).

It is known \((^3)\) that \(C(x)\geqslant 0\), if \(\Delta^2a_n\geqslant 0\) \((n=0,1,2,\ldots)\) and \(a_n\to 0\). Under the conditions of Theorem 3 (for \(N=0\)) it turns out that \(C(x)\geqslant 0\) also for

\(\Delta^2 a_0 < 0\), but nevertheless it must be that
\(\Delta^2 a_0 > -(\Delta^2 a_2 + \Delta^2 a_4 + \Delta^2 a_6 + \ldots)\) (the expression in parentheses is positive, if the case \(a_n \equiv 0\) \((n=2,3,\ldots)\) is excluded).

Remark. Let \(a_n \to 0\), \(\Delta^2 a_n \geqslant 0\), and \(p_n \geqslant 0\) for \(n=2,3,\ldots\). Then, for any integer \(k \geqslant 1\),

\[ C(x) \leqslant C(\pi) + \operatorname{ctg}\left(\frac{x}{2}\right)\sum_{n=1}^{\infty}\Delta a_n \sin nx -2\operatorname{ctg}^2\left(\frac{x}{2}\right)\sum_{n=1}^{k} p_{n+1}\sin^2\left(\frac{n}{2}x\right). \]

If \(k=\infty\), then this inequality becomes an equality.

Theorem 4. If \(b_n \to 0\) and \(\Delta^2 b_n \geqslant 0\) for \(n \geqslant N+1\), then for any integer \(m \geqslant N\) and \(0<x<\pi\),

\[ S_1(x) \geqslant \frac{b_1}{2}\sin x +\frac{1}{2\sin x}\sum_{n=1}^{m}\left(\Delta^2 b_n+4\beta_{n+1}\cos^2 x\right)\sin^2(nx), \tag{7} \]

where

\[ S_1(x)=\sum_{n=1}^{\infty} b_n\sin(2n-1)x \quad\text{and}\quad \beta_{n+1}=\Delta^2 b_{n+1}+\Delta^2 b_{n+3}+\Delta^2 b_{n+5}+\ldots \]

If \(m=\infty\), then (7) becomes an equality.

For example, putting \(m=1\) in (7), we find

\[ S_1(x)\geqslant \frac{b_1}{2}\sin x +\frac{1}{2}\left(\Delta^2 b_1+4\beta_2\cos^2 x\right)\sin x \quad (0\leqslant x\leqslant \pi). \]

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

Received
19 I 1965

REFERENCES

\(^{1}\) L. Fejér, Trans. Am. Math. Soc., 39, 18 (1936).
\(^{2}\) N. K. Bari, Trigonometric Series, Moscow, 1961.
\(^{3}\) W. H. Young, Proc. London Math. Soc., 12, 41 (1913).
\(^{4}\) I. N. Pak, DAN, 151, No. 1, 38 (1963).
\(^{5}\) I. N. Pak, Vestn. Leningrad Univ., No. 7 (1963).
\(^{6}\) I. N. Pak, Tr. uchebn. institutov svyazi, issue 12 (1963).