MATHEMATICS

V. F. VLASOV and A. F. TIMAN

ON A CERTAIN RELATION FOR INTEGRALS OF MODULI OF TRIGONOMETRIC POLYNOMIALS

(Presented by Academician S. N. Bernstein on 17 II 1961)

Let

\[ T_n(x)=\frac{a_0^{(n)}}{2}+\sum_{\nu=1}^{n} a_\nu^{(n)}\cos \nu x \tag{1} \]

be a sequence of even trigonometric polynomials with real coefficients, and let \(\rho_n(x)\) be a sequence of functions of bounded variation, defined on \([-\pi,\pi]\), for which
\[ \operatorname*{Var}_{x\in[-\pi,\pi]}\rho_n(x)=1. \]
Denote by

\[ c_n=\int_{-\pi}^{\pi} e^{int}\,d\rho_n(t) \tag{2} \]

the corresponding diagonal sequence of Fourier–Stieltjes coefficients for \(\rho_n(x)\).

In the present note we consider the integrals

\[ L(T_n;\rho_n)=\int_{-\pi}^{\pi}\left|\int_{-\pi}^{\pi}T_n(x+t)\,d\rho_n(t)\right|\,dx \tag{3} \]

and establish one relation between the sequences of constants (3) and (2).

Theorem. If the coefficients of the polynomials (1) form a convex or concave system of numbers
\[ \left(\Delta^2 a_k^{(n)}=a_k^{(n)}-2a_{k+1}^{(n)}+a_{k+2}^{(n)}\le 0 \quad\text{or}\quad \Delta^2 a_k^{(n)}\ge 0;\ k=0,1,\ldots,n-1;\ a_{n+1}^{(n)}=0\right) \]
and, for some positive sequence \(\varepsilon_n=O\left(\frac1n\right)\),

\[ \operatorname*{Var}_{t\in[-\varepsilon_n,\varepsilon_n]}\rho_n(t)\ge 1-O\left(\frac1n\right), \tag{4} \]

then the equality

\[ L(T_n;\rho_n)=|c_n|\int_{-\pi}^{\pi}|T_n(x)|\,dx +O\left(\max_{0\le \nu\le n}|a_\nu^{(n)}|\right) \tag{5} \]

holds, in which \(O(1)\) is a quantity uniformly bounded with respect to \(n\) over all polynomials \(T_n(x)\).

We note that condition (4), while not, generally speaking, necessary, is in a certain sense essential for the validity of relation (5). Without it, i.e. in the general case, the theorem stated would already be false. To see this, it suffices to put \(a_\nu^{(n)}=1\) \((\nu=0,1,\ldots,n)\)

and consider the sequence of functions

\[ \rho_n(t)= \begin{cases} 0, & -\pi \leq t < -\dfrac{2k+1}{2n}\pi,\\[4pt] \dfrac12, & -\dfrac{2k+1}{2n}\pi \leq t < \dfrac{2k+1}{2n}\pi,\\[6pt] 1, & \dfrac{2k+1}{2n}\pi \leq t \leq \pi, \end{cases} \]

where \(k=k(n)\) is a nonnegative integer-valued function of \(n\), increasing without bound together with \(n\). In this case it is known (see (3), Lemma 1) that, as \(n\to\infty\),

\[ L(T_n;\rho_n) = \frac12 \int_{-\pi}^{\pi} \left| D_n\left(x+\frac{2k+1}{2n}\pi\right) + D_n\left(x-\frac{2k+1}{2n}\pi\right) \right|\,dx = \]

\[ = \frac4\pi \ln\{K(n)+1\}+O(1)\to\infty, \qquad D_n(x)=\frac{\sin (n+\tfrac12)x}{2\sin \tfrac12 x}, \]

whereas the right-hand side of (5) is bounded.

Thus, for the validity of relation (5) the condition \(\varepsilon_n=O\!\left(\dfrac1n\right)\) is necessary in the sense that, if it is not satisfied, then there exists a sequence of functions \(\rho_n(t)\) satisfying (4) and such that relation (5) does not hold.

On the other hand, the example

\[ \rho_n(t)= \begin{cases} 0, & -\pi \leq t < -\dfrac{\pi}{2n},\\[4pt] \dfrac12, & -\dfrac{\pi}{2n} \leq t < \dfrac{\pi}{2n},\\[6pt] 1, & \dfrac{\pi}{2n} \leq t \leq \pi, \end{cases} \]

\[ T_n(x)=\frac12+\sum_{\nu=1}^{m}\cos \nu x+\cos nx, \]

where \(m=m(n)\) is an arbitrary integer-valued function of \(n\) for which

\[ \lim_{n\to\infty} m(n)=\infty,\qquad \sup_{n\geq 1}\frac{m}{n}<1, \]

shows that, for the validity of relation (5), condition (4) alone, without the assumption of convexity or concavity of the coefficient system of the polynomial \(T_n(x)\), is still insufficient. Since in this case \(c_n=0\), the right-hand side of (5) is bounded, whereas (see (4), Theorem 11)

\[ L(T_n;\rho_n) = \frac12 \int_{-\pi}^{\pi} \left| D_m\left(x+\frac{\pi}{2n}\right) + D_m\left(x-\frac{\pi}{2n}\right) \right|\,dx +O(1) = \]

\[ = \frac4\pi \cos\frac{m\pi}{2n}\ln(m+1)+O(1), \]

i.e. \(L(T_n;\rho_n)\to\infty\) as \(n\to\infty\).

At the same time, there exist examples showing that relation (5) may sometimes hold also in cases not covered by the theorems, i.e., in fact it is valid for a broader class of sequences \(T_n(x)\) and \(\rho_n(x)\).

Let us also note that in some concrete cases the asymptotic estimate of the sequence \(L(T_n;\rho_n)\) is known.

In particular, for

\[ \rho_n(t)= \begin{cases} 0, & -\pi \leq t < 0,\\ 1, & 0 \leq t \leq \pi \end{cases} \]

the equality (6) is valid (see also (7), p. 8. 2. 33)

\[ L(T_n;\rho_n)=\int_{-\pi}^{\pi}|T_n(x)|\,dx = \frac{4}{\pi}\left|\sum_{\nu=0}^{n}\frac{a_{\nu}^{(n)}}{n-\nu+1}\right| + O\left(\max_{0\leq \nu\leq n}|a_{\nu}^{(n)}|\right). \tag{6} \]

From the theorem presented and equality (6) there follows directly a number of other results relating to particular sequences of polynomials \(T_n(x)\) and functions \(\rho_n(x)\) (see, for example, (5)).

Corollary. If \(a_{\nu}^{(n)}=O(1)\), then, under the conditions of the theorem, the sequence of constants \(L(T_n;\rho_n)\) is bounded if and only if

\[ c_n=O\left(\left[1+\left|\sum_{\nu=0}^{n}\frac{a_{\nu}^{(n)}}{n-\nu+1}\right|\right]^{-1}\right). \]

In special cases, for \(a_{\nu}^{(n)}=1\) \((\nu=0,1,\ldots,n)\), this corollary contains the known convergence conditions for summation processes of the S. N. Bernstein—Rogosinski type for Fourier series (see \((1^2,{}^4,{}^5)\)).

Received
10 II 1961

CITED LITERATURE

\(^{1}\) S. N. Bernstein, C. R., 191, 976 (1930); S. N. Bernstein, Collected Works, 1, 1952, p. 523.
\(^{2}\) W. Rogosinski, Math. Ann., 95, 110 (1925).
\(^{3}\) A. F. Timan, Izv. AN SSSR, ser. matem., 11, 263 (1947).
\(^{4}\) A. F. Timan, Izv. AN SSSR, ser. matem., 14, 85 (1950).
\(^{5}\) A. F. Timan, M. M. Ganzburg, DAN, 63, No. 6 (1948).
\(^{6}\) A. F. Timan, DAN, 101, No. 2 (1955).
\(^{7}\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.