V. G. PONOMARENKO

SUMMATION OF FOURIER INTEGRALS AND BEST APPROXIMATION BY ENTIRE FUNCTIONS

(Presented by Academician V. I. Smirnov on 11 VI 1962)

Let us consider the space \(L_p(-\infty,\infty)\), \(1 \leq p \leq \infty\), of functions \(f(x)\) defined on the entire real axis, i.e.,
\[ \|f(x)\|_{L_p} = \left\{\int_{-\infty}^{\infty} |f(x)|^p dx\right\}^{1/p} <\infty \qquad (1\leq p<\infty), \]
\[ \|f(x)\|_{L_\infty} = \operatorname{vrai\,sup}_{-\infty<x<\infty}|f(x)|<\infty. \]

Denote
\[ U_\lambda(f;x;\varphi) = \int_{-\lambda}^{\lambda} \varphi_\lambda(u)F(u)e^{iux}\,du, \]
where \(F(u)\) is the Fourier transform of the function \(f(x)\)*, and \(\varphi_\lambda(u)\) is a certain function satisfying the conditions:
\[ \varphi_\lambda(u)= \begin{cases} 1, & \text{for } u=0,\\ 0, & \text{for } |u|\geq \lambda, \end{cases} \]
\[ \varphi_\lambda(-u)=\varphi_\lambda(u),\qquad \varphi_\lambda''(u)\in L_p(-\infty,\infty). \]

Let
\[ R_\lambda(f;\varphi)_{L_p} = \|f(x)-U_\lambda(f;x;\varphi)\|_{L_p} \]
and let \(A_\sigma(f)_{L_p}\) denote the best approximation of the function in the corresponding metric by entire functions of degree \(\leq \sigma\) belonging to the space \(L_p(-\infty,\infty)\), i.e.,
\[ A_\sigma(f)_{L_p} = \inf_{g_\sigma}\|f(x)-g_\sigma(x)\|_{L_p}, \]
where \(g_\sigma(x)\in L_p\) is an entire function of degree \(\leq \sigma\).

Below are given some results concerning the order of decrease of the quantity \(R_\lambda(f;\varphi)_{L_p}\) \((\lambda\to\infty)\), depending on the behavior of the function \(A_\sigma(t)_{L_p}\) (as \(\sigma\to\infty\)).

Theorem 1. Let \(H_{\psi(\lambda)}\) denote the class of functions uniformly continuous on the entire axis, whose best approximation by entire functions of degree \(\leq \lambda\) satisfies the inequality
\[ A_\lambda(t)\leq \psi(\lambda), \]
where \(\psi(\lambda)\) is a function continuous from the right and decreasing to zero as \(\lambda\to\infty\). Then for \(p=\infty\) the inequality
\[ \sup_{f\in H_{\psi(\lambda)}} R_\lambda(t;\varphi)_{L_\infty} \geq \left| \int_0^\lambda \varphi_\lambda'(t)\psi(t)\,dt \right|. \tag{1} \]

* Under the assumption that \(F(u)\) exists and belongs to \(L_q(-\infty,\infty)\), \(1\leq q\leq\infty\).

In particular, it follows from this, for example, that for

\[ \varphi_\lambda(u)=1-\frac{|u|^r}{\lambda^r}\qquad (r\geqslant 1) \tag{2} \]

\[ \sup_{f\in H_\psi(\lambda)} R_\lambda\left(f;1-\frac{|u|^r}{\lambda^r}\right)_{L_\infty} \geqslant \begin{cases} \left|\dfrac{r}{\lambda^r}\displaystyle\int_0^\lambda t^{r-1}\psi(t)\,dt\right|, & \text{for } r>1,\\[1.2em] \left|\dfrac{1}{\lambda}\displaystyle\int_0^\lambda \psi(t)\,dt\right|, & \text{for } r=1. \end{cases} \tag{3} \]

The following theorem gives an upper estimate for the quantity \(R_\lambda(f;\varphi)_{L_p}\).

Theorem 2. If \(f(x)\in L_p(-\infty,\infty)\), \(1\leqslant p\leqslant\infty\), then

\[ R_\lambda(f;\varphi)_{L_p} \leqslant C\left\{ |\varphi_\lambda'(0)|\int_0^\lambda A_u(f)_{L_p}\,du + \int_0^\lambda |\varphi_\lambda''(u)|A_u(f)_{L_p}(\lambda-u) \ln\frac{\lambda+u}{\lambda-u}\,du \right\}. \tag{4} \]

It follows directly from Theorem 2 that for the function \(\varphi_\lambda(u)\) defined by equality (2), the estimate

\[ R_\lambda\left(f;1-\frac{|u|^r}{\lambda^r}\right)_{L_p} \leqslant \frac{C_r}{\lambda^r}\int_0^\lambda u^{r-1}A_u(f)_{L_p}\,du. \tag{5} \]

Estimates (3) and (5) give the order relation

\[ \sup_{f\in H_\psi(\lambda)} R_\lambda\left(f;1-\frac{|u|^r}{\lambda^r}\right)_{L_\infty} \asymp \frac{1}{\lambda^r}\int_0^\lambda u^{r-1}\psi(u)\,du. \tag{6} \]

Theorem 2 is proved with the aid of inverse theorems of the constructive theory of functions defined on the whole real axis (see \((^4)\), Theorem 2), the integral representation of Bochner–Fejér sums (see \((^5)\)) and the known estimate \((^2)\)

\[ \frac{1}{\pi u}\int_{-\infty}^{\infty} \left| \frac{2\sin\dfrac{2\lambda-u}{2}t\,\sin\dfrac{u}{2}t}{t^2} \right|\,dt \leqslant \frac{\pi}{4}+\frac{2}{\pi}\ln\frac{2\lambda-u}{u}. \]

As relation (6) shows, inequality (5) for \(p=\infty\) cannot be improved in order.

We give one assertion showing that in the case \(1<p<\infty\) inequality (5) can be replaced by a sharper one.

Theorem 3. For any function \(f(x)\in L_p(-\infty,\infty)\), with \(1<p<\infty\), the inequality

\[ R_\lambda\left(f;1-\frac{|u|^r}{\lambda^r}\right)_{L_p} \leqslant \frac{C_{p,r}}{\lambda^r} \left( \sum_{\nu=0}^{[\lambda]}(\nu+1)^{\gamma r-1}A_\nu^\gamma(f)_{L_p} \right)^{1/\gamma}, \tag{7} \]

holds, where \(\gamma=p\) for \(1<p\leqslant 2\) and \(\gamma=2\) for \(2\leqslant p<\infty\).

The proof of Theorem 3 is substantially based on the results of the work of M. F. Timan \((^4)\).

Using Parseval’s equality for \(p=2\), it is not hard to verify that for every function \(f(x)\in L_2(-\infty,\infty)\) the order relation

\[ R_\lambda\left(f;1-\frac{|u|^r}{\lambda^r}\right)_{L_2} \asymp \frac{1}{\lambda^2}\left(\sum_{\nu=0}^{[\lambda]}(\nu+1)^{2r-1}A_\nu^2(f)_{L_2}\right)^{1/2}. \]

The method of proof of Theorem 2 makes it possible to obtain the following assertion for uniformly almost-periodic functions whose Fourier exponents have no finite limit points.

Theorem 4*. Let \(f(x)\) be a uniformly almost-periodic function whose Fourier exponents have no finite limit points.

Then

\[ \max_x |f(x)-\sigma_\lambda(f;x)| \le \frac{C}{\lambda}\int_0^\lambda A_u(f)\,du, \]

where

\[ \sigma_\lambda(f;x)= \sum_{|\Lambda_k|<\lambda} \left(1-\frac{|\Lambda_k|}{\lambda}\right)a_k e^{i\Lambda_k x}; \]

\(a_k\) are the Fourier coefficients; \(\Lambda_k\) are the Fourier exponents of the function \(f(x)\)
\((\Lambda_0=0,\ \Lambda_{-k}=-\Lambda_k,\ \Lambda_k<\Lambda_{k+1}\) for \(k=0,1,2,\ldots;\ \lim_{k\to\infty}\Lambda_k=\infty)\).

In particular, when

\[ A_u(f)=O\left[\frac{1}{(u+1)^\alpha}\right]\qquad (0<\alpha\le 1), \]

we obtain the estimate

\[ |f(x)-\sigma_\lambda(f;x)| \le \begin{cases} C\dfrac{1}{\lambda^\alpha}, & \text{for } 0<\alpha<1,\\[6pt] C'\dfrac{\ln\lambda}{\lambda}, & \text{for } \alpha=1, \end{cases} \]

which constitutes the content of Theorems 2 and 3 of paper (¹).

Dnepropetrovsk Agricultural Institute

Received
7 VI 1962

CITED LITERATURE

¹ E. A. Bredikhina, Izv. vyssh. uchebn. zaved., Mathematics, 5(18), 33 (1960).
² B. M. Levitan, Almost-Periodic Functions, Moscow, 1953.
³ S. B. Stechkin, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 62 (1961).
⁴ M. F. Timan, Izv. vyssh. uchebn. zaved., Mathematics, No. 6 (25), 108 (1961).
⁵ E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Moscow–Leningrad, 1948.

* For the case of continuous \(2\pi\)-periodic functions, an analogous theorem was obtained by different methods by S. B. Stechkin (³) and M. F. Timan. We note that M. F. Timan obtained a more general result, which he reported at the seminar of the Department of Higher Mathematics of the Dnepropetrovsk Agricultural Institute on 12 X 1961.