UDC 517.512

MATHEMATICS

Academician of the Academy of Sciences of the Azerbaijan SSR I. I. Ibragimov, F. T. Nasibov

ON ESTIMATING THE BEST APPROXIMATION OF AN INTEGRABLE FUNCTION ON THE REAL AXIS BY ENTIRE FUNCTIONS OF FINITE DEGREE

Let \(W_\sigma^{(p)}(p \geqslant 1)\) be the class of entire functions \(g_\sigma(z)\) of finite degree \(\leqslant \sigma\), for which \(g_\sigma(x) \in L_p(-\infty,\infty)\), \(p \geqslant 1\). Denote by \(A_\sigma(f)_p\) the best approximation of a function \(f(x) \in L_p(-\infty,\infty)\) in the metric of the space \(L_p(-\infty,\infty)\) by entire functions from the class \(W_\sigma^{(p)}\), i.e.

\[ A_\sigma(f)_p=\inf_{g_\sigma \in W_\sigma^{(p)}}\left(\int_{-\infty}^{\infty}|f(x)-g_\sigma(x)|^p\,dx\right)^{1/p}, \]

and introduce for consideration the moduli of continuity

\[ \omega_1(f;\delta)_p=\sup_{|h|\leqslant \delta} \left(\int_{-\infty}^{\infty}|f(x+h)-f(x)|^p\,dx\right)^{1/p}, \]

\[ \omega_2(f;\delta)_p=\sup_{|h|\leqslant \delta} \left(\int_{-\infty}^{\infty}|f(x+h)-2f(x)+f(x-h)|^p\,dx\right)^{1/p}. \]

It is known that for a function \(f(x)\in L_p(-\infty,\infty)\) there exist constants \(K_1\) and \(K_2\) satisfying the inequalities:

\[ A_\sigma(f)_p \leqslant K_i\omega_i(f;\delta)_p \qquad (i=1,2), \]

which are analogues of the classical Jackson theorem for periodic functions (see \((^1)\), p. 274).

The present article is devoted* to finding the possible least values \(K_i^0\) of the constants \(K_i\) \((i=1,2)\), mainly for \(p=2\). First of all, two auxiliary lemmas are proved.

Lemma 1. Let \(f(x)\in L_2(-\infty,\infty)\), and let \(\varphi(x)\) be its Fourier transform in the sense of \(L_2(-\infty,\infty)\), i.e.

\[ f(x)=\frac{1}{\sqrt{2\pi}}\frac{d}{dx} \int_{-\infty}^{\infty}\varphi(t)\frac{e^{ixt}-1}{it}\,dt, \tag{1} \]

where \(\varphi(x)\in L_2(-\infty,\infty)\). Then the function

\[ g_\sigma^0(x)=\frac{1}{\sqrt{2\pi}}\int_{-\sigma}^{\sigma}\varphi(t)e^{ixt}\,dt \tag{2} \]

is an entire function from the class \(W_\sigma^{(2)}\), least deviating from \(f(x)\) in the metric \(L_2(-\infty,\infty)\). Moreover,

\[ A_\sigma(f_2)=\|f(x)-g_\sigma^0(x)\|_2 =\left(\int_{|t|>\sigma}|\varphi(t)|^2\,dt\right)^{1/2}. \tag{3} \]

Indeed, applying the known Plancherel theorem to the function

\[ f(x)-g_\sigma(x)=\frac{1}{\sqrt{2\pi}}\frac{d}{dx} \int_{-\infty}^{\infty}[\varphi(x)-\psi(x)]\frac{e^{ixt}-1}{it}\,dt, \]

* Works \((^{2-5})\) are devoted to similar questions in the periodic case.

where \(\psi(t)=0\) for \(|t|>\sigma\), we find

\[ \inf_{g_\sigma\in W_\sigma^{(2)}} \|f(x)-g_\sigma(x)\|_2^2 = \inf_{\psi\in L_2(-\sigma,\sigma)} \int_{-\sigma}^{\sigma} |\varphi(t)-\psi(t)|^2\,dt + \int_{|t|>\sigma} |\varphi(t)|^2\,dt. \]

This implies equality (3). Applying Lemma 1 to the function

\[ f^{(r)}(x)=\frac{1}{\sqrt{2\pi}}\frac{d}{dx} \int_{-\infty}^{\infty}\frac{e^{ixt}-1}{it}(it)^r\varphi(t)\,dt \in L_2(-\infty,\infty), \]

we find

\[ A_\sigma(f^{(r)})_2 = \left\{\int_{|t|>\sigma}|(it)^r\varphi(t)|^2\,dt\right\}^{1/2} > \sigma^r \left\{\int_{|t|>\sigma}|\varphi(t)|^2\,dt\right\}^{1/2} = \sigma^r A_\sigma(f)_2. \]

Lemma 2. If \(\varphi(t)\) is any function from \(L_2(-\infty,\infty)\), then

\[ \inf_{|y|\le \pi/\sigma}\int_{|t|>\sigma}|\varphi(t)|^2\cos yt\,dt<0. \tag{4} \]

Proof. Extending evenly to \((-\infty,0)\) the function

\[ \varphi_\sigma(y)=-\sin\sigma y \quad \text{for } 0\le y\le \pi/\sigma; \qquad \varphi_\sigma(y)=0 \quad \text{for } y>\pi/\sigma, \]

we represent it in the form of a Fourier integral

\[ \varphi_\sigma(y)=\sqrt{\frac{2}{\pi}}\int_0^\infty \widetilde{\varphi}_\sigma(t)\cos yt\,dt. \]

From this we find

\[ \widetilde{\varphi}(y) = \sqrt{\frac{2}{\pi}}\int_0^\infty \varphi_\sigma(y)\cos yt\,dt = \frac{1}{\sqrt{2\pi}}\frac{4\sigma}{t^2-\sigma^2}\cos^2\frac{\pi t}{2\sigma}. \]

Let us note that the function

\[ F_\sigma(y)=\sqrt{\pi/2}\,|\varphi(t)|^2,\quad |t|>\sigma; \qquad F_\sigma(y)=0,\quad |t|\le \sigma, \]

is the cosine transform of the function

\[ F_\sigma(y)=\int_\sigma^{+\infty}|\varphi(t)|^2\cos yt\,dt, \qquad 0\le y\le \pi/\sigma. \]

It is not difficult to show that

\[ \int_0^\infty F_\sigma(y)\varphi_\sigma(y)\,dy = -\int_0^\infty \widetilde{F}_\sigma(y)\widetilde{\varphi}_\sigma(y)\,dy = 2\sigma\int_\sigma^\infty |\varphi(t)|^2 \frac{\cos^2\pi t/2\sigma}{t^2-\sigma^2}\,dt \ge 0. \tag{5} \]

On the other hand, if \(F_\sigma(y)\ge 0\) everywhere on \([0,\pi/\sigma]\), then

\[ \int_0^\infty F_\sigma(y)\varphi_\sigma(y)\,dy = -\int_0^{\pi/\sigma} F_\sigma(y)\sin\sigma y\,dy <0. \tag{6} \]

Inequality (6) contradicts (5), and, consequently, \(F_\sigma(y)\) assumes both positive and negative values on \([0,\pi/\sigma]\), i.e. (4) is true. Owing to these lemmas, the following is proved.

Theorem 1. If the function \(f(x)\in L_2(-\infty,\infty)\), then for the best approximation \(A_\sigma(f)_2\) the inequalities

\[ A_\sigma(f)_2 < \frac{1}{\sqrt{2}}\,\omega_1(f;\pi/\sigma)_2, \tag{7} \]

\[ A_\sigma(f)_2 < \frac{1}{2}\,\omega_2(f;\pi/\sigma)_2 \tag{8} \]

hold.

Proof. From equality (1) we find

\[ \omega_1\left(f;\frac{\pi}{\sigma}\right)_2 = \sup_{|y|\le \pi/\sigma}\|f(x+y)-f(x)\|_2 = \sqrt{2}\left\{ \int_{|t|>\sigma}|\varphi(t)|^2\,dt - \inf_{0\le y\le \pi/\sigma}F_\sigma(y) \right\}. \]

Hence, by Lemmas 1 and 2, (7) follows. Moreover, for \(|t|\leq \pi/\sigma\),

\[ \sup_{|t|\leq \pi/\sigma}\|f(x+t)-2f(x)+f(x-t)\|_2^2 \geq 4\sup_{|t|\leq \pi/\sigma}\int_{|u|>\sigma}|\varphi(u)|^2(1-\cos ut)^2\,du \geq \]

\[ \geq 4\int_{|u|>\sigma}|\varphi(u)|^2\,du - 8\inf_{0\leq y\leq \pi/\sigma}F_\sigma(y). \]

Hence inequality (8) follows.

Corollary. If \(f(x)\) has a derivative \(f^{(r)}(x)\in L_2(-\infty,\infty)\), then the estimates

\[ A_\sigma(f)_2\leq \frac{1}{\sqrt{2}\,\sigma^r}\, \omega_1(f^{(r)};\pi/\sigma)_2, \qquad A_\sigma(f)_2\leq \frac{1}{2\sigma^r}\, \omega_2(f^{(r)};\pi/\sigma)_2 \]

are valid.

Along the way it is established that, for the best approximation of an entire function

\[ f_\lambda(x)=\frac{1}{\sqrt{2\pi}}\int_{-\lambda}^{\lambda}\varphi(t)e^{ixt}\,dt \]

of degree \(\lambda\), where \(\varphi(t)\in L_2(-\lambda,\lambda)\), by entire functions from the class \(W_\sigma^{(2)}\), we have

\[ A_\sigma(f_\lambda)_2 = \left( \int_{\sigma<|t|\leq \lambda}|\varphi(t)|^2\,dt \right)^{1/2}. \]

As an example, the function \(f(x)=e^{-|x|}\in L_2(-\infty,\infty)\) with Fourier transform

\[ \varphi(t)=\sqrt{\frac{2}{\pi}}\,\frac{1}{1+t^2} \]

is considered. According to Lemma 1, we have

\[ A_\sigma(e^{-|x|})_2 = \left[ 1-\frac{2}{\pi}\left(\operatorname{arctg}\sigma-\frac{\sigma}{1+\sigma^2}\right) \right]^{1/2} < \sqrt{\frac{2}{\sigma}}. \]

Now let \(f(x)\) be an arbitrary function from the class \(L_p(-\infty,\infty)\) \((1<p\leq 2)\). Then

\[ F(x,a)=\frac{1}{\sqrt{2\pi}}\int_{-a}^{a}f(t)e^{ixt}\,dt \]

as \(a\to\infty\) converges in the mean with exponent \(q\), where \(q=p/(p-1)\). The mean limit \(F(x)\), called the Fourier transform of the function \(f(x)\), satisfies the inequality \((^6)\)

\[ \left(\int_{-\infty}^{\infty}|F(x)|^q\,dx\right)^{1/q} \leq B_{p,q} \left(\int_{-\infty}^{\infty}|f(x)|^p\,dx\right)^{1/p}, \]

where

\[ B_{p,q}=(2\pi/q)^{1/2q}(p/2\pi)^{1/2p}. \]

Moreover, for almost all \(x\) we have

\[ F(x)=\frac{1}{\sqrt{2\pi}}\frac{d}{dx} \int_{-\infty}^{\infty} f(x)\frac{e^{ixt}-1}{it}\,dt, \qquad f(x)=\frac{1}{\sqrt{2\pi}}\frac{d}{dx} \int_{-\infty}^{\infty} F(t)\frac{e^{-itx}-1}{-it}\,dt. \]

Thanks to these assertions, the following is proved.

Theorem 2. If \(f(x)\in L_p(-\infty,\infty)\) \((1<p\leq 2)\) and \(f(x)\) is its Fourier transform in the sense of \(L_p(-\infty,\infty)\), then the estimates

\[ A_\sigma(f)_p \geq B_{p,q}^{-1} \left( \int_{|x|>\sigma}|F(x)|^q\,dx \right)^{1/q} \qquad (1/p+1/q=1), \]

\[ \sup_{|y|\leq \pi/\sigma} \left( \int_{-\infty}^{\infty}|F(x)|^q \left|\sin\frac{yx}{2}\right|^q\,dx \right)^{1/q} \leq \frac{1}{2}B_{p,q}\omega_1\left(f;\frac{\pi}{\sigma}\right)_p. \]

The arguments used in proving the assertions given above can also be applied in the multidimensional case. For example, consider a function admitting the representation

\[ f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{ixt+iy\tau}\varphi(t,\tau)\,dt\,d\tau, \tag{9} \]

and entire functions \(g_{\sigma_1,\sigma_2}(x,y)\in W_{\sigma_1,\sigma_2}^{(p)}\), admitting the spectral representation

\[ g_{\sigma_1,\sigma_2}(x,y)=\frac{1}{2\pi} \int_{-\sigma_1}^{\sigma_1}\int_{-\sigma_2}^{\sigma_2} \psi(t,\tau)e^{ixt+iy\tau}\,dt\,d\tau, \]

where \(\psi(t,\tau)\in L_2\binom{-\sigma_1,\sigma_1}{-\sigma_2,\sigma_2}\) and \(\psi(t,\tau)=0\) for \((x,y)\notin \begin{bmatrix}-\sigma_1,\sigma_1\\ -\sigma_2,\sigma_2\end{bmatrix}\).

Theorem 3. If the function \(f(x,y)\in L_2\binom{-\infty,\infty}{-\infty,\infty}\) is defined by equality (9), then the entire function from the class \(W_{\sigma_1,\sigma_2}^{(2)}\) least deviating from it in the sense of the metric \(L_2\binom{-\infty,\infty}{-\infty,\infty}\) is

\[ g_{\sigma_1,\sigma_2}^{0}(x,y)=\frac{1}{2\pi} \int_{-\sigma_1}^{\sigma_1}\int_{-\sigma_2}^{\sigma_2} e^{ixt+iy\tau}\varphi(t,\tau)\,dt\,d\tau, \tag{10} \]

where \(\varphi(x,y)\) is the Fourier transform of the function \(f(x,y)\) in the sense of (9); moreover, the value of the best approximation \(A_{\sigma_1,\sigma_2}(f)_2\) of the function \(f(x,y)\) by entire functions from the class \(W_{\sigma_1,\sigma_2}^{(2)}\) is computed from one of the equalities

\[ A_{\sigma_1,\sigma_2}(f)_2= \left\{ \int_{-\infty}^{\infty}\int_{|\tau|\geq\sigma_2}|\varphi(t,\tau)|^2\,dt\,d\tau + \int_{|t|>\sigma_1}\int_{|\tau|\leq\sigma_2}|\varphi(t,\tau)|^2\,dt\,d\tau \right\}^{1/2}, \tag{11} \]

or

\[ A_{\sigma_1,\sigma_2}(f)_2= \left\{ \int_{|t|>\sigma_1}\int_{-\infty}^{\infty}|\varphi(t,\tau)|^2\,dt\,d\tau + \int_{|t|\leq\sigma_1}\int_{|\tau|>\sigma_2}|\varphi(t,\tau)|^2\,dt\,d\tau \right\}^{1/2}. \tag{12} \]

Introduce the notation:

\[ A_{\sigma_1,\infty}(f)_p= \inf_{g_{\sigma_1,\infty}\in W_{\sigma_1,\infty}^{(p)}} \left\{ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} |f(x,y)-g_{\sigma_1,\infty}(x,y)|^p\,dx\,dy \right\}^{1/p} \quad (p\geq 1), \]

where

\[ g_{\sigma_1,\infty}(x,y)= \int_{-\sigma_1}^{\sigma_1}\int_{-\infty}^{\infty} e^{ixt+iy\tau}\varphi(t,\tau)\,dt\,d\tau \]

is an entire function in \(x\) from the class \(W_{\sigma_1}^{(p)}\). The quantity \(A_{\infty,\sigma_2}(f)_p\) is defined analogously, and it is proved that

\[ A_{\sigma_1,\sigma_2}(f)_2\leq \left[A_{\sigma_1,\infty}^{2}(f)_2+A_{\infty,\sigma_2}^{2}(f)_2\right]^{1/2}. \]

Owing to this, from the inequalities

\[ A_{\sigma_1,\infty}^{2}(f)_2< \frac{1}{2}\omega_1^2\left(f;\frac{\pi}{\sigma_1};0\right)_2; \qquad A_{\infty,\sigma_2}^{2}(f)_2< \frac{1}{2}\omega_1^2\left(f;0;\frac{\pi}{\sigma_2}\right)_2 \]

we obtain

\[ A_{\sigma_1,\sigma_2}(f)_2< \frac{1}{\sqrt{2}} \left\{ \omega_1^2\left(f;\frac{\pi}{\sigma_1};0\right)_2 + \omega_1^2\left(f;0;\frac{\pi}{\sigma_2}\right)_2 \right\}^{1/2}. \]

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Baku

Received
25 V 1970

References

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