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Reports of the Academy of Sciences of the USSR
- Volume 191, No. 2
MATHEMATICS
GEZA FREUD
ON WEIGHTED APPROXIMATION BY POLYNOMIALS ON THE REAL AXIS
(Presented by Academician L. S. Pontryagin, 8 XII 1969)
Approximation by polynomials with the weight \(\rho(x)=e^{-x^2/2}\) is considered. Denote by \(\|f\|_p\) the norm in the space \(\mathcal L_p=\mathcal L_p(-\infty,+\infty)\), and let \(\mathcal L_p^*\) be the set of such functions \(f(t)\) that \(f\rho\in\mathcal L_p\) and \(\|f\|_p^*=\|\rho f\|_p\). Let \(n\) be an arbitrary integer \(\geq 4\), \(\nu=[n/2]\), \(\pi_n\) the set of algebraic polynomials of degree not exceeding \(n\), and \(c_1,c_2,\ldots\) absolute constants.
We consider the behavior of the quantities
\[ \varepsilon_n^{(p)*}(f)=\inf_{\varphi_n\in\pi_n}\|f-\varphi_n\|_p^* . \tag{1} \]
Theorem 1. If \(F(t)=\int f(\tau)d\tau\), and \(f\in\mathcal L_p^*\), then
\[ \varepsilon_n^{(p)*}(F)\leq c_1 n^{-1/2}\varepsilon_{n-1}^{(p)*}(f). \tag{2} \]
Corollary. The relation
\[ \varepsilon_n^{(p)*}(F)\leq c_2 n^{-1/2}\|f\|_p^* \tag{3} \]
holds.
Already the corollary (3) refines some known results. For \(n=+\infty\) and under the additional condition \(\|F\|_\infty+\|f\|_\infty<\infty\), (3) follows from a theorem of M. M. Dzhrbashyan \((^2)\), and under the somewhat weaker condition \(\|f\|_\infty<\infty\)—from a result of the author \((^5)\). A. S. Dzhafarov \((^1)\) considers the case \(p<+\infty\). We note that in the works \((^1,^2,^5)\) more general weight functions are considered.
We shall prove Theorem 1 at the end of the article, after first establishing the corollary (3).
Theorem 2. Suppose \(F(t)\) has bounded variation on every finite interval; then (the right-hand side is assumed finite)
\[ \varepsilon_n^{(1)*}(F)\leq c_3 n^{-1/2}\int_{-\infty}^{+\infty}\rho(t)\,|dF(t)|. \tag{4} \]
Proof of Theorem 2. By the duality theorem of S. M. Nikolsky \((^3)\), we have
\[ \varepsilon_n^{(1)*}(F)= \sup_{g\in B_n}\int F(t)g(t)\rho(t)\,dt = \sup_{\rho^{-1}G\in B_n}\int G(t)\,dF(t), \tag{5} \]
where \(B_n\) is the set of such functions that \(g(t)\in\mathcal L_\infty\), \(\|g\|_\infty\leq 1\), and \(\int g\varphi_n\rho\,dt=0\) for any \(\varphi_n\in\pi_n\), and
\[ G(x)=\int_x^\infty g(t)\rho(t)\,dt = \int_{-\infty}^{+\infty}\Gamma_x(t)g(t)\rho(t)\,dt = \int(\Gamma_x-\varphi_n)g\rho\,dt\quad(\varphi_n\in\pi_n), \tag{6} \]
where \(\Gamma_x(t)=0\) for \(t<x\), and \(\Gamma_x(t)=1\) if \(t\geq x\).
It follows from \((^6)\) that for any \(x\) there exists \(\varphi_{nx}\in\pi_n\) such that
\[ {}^*\|\Gamma_x-\varphi_{nx}\|_1^*\leq c_3 n^{-1/2}\rho(x). \]
From relation (6), taking \(\varphi_n=\varphi_{nx}\), we obtain that \(|G(x)|\leq c_3 n^{-1/2}\rho(x)^*\) for every admissible \(G\). Thus relation (4) follows from (5), as was required to prove.
Let \(F_n(f;t)\) be the \((C,1)\)-means of order \(n\) of the expansion of a certain funct—
* In the case \(|x|\leq \sqrt n/4\), the lemma of § 2 of the work \((^6)\) is used; if \(x>\sqrt n/4\), then \(\varphi_{nx}(t)\equiv 0\), and if \(x<-\sqrt n/4\), then \(\varphi_{nx}(t)\equiv 1\).
the expansion of \(f \subset {\mathcal L}_1^*\) in orthogonal Hermite polynomials, and
\[ v_n(f;t)=(n-\nu+1)^{-1}\bigl[(n+1)f_n(f;t)-\nu F_\nu(f;t)\bigr] \tag{7} \]
are the Vallée-Poussin means. Then \(v_n(\varphi_\nu;t)\equiv \varphi_\nu(t)\) for every \(\varphi_\nu\in \pi_\nu\). From § 5 of paper \((^4)\) it follows that \(\|F_n(f;t)\|_\infty^*\le c_4\|f(t)\|_\infty^*\); thus from (7) we obtain
\[ \|v_n(f;t)\|_\infty^* \le c_5\|f\|_\infty^* \tag{8} \]
and further
\[ \|v_n(f;t)\|_1^* = \sup_{\|g\|_\infty^*\le 1}\int g(t)v_n(f;t)\rho^2(t)\,dt = \]
\[ = \sup_{\|g\|_\infty^*\le 1}\int f(t)v_n(g;t)\rho^2(t)\,dt \le c_5\|f\|_1^*. \tag{9} \]
Proof of inequality (3). Let first \(p=\infty\). Put \(\psi_x(t)=e^{t^2}\) for \(t\in[0,x]\) and \(\psi_x(t)=0\) for \(t\notin[0,x]\). By Theorem 2 there exists a polynomial \(\varphi_{\nu x}\in \pi_{\nu-1}\) such that
\[ \|\psi_x-\varphi_{\nu x}\|_1^* \le c_6 n^{-1/2}\rho^{-1}(x). \]
If \(\varphi_\nu\in\pi_{\nu-1}\), then \(\int (f-v_{n-1})\varphi_\nu \rho^2\,dt=0\), and
\[ \left|\int_0^x [f(t)-v_{n-1}(f;t)]\,dt\right| = \left|\int_{-\infty}^{+\infty}[f(t)-v_{n-1}(f;t)][\psi_x(t)-\varphi_{\nu x}(t)]\rho^2(t)\,dt\right| \le \]
\[ \le \|f-v_{n-1}(f)\|_\infty^*\,\|\psi_x-\varphi_{\nu x}\|_1^* \le (c_5+1)\|f\|_\infty^*\,\|\psi_x-\varphi_{\nu x}\|_1^* \le c_6 n^{-1/2}\|f\|_\infty^*\rho^{-1}(x). \]
Thus,
\[ \varepsilon_n^{(\infty)*}(F)\le c_6 n^{-1/2}\|f\|_\infty^*, \]
and since \(v_n(\varphi_\nu;t)\equiv \varphi_\nu(t)\) for any \(\varphi_\nu\in\pi_\nu\), from relation (8) we obtain
\[ \|F(t)-v_n(F;t)\|_\infty^* \le (c_5+1)\varepsilon_\nu^{(\infty)*}(F) \le c_7 n^{-1/2}\|f\|_\infty^*, \tag{10} \]
whence inequality (3) follows for \(p=+\infty\).
From inequality (9) and Theorem 2 we obtain
\[ \|F(t)-v_n(F;t)\|_1^* = \inf \|F(t)-\varphi_\nu(t)+v_n(F-\varphi_\nu;t)\|_1^* \le \]
\[ \le (1+c_4)\inf \|F(t)-\varphi_\nu(t)\|_1^* \le c_8 n^{-1/2}\|f\|_1^*. \tag{11} \]
By the Riesz-Thorin interpolation theorem (see \((^7)\), vol. II), from relations (10) and (11) we obtain
\[ \|F(t)-v_n(F;t)\|_p^* \le c_2 n^{-1/2}\|f\|_p^*. \tag{12} \]
Inequality (3) is proved.
Proof of Theorem 1. Let \(\varphi_{n-1}\in\pi_{n-1}\) and
\[ \|f-\varphi_{n-1}\|_p^* < 2\varepsilon_n^{(p)}(f). \]
If in inequality (3) we replace the function \(f\) by \(f-\varphi_{n-1}\), then for a suitably chosen polynomial we have
\[ \left\|F(t)-\int^t \varphi_{n-1}(\tau)\,d\tau-\psi_n(t)\right\|_p^* \le c_2 n^{-1/2}\|f-\varphi_{n-1}\|_p^* \le 2c_2 n^{-1/2}\varepsilon_{n-1}^{(p)*}(f), \]
which completes the proof of the theorem.
Mathematical Institute
Hungarian Academy of Sciences
Budapest
Received
3 XII 1969
References
\(^1\) A. S. Dzhafarov, Tr. Inst. Fiz. i Matem. AN AzerbSSR, 8, 117 (1959).
\(^2\) M. M. Dzhrbashyan, Matem. sborn., 36, 353 (1955).
\(^3\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 10, 207 (1946).
\(^4\) G. Freud, S. Knapowski, Studia Math., 25, 373 (1965).
\(^5\) G. Freud, Acta Math. Acad. Sci. Hung., 20, 223 (1969).
\(^6\) G. Freud, J. Szabados, Acta Sci. Math. (Szeged), in press.
\(^7\) A. Zygmund, Trigonometric Series, 2nd ed., Cambridge, 1959.