Abstract Generated abstract
This paper studies weighted approximation of functions on the real line by algebraic polynomials with the Gaussian weight exp(-x²/2), using weighted Lp norms. It proves an inverse-type estimate for an antiderivative F of a function f, showing that the best weighted polynomial approximation error of F of degree n is bounded by a constant times n⁻¹/² times the corresponding approximation error of f of degree n minus 1. The argument first establishes a related L1 estimate for functions of bounded variation via Nikolsky duality, then uses Hermite polynomial expansions, Vallée-Poussin means, and interpolation to obtain the result for all p. These estimates refine earlier results in weighted polynomial approximation under weaker or different hypotheses.
Full Text
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.