Abstract
Full Text
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS BY VALLEE-POUSSIN SUMS
S. A. TELYAKOVSKII
(Presented by Academician N. N. Bogolyubov on 27 III 1958)
We shall consider continuous functions (f(x)) of period (2\pi). Let
[
s_n(f,x)=\frac{a_0}{2}+\sum_{k=1}^{n}(a_k\cos kx+b_k\sin kx)
\quad (n=0,1,2,\ldots),
]
[
\sigma_n(f,x)=\frac{1}{n}\sum_{k=0}^{n-1}s_k(f,x)
\quad (n=1,2,\ldots)
]
be the partial sums of the Fourier series and the Fejér sums of the function (f(x)). Vallée-Poussin ((^{2,3})) first considered the polynomials
[
v_{n,m}(f,x)=\frac{1}{m}\sum_{k=n-m}^{n-1}s_k(f,x)
=\frac{n}{m}\sigma_n(f,x)-\frac{n-m}{m}\sigma_{n-m}(f,x)
\tag{1}
]
[
[(m=1,2,\ldots,n;\ n=1,2,\ldots)],
]
which received the name of Vallée-Poussin sums, and there also indicated the formula
[
|f(x)-v_{n,m}(f,x)|\leq 2\frac{n}{m}E_{n-m}(f),
\tag{2}
]
where (E_k(f)), (k=0,1,2,\ldots), is the best approximation of the function (f(x)) by polynomials of order (k-1).
Let (W^r) be the class of functions (f(x)) for which the derivative (f^{(r-1)}(x)) is absolutely continuous and the inequality (|f^{(r)}(x)|\leq 1) holds almost everywhere, and let (\overline{W}^r) be the class of functions conjugate to functions of (W^r), (r=1,2,\ldots). In this paper we determine the asymptotic behavior as (n\to\infty) of the quantity
[
V_{n,m}(\mathfrak{M})=\sup_{f\in\mathfrak{M}}|f(x)-v_{n,m}(f,x)|_C
]
for (\mathfrak{M}=W^r) and (\mathfrak{M}=\overline{W}^r), under the assumption that (\lim \frac{m}{n}) exists and is equal to (\theta), (0\leq \theta\leq 1).
For Fourier sums ((m=1)), Vallée-Poussin sums close to them (the case (m=o(n))), and Fejér sums ((m=n)), the asymptotic behavior of (V_{n,m}(W^r)) and (V_{n,m}(\overline{W}^r)) is known.
If (m=o(n)), then
[
V_{n,m}(W^r)=\frac{4}{\pi^2}\frac{1}{n^r}\log\frac{n}{m}+O\left(\frac{1}{n^r}\right),
\tag{3}
]
[
V_{n,m}(\overline{W}^r)=\frac{4}{\pi^2}\frac{1}{n^r}\log\frac{n}{m}+O\left(\frac{1}{n^r}\right).
\tag{4}
]
For Fourier sums, formula (3) was obtained by A. N. Kolmogorov ((^4)), and formula (4) by S. M. Nikol’skii ((^{8,9})); for the case (m=o(n)) these formulas were obtained by A. F. Timan ((^{11,12})).
For approximations by Fejér sums, S. M. Nikol’skii ((^{5,7,9})) obtained the following asymptotic formulas:
[
V_{n,n}(W^1)=\frac{2}{\pi}\frac{1}{n}\log n+O\left(\frac{1}{n}\right)
]
and for (r>1)
[
V_{n,n}(W^r)=\frac{c_r}{n}+O\left(\frac{1}{n^r}\right),\qquad
V_{n,n}(\overline{W}^r)=\frac{\overline{c}_r}{n}+O\left(\frac{1}{n^r}\right),
]
where
[
c_r=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(-1)^{k(r-1)}}{(2k+1)^r},\qquad
\overline{c}r=\frac{4}{\pi}\sum.}^{\infty}\frac{(-1)^{kr}}{(2k+1)^r
]
S. B. Stechkin communicated to me the formula*
[
V_{n,n}(\overline{W}^1)=\frac{2}{\pi n}\int_{0}^{\infty}\left|\int_{u}^{\infty}\frac{\sin t}{t^2}\,dt\right|\,du
+O\left(\frac{1}{n^2}\right).
\tag{5}
]
It is also known that in the case (0<\theta<1)
[
V_{n,m}(W^r)=O\left(\frac{1}{n^r}\right),\qquad
V_{n,m}(\overline{W}^r)=O\left(\frac{1}{n^r}\right).
]
These relations follow from inequality (2) and theorems on the order of best approximations of functions from (W^r) and (\overline{W}^r).
The method used in this paper gives the asymptotic behavior of (V_{n,m}(W^r)) and (V_{n,m}(\overline{W}^r)) for (m=1,2,\ldots,n-1), for all (r=1,2,\ldots); for (m=n) (Fejér sums), only for (r=1).
Theorem. For (V_{n,m}(W^r)) and (V_{n,m}(\overline{W}^r)) the following asymptotic formulas hold:
-
If (\theta=0), formulas (3) and (4).
-
If (0<\theta<1), then
[
V_{n,m}(W^r)=c(r,\theta)\frac{1}{n^r}
+O\left(\frac{1}{n^{r+1}}\right)
+O\left(\frac{\varepsilon_n}{n^r}\right);
\tag{6}
]
[
V_{n,m}(\overline{W}^r)=\overline{c}(r,\theta)\frac{1}{n^r}
+O\left(\frac{1}{n^{r+1}}\right)
+O\left(\frac{\varepsilon_n}{n^r}\right),
\tag{7}
]
where
[
\left.
\begin{array}{c}
c(r,\theta)\
\overline{c}(r,\theta)
\end{array}
\right}
=
\frac{2}{\pi\theta}
\int_{0}^{\infty}
\left|
\int_{u_r}^{\infty}\cdots
\int_{u_1}^{\infty}
\left{
\begin{array}{c}
\cos\
\sin
\end{array}
u-\frac{
\begin{array}{c}
\cos\
\sin
\end{array}
(1-\theta)u}{}
\right}
u^{-2}\,du\ldots du_{r-1}
\right|\,du_r,
]
[
\varepsilon_n=
\left|\frac{m}{n}-\theta\right|
\log\frac{1}{\left|m/n-\theta\right|}
\quad \text{for } \frac{m}{n}\ne\theta,
\qquad
\varepsilon_n=0 \quad \text{for } \frac{m}{n}=\theta.
]
- If (\theta=1), then for (r>1) the case (n-m) fixed and (n-m\to\infty) are considered separately.
The case (n-m=p) fixed, (p\ge1):
[
V_{n,m}(W^r)=c(r,p)
\left[
\frac{1}{n}+\frac{p}{n^2}+\ldots+\frac{p^{r-2}}{n^{r-1}}
\right]
+O\left(\frac{1}{n^r}\right);
\tag{8}
]
[
V_{n,m}(\overline{W}^r)=\overline{c}(r,p)
\left[
\frac{1}{n}+\frac{p}{n^2}+\ldots+\frac{p^{r-2}}{n^{r-1}}
\right]
+O\left(\frac{1}{n^r}\right),
\tag{9}
]
[
\text{* Published here with the author’s permission.}
]
where
[
\left.
\begin{matrix}
c(r,p)\
\bar c(r,p)
\end{matrix}
\right}
=
\frac{2}{\pi}\sup_f
\left|
\int_0^\infty f^{(r)}(u_r)\int_{u_r}^\infty\cdots\int_{u_1}^\infty
\left{
\begin{matrix}
\cos\
\sin
\end{matrix}
\right}
pu\,u^{-2}\,du\ldots du_r
\right|
]
and the upper bound is taken over functions (f\in W^r), even for (c(r,p)) and odd for (\bar c(r,p)).
The case (n-m\to\infty):
[
V_{n,m}(W^r)=c(r,\infty)\left[
\frac{1}{n(n-m)^{r-1}}+\frac{1}{n^2(n-m)^{r-2}}+\cdots
\right.
]
[
\left.
\cdots+\frac{1}{n^{r-1}(n-m)}
\right]
+O\left(\frac{1}{n^r}\right)
+O\left(\frac{1}{n(n-m)^r}\right);
\tag{10}
]
[
V_{n,m}(\overline W^r)=\bar c(r,\infty)\left[
\frac{1}{n(n-m)^{r-1}}+\frac{1}{n^2(n-m)^{r-2}}+\cdots
\right.
]
[
\left.
\cdots+\frac{1}{n^{r-1}(n-m)}
\right]
+O\left(\frac{1}{n^r}\right)
+O\left(\frac{1}{n(n-m)^r}\right),
\tag{11}
]
where
[
\left.
\begin{matrix}
c(r,\infty)\
\bar c(r,\infty)
\end{matrix}
\right}
=
\frac{2}{\pi}
\int_0^\infty
\left|
\int_{u_r}^\infty\cdots\int_{u_1}^\infty
\left{
\begin{matrix}
\cos\
\sin
\end{matrix}
\right}
u\,u^{-2}\,du\ldots du_{r-1}
\right|\,du_r.
]
For (r=1)
[
V_{n,m}(W^1)=\frac{2}{\pi}\frac{1}{n}\log\frac{n}{\,n-m+1\,}+O\left(\frac{1}{n}\right).
\tag{12}
]
In the case (n-m=p) fixed, (p\ge 0):
[
V_{n,m}(\overline W^1)=\bar c(1,p)\frac{1}{n}
+O\left(\frac{1}{n}\sqrt{\frac{\log n}{n}}\right),
\tag{13}
]
where
[
\bar c(1,p)=\frac{2}{\pi}\int_0^\infty
\left|\int_u^\infty\frac{\sin t}{t^2}\,dt\right|\,du
+pV_{p,p}(\overline W^1).
]
In the case (n-m\to\infty)
[
V_{n,m}(\overline W^1)=\bar c(1,\infty)\frac{1}{n}
+O\left(\frac{1}{n}\sqrt{\frac{n-m}{n}\log\frac{n}{n-m}}\right)
+O\left(\frac{1}{n(n-m)}\right),
\tag{14}
]
where
[
\bar c(1,\infty)=\frac{4}{\pi}\int_0^\infty
\left|\int_u^\infty\frac{\sin t}{t^2}\,dt\right|\,du.
]
For comparison we give the asymptotic formulas obtained by S. M. Nikol’skii ((^6)) for the norms of the de la Vallée Poussin sums,
[
V_{n,m}=\frac{4}{\pi^2}\log\frac{n}{m}+O(1)
\quad\text{for }\theta=0;
\tag{15}
]
[
V_{n,m}=\frac{2}{\pi}\int_0^\infty
\frac{|\cos u-\cos(1-\theta)u|}{u^2}\,du
+O(\varepsilon_n)
\quad\text{for }0<\theta\le 1.
\tag{16}
]
In ((^6)) the order of decrease of the remainder term in (16) is not indicated; this refinement is easily obtained from the work of S. B. Stechkin ((^{10})).
For the proof of the theorem one uses the representation of de la Vallée Poussin sums by Fejér sums (((^3)), see also ((^1))), from which it follows that
[
f(x)-v_{n,m}(f,x)=\frac{1}{\pi m}\int_{-\infty}^{\infty}
[f(x+u)-f(x)]\,
\frac{\cos nu-\cos(n-m)u}{u^2}\,du.
]
In determining (V_{n,m}(W^r)) it suffices to consider only the deviation:
to zero for even functions. In this case we have
[
f(0)-v_{n,m}(f,0)=-\frac{2}{\pi m}\int_0^\infty f^{(r)}(u_r)\int_{u_r}^{\infty}\cdots\int_{u_1}^{\infty}
\frac{\cos nu-\cos(n-m)u}{u^2}\,du\ldots du_r .
\tag{17}
]
From this (8) follows immediately, as well as the inequality
[
V_{n,m}(W^r)\leqslant \frac{2}{\pi m}\int_0^\infty
\left|\int_{u_r}^{\infty}\cdots\int_{u_1}^{\infty}
\frac{\cos nu-\cos(n-m)u}{u^2}\,du\ldots du_{r-1}\right|du_r .
\tag{18}
]
Next one constructs a function showing that
[
V_{n,m}(W^r)= \frac{2}{\pi m}\int_0^\infty
\left|\int_{u_r}^{\infty}\cdots\int_{u_1}^{\infty}
\frac{\cos nu-\cos(n-m)u}{u^2}\,du\ldots du_{r-1}\right|du_r
+O!\left(\frac{1}{m(n-m)^r}\right).
\tag{19}
]
If (\bar f(x)) is the function conjugate to (f(x)), then
[
V_{n,m}(\overline{W^r})=\sup_{f\in W^r}\left|\bar f(0)-v_{n,m}(\bar f,0)\right|,
]
and by analogous arguments we obtain
[
V_{n,m}(\overline{W^r})= \frac{2}{\pi m}\int_0^\infty
\left|\int_{u_r}^{\infty}\cdots\int_{u_1}^{\infty}
\frac{\sin nu-\sin(n-m)u}{u^2}\,du\ldots du_{r-1}\right|du_r
+O!\left(\frac{1}{m(n-m)^r}\right).
\tag{20}
]
From (19), (20) we obtain (10), (11), and, since for small (\alpha)
[
\int_0^\infty\left|\int_{u_r}^{\infty}\cdots\int_{u_1}^{\infty}
\left{\begin{array}{c}
\cos\
\sin
\end{array}\right}u-
\left{\begin{array}{c}
\cos\
\sin
\end{array}\right}(1+\alpha)u
\right|u^{-2}du\ldots du_{r-1}\right|du_r
=\frac{2}{\pi}|\alpha|\log\frac{1}{|\alpha|}+O(|\alpha|),
]
also formulas (3), (4), (6), (7).
To obtain formulas (12) and (13) the arguments given above are not suitable. For (\theta=1) we have
[
V_{n,m}(W^1)=\frac{n}{m}V_{n,n}(W^1)-\frac{n-m}{m}V_{n-m,n-m}(W^1)+O!\left(\frac{1}{m}\right);
\tag{21}
]
[
V_{n,m}(\overline{W^1})=\frac{n}{m}V_{n,n}(\overline{W^1})+\frac{n-m}{m}V_{n-m,n-m}(\overline{W^1})+
]
[
+O!\left(\frac{1}{m}\sqrt{\frac{n-m}{n}\log\frac{n}{n-m}}\right).
\tag{22}
]
Formula (21) is easily obtained from the work of S. M. Nikol’skii ((^9)). From these formulas follow (12), (13), and (14).
I express my deep gratitude to S. B. Stechkin, under whose supervision this work was carried out.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
26 III 1958
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