Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1967. Volume 172, No. 6
UDC 517.5
MATHEMATICS
T. A. Azlarov, Kh. Mansurov
On a One-Parameter Apparatus for the Approximation of Functions
(Presented by Academician A. N. Kolmogorov, 18 IV 1966)
- Let (C) be the set of continuous functions defined on the interval ([0,1]). In the present paper, for the approximation of (f(x)\in C), the following one-parameter apparatus is considered and studied:
[
P_{n,\alpha}(f;x)=\frac{1}{\sigma_\alpha n^\beta}
\sum_{k=0}^{n} f\left(\frac{k}{n}\right)
\left[1-\left|\frac{k}{n}-x\right|^\alpha\right]^{n\gamma},
\tag{1}
]
where (\alpha \geqslant 1,\ \beta=\dfrac{\alpha}{\alpha+2},\ \gamma=\dfrac{2\alpha}{\alpha+2},\ \sigma_\alpha=\dfrac{2}{\alpha}\Gamma\left(\dfrac{1}{\alpha}\right)). For (\alpha=2) this apparatus was introduced in work ((^1)).
- It is not difficult to prove that if (f\in C), then
[
\lim_{n\to\infty}\sup_x |P_{n,\alpha}(f;x)-f(x)|=0.
\tag{2}
]
The order of approximation in (2) is established by the following theorem:
Theorem 1. If (f\in C) and (\omega_f(\delta)) is its modulus of continuity, then
[
|P_{n,\alpha}(f;x)-f(x)|
\leqslant
K_1(\alpha)\omega_f\left(\frac{1}{n^{1-\beta}}\right)
+\frac{K_2}{n^{2\beta}}
+\frac{K_3}{n^\beta}\omega_f\left(\frac{1}{n^{1-\beta}}\right),
\tag{3}
]
where
[
K_1(\alpha)=1+\frac{2}{\sigma_\alpha}\int_1^\infty y e^{-y^\alpha}\,dy,
\tag{4}
]
(K_2, K_3) are constants independent of (n).
The obtained estimate of approximation (3) cannot be improved in order (see Theorem 3).
For (\alpha=2), from (4) we obtain
[
K_1(2)=1+\frac{1}{e\sqrt{\pi}}.
]
(We note that the corresponding constant given in Theorem 1 of work ((^2)) was equal to 1.5.)
The proof of Theorem 1 is based essentially on the following asymptotic equalities:
for (\alpha\ne 1)
[
P_{n,\alpha}(1;x)=
1-\frac{\alpha+1}{2\alpha^2 n^{2\beta}}
+O\left(n^{-\beta(1+\alpha)}+n^{-4\beta}\right);
\tag{5}
]
for (\alpha=1)
[
P_{n,1}(1,x)=
1-\left[\frac{25}{24}-\frac{1}{2}\rho^2\left(nx+\frac{1}{2}\right)\right]\frac{1}{n^2}
+O\left(n^{-4\beta}\right),
]
where (\rho(t)) is the distance from the point (t) to the nearest integer.
Let (\omega_0(\delta)) be some fixed modulus of continuity. Introduce the class (C_{\omega_0}) of functions (f(x)) for which
[
\omega_f(\delta)\leqslant \omega_0(\delta),\qquad 0\leqslant \delta\leqslant 1.
]
Theorem 2. Put
[
E_{n,\alpha}(x,C_{\omega_0})=\sup_{f\in C_{\omega_0}}\left|P_{n,\alpha}(f;x)-f(x)\right|.
]
Then, as (n\to\infty),
[
E_{n,\alpha}(x,C_{\omega_0})=
\frac{1}{\sigma_\alpha n^\beta}
\sum_{k=0}^{n}
\omega_0\left(\left|\frac{k}{n}-x\right|\right)
\left[1-\left|\frac{k}{n}-x\right|^{\alpha} n^\gamma\right]
+O(n^{-2\beta}).
\tag{6}
]
Theorem 3. The following asymptotic formula holds ((n\to\infty)):
[
E_{n,\alpha}(xC_{\omega_0})=
\frac{2}{\sigma_\alpha}
\int_{0}^{\infty}
\widetilde{\omega}\left(\frac{y}{n^{1-\beta}}\right)e^{-y^\alpha}\,dy
+O\left(\frac{1}{n^{2\beta}}\right)
+O\left(\omega_0\left(\frac{1}{n}\right)\right).
\tag{7}
]
Here (\widetilde{\omega}(\delta)=\omega_0(\delta)) if (0\leqslant \delta\leqslant 1), and (\widetilde{\omega}(\delta)=0) for (\delta>1).
Proof of Theorem 3. Denoting
[
\widetilde{E}{n,\alpha}(x,C)=
\frac{1}{\sigma_\alpha n^\beta}
\sum_{k=0}^{n}
\omega_0\left(\left|\frac{k}{n}-x\right|\right)
\left[1-\left|\frac{k}{n}-x\right|^{\alpha}n^\gamma\right],
]
from (6) we obtain
[
E_{n,\alpha}(x,C_{\omega_0})
=
\widetilde{E}{n,\alpha}(x,C)
+O(n^{-2\beta}).
]
Next, put (\delta_n=n^{-\beta/\alpha}) and (\widetilde{E}{n,\alpha}(x,C)=S_1+S_2), where in (S_1) the summation is over all (k) for which (|k/n-x|<\delta_n), and in (S_2) over the remaining (k). It is not hard to see that
[
S_2=O(n^{1-\beta}e^{-n^\beta}).
]
Now consider
[
S_1=\frac{1}{\sigma_\alpha}
\sum_{|k/n-x|<\delta_n}
\omega_0\left(\frac{|\xi_k|}{n^{1-\beta}}\right)
\left[1-\frac{|\xi_k|^\alpha}{n^\gamma}\right]^{n^\gamma}
\Delta \xi_k,
]
where
[
\xi_k=(k-nx)/n^\beta,\qquad \Delta \xi_k=\xi_{k+1}-\xi_k.
]
Since, as (n\to\infty), (|\xi|^\alpha n^{-\gamma}\leqslant \delta_n^\alpha=n^{-\beta}\to 0), we may use the expansion
[
\left(1-\frac{|\xi_k|^\alpha}{n^\gamma}\right)^{n^\gamma}
=
e^{-|\xi_k|^\alpha}\left[1+O\left(|\xi_k|^{2\alpha}n^{-\gamma}\right)\right].
]
We have
[
S_1=
\frac{1}{\sigma_\alpha}
\sum_{|\xi_k|<\delta_n n^{1-\beta}}
\omega_0\left(\frac{|\xi_k|}{n^{1-\beta}}\right)
e^{-|\xi_k|^\alpha}\Delta \xi_k
+
O\left(
n^{-\gamma}
\sum_{k=0}^{n}
\omega_0\left(\frac{|\xi_k|}{n^{1-\beta}}\right)
e^{-|\xi_k|^\alpha}
|\xi_k|^{2\alpha}\Delta \xi_k
\right).
\tag{8}
]
The second term on the right-hand side of the last equality is a quantity of order (O(n^{-2\beta})).
Let
[
R_k=
\omega_0\left(\frac{|\xi_k|}{n^{1-\beta}}\right)
e^{-|\xi_k|^\alpha}\Delta \xi_k
-
\int_{\xi_k-\frac12 n^{-\beta}}^{\xi_k+\frac12 n^{-\beta}}
\omega_0\left(\frac{|t|}{n^{1-\beta}}\right)e^{-|t|^\alpha}\,dt,
\tag{9}
]
[
R=\sum_{|\xi_k|<n^{\beta/\alpha}} R_k.
\tag{10}
]
Using the known properties of the modulus of continuity: (\omega(\lambda\delta) \leq (1+\lambda)\omega(\delta)) and (\omega(t_1)-\omega(t_2)\leq \omega(|t_1-t_2|)), one can show that
[
|R_k| \leq \omega_0\left(\frac{1}{n}\right)e^{-|\xi_k|^\alpha}Q(|\xi_k|)\Delta \xi_k,
\tag{11}
]
where (Q(z)) is some polynomial in (z).
From relations (8), (9), (10), and (11) it follows that
[
S_1=\frac{1}{\sigma_\alpha}
\int_{-n^{\beta/\alpha}}^{n^{\beta/\alpha}}
\omega_0\left(\frac{|y|}{n^{1-\beta}}\right)e^{-|y|^\alpha}\,dy
+
O\left(\omega_0\left(\frac{1}{n}\right)\right)
+
O(n^{-2\beta}).
]
Hence, taking into account the estimate for (S_2), it is easy to obtain the assertion of the theorem.
Remark. 1) As is known, the problem of the asymptotics of the upper bound for the deviation of functions from their approximating apparatus for various classes of functions was considered by A. N. Kolmogorov, S. M. Nikol’skii, and others. The formulated theorem solves this problem for the apparatus (1).
2) It is interesting to note that the principal term of expansion (7) does not depend on (x).
3) If, for example, for small (x) we have the expansion
[
\omega_0(x)=x^\lambda(a_0+a_1x+\cdots), \qquad 0<\lambda\leq 1,
]
then for any integer (s) ((s\geq 0))
[
\frac{2}{\sigma_\alpha}\int_0^\infty
\widetilde{\omega}\left(\frac{y}{n^{1-\beta}}\right)e^{-y^\alpha}\,dy
=
\frac{2}{\sigma_\alpha}n^{1-\beta}\int_0^1
\omega_0(x)e^{-n^{2\beta}x^\alpha}\,dx
=
]
[
\frac{1}{\Gamma(1/\alpha)}
\sum_{k=0}^{s}
a_k\Gamma\left(\frac{\lambda+k+1}{\alpha}\right)
n^{-2\frac{\lambda+k}{\alpha+2}}
+
o\left(n^{-2\frac{\lambda+s}{\alpha+2}}\right).
]
Hence it is not difficult to obtain that:
for (\lambda \geq \dfrac{2\alpha}{\alpha+2}),
[
E_{n,\alpha}(x,C_{\omega_0})
=
\frac{1}{\Gamma(1/\alpha)}
a_0\Gamma\left(\frac{\lambda+1}{\alpha}\right)
n^{-2\frac{\lambda}{\alpha+2}}
+
O\left(n^{-2\frac{\alpha}{\alpha+2}}\right);
]
for (\lambda < \dfrac{2\alpha}{\alpha+2}),
[
E_{n,\alpha}(x,C_{\omega_0})
=
\frac{1}{\Gamma(1/\alpha)}
\sum_{k=0}^{s}
a_k\Gamma\left(\frac{\lambda+k+1}{\alpha}\right)
n^{-\frac{2(\lambda+k)}{\alpha+2}}
+
O(n^{-\lambda}),
]
where (s=[\lambda\alpha/2]).
- Questions on the convergence of (P_{n,2}(f;x)) to (f(x)) and on the estimate of this approximation in terms of the modulus of continuity (\omega_f) were considered in ((^{1,2})). In ((^2)), moreover, for (f\in C^2) ((C^m) is the class of functions having an (m)-th continuous derivative), the asymptotic formula
[
P_{n,2}(f;x)=f(x)+\frac{f''(x)-1.5f(x)}{4n}+o\left(\frac{1}{n}\right).
\tag{*}
]
was proved. If we consider
[
\overline{P}{n,2}(f;x)=\frac{8n}{8n-3}P(f;x),
]
then formula (*) takes the form
[
\overline{P}_{n,2}(f;x)=f(x)+\frac{f''(x)}{4n}+o\left(\frac{1}{n}\right).
]
If one carries out an analogous “shift of normalization” in the general case and considers the operator
[
\overline{P}{n,\alpha}(f;x)=
\frac{P}(f;x)
{1-\dfrac{\alpha+1}{2\alpha^2}\,n^{-2\beta}},
]
then one can prove the theorem:
Theorem 4. Let (f\in C^2) and (\alpha>(\sqrt{17}-1)/2). Then
[
\overline{P}_{n,\alpha}(f;x)
=
f(x)+
\frac{\Gamma(3/\alpha)}{2\Gamma(1/\alpha)}
\frac{f''(x)}{n^{2-2\beta}}
+
o!\left(\frac{1}{n^{2-2\beta}}\right).
]
- For (\alpha=2l,\ l=1,2,\ldots,) it is not difficult to prove that
[
\lim_{n\to\infty}\sup_x
\left|
\frac{\partial P_{n,\alpha}(f;x)}{\partial x}
-
\frac{df}{dx}
\right|
=0
]
and to estimate the error of this approximation. Under the assumption that (f\in C^1) and (\omega_{1f}) is the modulus of continuity of (f'(x)), Theorems 1–3 can be formulated in terms of (\omega_{1f}).
In conclusion, we note that the results obtained also extend to functions depending on many variables.
The authors express their deep gratitude to Prof. S. Kh. Sirazhdinov for valuable advice and suggestions.
Tashkent State University
named after V. I. Lenin
Received
3 IV 1966
REFERENCES
- L. C. Hsu, L. P. Hsu, On a New Class of Approximating Polynomials for Real Functions, Sci. Record, New Ser., 3, No. 2, 1959.
- T. A. Azlarov, Kh. Mansurov, Izv. AN UzSSR, Ser. Phys.-Math. Sci., No. 2 (1966).