Abstract
Full Text
Mathematics
A. V. Efimov
ON THE APPROXIMATION OF CERTAIN CLASSES OF CONTINUOUS FUNCTIONS BY FOURIER SUMS AND FEJÉR SUMS
(Presented by Academician M. A. Lavrent'ev on 4 I 1957)
Let (\mathfrak M) be some class of continuous functions and let (U_n) be a linear operator (method of approximation) assigning to each function (f \in \mathfrak M) some polynomial of order (n), whose value at the point (x) we denote by (U_n(f,x)).
We consider two problems:
I. To find the principal term of the deviation of the function (f(x)) from (U_n(f,x)), with a remainder term uniform with respect to the whole class (\mathfrak M), i.e., to obtain a representation:
[
f(x)-U_n(f,x)=AU_n(f,x)+O\bigl(BU_n(\mathfrak M)\bigr).
]
II. To investigate the behavior of the upper bound
[
\mathcal E_{U_n}(\mathfrak M)=\sup_{f\in\mathfrak M}|f(x)-U_n(f,x)|
=\sup_{f\in\mathfrak M}\max_x |f(x)-U_n(f,x)|,
]
i.e., the upper bound of the deviations of the function (f(x)) from (U_n(f,x)), extended over the whole class (\mathfrak M).
In a number of cases the solution of problem I makes it possible to find an asymptotically exact solution of problem II. Indeed, we have
[
\mathcal E_{U_n}(\mathfrak M)
=\sup_{f\in\mathfrak M}|AU_n(f,x)|+O\bigl(BU_n(\mathfrak M)\bigr).
]
Therefore, if
[
BU_n(\mathfrak M)=o\bigl(\mathcal E_{U_n}(\mathfrak M)\bigr),
]
then
[
\mathcal E_{U_n}(\mathfrak M)\asymp \sup_{f\in\mathfrak M}|AU_n(f,x)|.
]
The first problem was first solved by E. V. Voronovskaya ((^1)) for the approximation of twice differentiable functions by S. N. Bernstein polynomials. The order of decrease of (\mathcal E_{U_n}(\mathfrak M)) for certain important classes of functions and methods of approximation was given by Lebesgue ((^2)), Jackson ((^3)), and S. N. Bernstein ((^4)) in 1910–1912. The method of obtaining an asymptotically exact equality from the expression for the principal term of the deviation was used earlier by S. B. Stechkin ((^5)) for approximations of certain classes of analytic functions by Taylor sums.
Let us define some classes of continuous functions of period (2\pi). For any integer (k \geq 0) put
[
\omega_k(h,f)=\sup_{|\delta|\leq h}|\Delta_\delta^k f(x)|,
]
where
[
\Delta_\delta^k f(x)=\sum_{i=0}^k (-1)^{k-i}\binom{k}{i} f\left(x+(k-2i)\frac{\delta}{2}\right),\qquad k\geqslant 0
\quad \left(\Delta_\delta^0 f(x)=f(x)\right).
]
We shall say that the function (f(x)) belongs to the class (MW^r H_k^\alpha) if its Weyl derivative (f^{(r)}(x)) of order (r\geqslant 0) (\bigl(f^{(0)}(x)=f(x)\bigr)) satisfies the condition
[
\omega_k\bigl(h,f^{(r)}\bigr)\leqslant Mh^\alpha,\qquad 0<\alpha\leqslant 1 .
]
For brevity we shall write (MW^r) if (k=0), and (MH_k^\alpha) if (r=0) and (k\geqslant 1). The classes (MH_2^\alpha) were introduced by A. Zygmund ((^6)). The conjugate classes will be denoted respectively by (M\overline W^{\,r}H_k^\alpha) for (r>0), and by (M\overline H_k^\alpha) for (r=0). In addition, by analogy with the classes (MW_\beta^r), introduced in the work of S. B. Stechkin ((^7)), we introduce the classes (MW_\beta^r H_k^\alpha). We shall say that (f(x)\in MW_\beta^r H_k^\alpha), (r\geqslant 0,\ 0<\alpha\leqslant 1), if (f(x)) can be represented in the form
[
f(x)=\sum_{m=1}^{\infty}\frac{1}{\pi m^r}\int_{-\pi}^{\pi}\varphi(x+t)\cos\left(mt+\frac{\beta\pi}{2}\right)\,dt,
\tag{1}
]
where (\varphi(x)\in MH_k^\alpha) ((0<\alpha\leqslant 1)) and
[
\int_{-\pi}^{\pi}\varphi(x)\,dx=0.
]
For (\beta=r) and (\beta=r+1) we obtain, respectively, the classes (MW^rH_k^\alpha) and (M\overline W^{\,r}H_k^\alpha). For brevity we shall omit the constant (M=1). We shall consider the following methods of approximation: partial Fourier sums, i.e.
[
U_n(f,x)=S_n(f,x)=\frac{a_0}{2}+\sum_{k=1}^{n}(a_k\cos kx+b_k\sin kx)
\qquad (n=0,1,2,\ldots),
]
and Fejér sums, i.e.
[
U_n(f,x)=\sigma_n(f,x)=\frac{1}{n+1}\sum_{k=0}^{n}S_k(f,x).
]
Problem I for the classes (H_1^1) and (H_2^1) and approximation by Fejér sums was solved by Zamanskii ((^8)).
The order of the quantities (\mathcal E_{\sigma_n}(H_1^\alpha)) was given by S. N. Bernstein ((^4)), while the order of the quantities (\mathcal E_{\sigma_n}(\overline H_1^\alpha)) for (0<\alpha<1) follows from results of I. I. Privalov ((^9)) and S. N. Bernstein ((^4)), and the order of (\mathcal E_{\sigma_n}(\overline H_1)) was given by Aleksich ((^{10})). Asymptotically exact results for (\mathcal E_{\sigma_n}(H_1^\alpha)) ((0<\alpha\leqslant 1)) were given by S. M. Nikol’skii ((^{11})). The first asymptotically exact result for (\mathcal E_{S_n}(\mathfrak M)) belongs to A. N. Kolmogorov ((^{12})), who considered (\mathcal E_{S_n}(W^r)) in the case of integer (r\geqslant 1). V. T. Pinkevich ((^{13})) considered (\mathcal E_{S_n}(W^r)) for arbitrary (r>0). The most general result in this direction belongs to S. M. Nikol’skii ((^{14,15})), who proved that for arbitrary (r\geqslant 0) and (0<\alpha\leqslant 1)
[
\left.
\begin{array}{l}
\mathcal E_{S_n}\left(W^rH_1^\alpha\right)\[2mm]
\mathcal E_{S_n}\left(W^r\overline H_1^\alpha\right)
\end{array}
\right}
=
\frac{C_1(\alpha)}{\pi}\frac{\ln n}{n^{r+\alpha}}
+
O\left(\frac{1}{n^{r+\alpha}}\right),
\tag{2}
]
where
[
C_1(\alpha)=\sup_{f\in H_1^\alpha}|a_1(f)|
=\sup_{f\in H_1^\alpha}\left|\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos x\,dx\right|
=
\frac{2^{1+\alpha}}{\pi}\int_{0}^{\pi/2} t^\alpha\sin t\,dt.
]
Theorem 1. Let (f(x)) be a continuous function of period (2\pi), and let (\omega_2(h,f)) be its modulus of smoothness. Then
[
f(x)-\sigma_{n-1}(f,x)
=
-\frac{1}{2\pi}\int_a^\infty
\frac{
f!\left(x+\frac{2t}{n}\right)-2f(x)+f!\left(x-\frac{2t}{n}\right)
}{t^2}\,dt
+
O!\left(\omega_2!\left(\frac1n,f\right)\right),
]
where (a>0) is an arbitrary constant.
For (\omega_2(h,f)=h) we obtain Zamansky’s result ((8)).
Theorem 2. Let (f(x)\in H_2^1), and let (\overline f(x)) be the conjugate function. Then
[
\overline f(x)-\overline{\sigma}_{n-1}(f,x)
=
-\frac{1}{\pi}\int_0^{a_1}
\left[
f!\left(x+\frac{t}{n}\right)-f!\left(x-\frac{t}{n}\right)
\right]\frac{\sin t}{t^2}\,dt
+
O!\left(\frac1n\right),
]
where (a_1) is the smallest root of the equation
[
\int_0^u \frac{\sin t}{t}\,dt=\frac{\pi}{2}.
]
From Theorem 2 and the author’s results ((17)) it follows that:
Theorem 3. The following asymptotic equality holds:
[
\mathcal E_{\sigma_n}!\left(\overline{H}2^1\right)
=
\sup_n(f,x)|}|\overline f(x)-\overline{\sigma
=
\frac{1}{2\ln(\sqrt2+1)}\frac{\ln n}{n}
+
O!\left(\frac1n\right).
]
Denote by (R_n(f,x)) the remainder of the Fourier series (1) for a function (f(x)\in W_\beta^r H_k^\alpha) when (r>0), and by (r_{n,\beta}(\varphi,x)) the remainder of the Fourier series for a function (f(x)\in W_\beta^0 H_k^\alpha), i.e.,
[
r_{n,\beta}(\varphi,x)
=
\cos\frac{\beta\pi}{2}\,[\varphi(x)-S_n(\varphi,x)]
+
\sin\frac{\beta\pi}{2}\,[\overline\varphi(x)-\overline S_n(\varphi,x)]
=
]
[
\frac{1}{\pi}\int_{-\pi}^{\pi}
[\varphi(x)-\varphi(x+t)]
\frac{
\sin!\left(\frac{2n+1}{2}t+\frac{\beta\pi}{2}\right)
}{
2\sin\frac{t}{2}
}\,dt,
]
where (S_n(\varphi,x)) and (\overline S_n(\varphi,x)) are the partial sums, respectively, of the Fourier series of the function (\varphi(x)) and of the conjugate series.
From the results of S. B. Stechkin ((16)) it follows that, for (0<\alpha\le 1) and (k>2), the inclusion
[
C_1H_k^\alpha\subset H_2^\alpha\subset C_2H_k^\alpha
]
holds, where (0<C_1<C_2) are certain constants. Hence, using Theorems 1 and 2, we obtain the following theorem.
Theorem 4. Let (f(x)\in W_\beta^r H_k^\alpha). Then, for any (r>0) and (0<\alpha\le 1), the equality
[
R_n(f,x)
=
f(x)
-
\sum_{m=1}^{n}\frac{1}{\pi m^r}
\int_{-\pi}^{\pi}
\varphi(x+t)\cos!\left(mt+\frac{\beta\pi}{2}\right)\,dt
=
]
[
\frac{1}{(n+1)^r}r_{n,\beta}(\varphi,x)
+
O!\left(\frac{1}{n^{r+\alpha}}\right).
]
Putting here (\beta=r) and (\beta=r+1), we obtain the solution of Problem I for the approximation of functions of the classes (W^r H_k^\alpha) and (\overline W^r H_k^\alpha) by Fourier sums.
Theorem 5. Let (f(x)\in W_\beta^0 H_k^\alpha). Then for any (0<\alpha\leqslant 1)
[
r_{n,\beta}(\varphi,x)
=
\frac{1}{2\pi^2}
\sum_{m=1}^{\left[\frac n2\right]-3}
\frac{1}{m^2}
\left{
\int_0^{2m\pi}
\left[
\varphi(x)-\varphi\left(x+\frac{t}{n}+\frac{5-\beta}{2n}\pi\right)
\right]\cos t\,dt
-\right.
]
[
\left.
-\int_{-2m\pi}^{0}
\left[
\varphi(x)-\varphi\left(x+\frac{t}{n}-\frac{3+\beta}{2n}\pi\right)
\right]\cos t\,dt
\right}
+
O\left(\frac{1}{n^\alpha}\right).
]
Put
[
C_k(\alpha)=\sup_{f\in H_k^\alpha}|a_1(f)|
=
\sup_{f\in H_k^\alpha}
\left|
\frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos x\,dx
\right|.
]
Theorem 6. For any (0<\alpha\leqslant 1) the asymptotic equality
[
\mathcal E_{s_n}\left(W_\beta^0 H_k^\alpha\right)
=
\frac{C_k(\alpha)}{\pi}\frac{\ln n}{n^\alpha}
+
O\left(\frac{\ln\ln n}{n^\alpha}\right)
]
holds.
From Theorems 4 and 6 there follows
Theorem 7. For any (r\geqslant 0) and (0<\alpha\leqslant 1) the asymptotic equality
[
\mathcal E_{s_n}\left(W_\beta^r H_k^\alpha\right)
=
\frac{C_k(\alpha)}{\pi}\frac{\ln n}{n^{r+\alpha}}
+
O\left(\frac{\ln\ln n}{n^{r+\alpha}}\right)
]
holds.
Putting in Theorems 5—7 (\beta=r) and (\beta=r+1), we obtain the solution of problems I and II for the approximation of functions of the classes (H_k^\alpha) and (\overline H_k^\alpha) by Fourier sums, and the solution of problem II for the approximation of functions of the classes (W^r H_k^\alpha) and (W^r\overline H_k^\alpha) by Fourier sums.
I express my deep gratitude to S. B. Stechkin for posing the problem and for valuable advice and guidance in carrying out the present work.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
3 I 1957
REFERENCES
- E. V. Voronovskaya, DAN, 79 (1932).
- H. Lebesgue, Bull. Soc. Math. de France, 38, 184 (1910).
- D. Jackson, Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen, Göttingen, 1911.
- S. N. Bernstein, Soobshch. Kharkovsk. matem. obshch., ser. 2, 13, 49 (1912).
- S. B. Stechkin, Izv. AN SSSR, ser. matem., 17, 461 (1953).
- A. Zygmund, Duke Math. J., 12, 47 (1945).
- S. B. Stechkin, Izv. AN SSSR, ser. matem., 20, 643 (1956).
- M. Zamansky, Ann. Sci. Ecole Norm. Sup., 66, 19 (1949).
- I. Privaloff, Bull. Soc. Math. de France, 44, 100 (1916).
- G. Alexits, Matematikai es Fizikai Lapok, 48, 410 (1941).
- S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 4, 501 (1940).
- A. Kolmogoroff, Ann. of Math., 36, 521 (1935).
- V. T. Pinkevich, Izv. AN SSSR, ser. matem., 4, 521 (1940).
- S. M. Nikol’skii, DAN, 32, 386 (1941).
- S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 15 (1945).
- S. B. Stechkin, Izv. AN SSSR, ser. matem., 15, 219 (1951).
- A. V. Efimov, Izv. AN SSSR, ser. matem., 21, 283 (1957).