UDC 517.525

MATHEMATICS

V. P. MOTORNYY

SOME QUESTIONS ON APPROXIMATION OF FUNCTIONS BY ALGEBRAIC POLYNOMIALS IN THE INTEGRAL METRIC

(Presented by Academician L. S. Pontryagin on 28 III 1966)

1. In the present paper we give some results connected with the question of a constructive characterization of nonperiodic functions in the integral metric. In the metric \(C\) this question was completely solved in the works of S. M. Nikol’skii \((^1)\), A. F. Timan \((^{2,3})\), and V. K. Dzyadyk \((^4)\). In \((^{2-4})\) it was established that, in order that a function \(f(x)\), given on the segment \([-1,1]\), have there, for some nonnegative integer \(r\), a derivative \(f^{r}(x)\) of order \(r\) belonging to the class \(\operatorname{Lip}\alpha\) \((\alpha<1)\), it is necessary and sufficient that for every natural \(n\) there be an ordinary polynomial \(P_n(x)\) of degree not exceeding \(n\) such that, for all \(x\in[-1,1]\), the inequality
   \[ |f(x)-P_n(x)|\le C\left(\sqrt{1-x^2}/n+1/n^2\right)^{r+\alpha}. \]
   holds.

In the integral metric the first works in this direction were carried out by G. K. Lebed’ and M. K. Potapov \((^{5,6})\).

1. In what follows we shall everywhere consider classes of functions integrable to the \(p\)-th power with norm
   \[ \|f\|_{p}=\left\{\int_{-1}^{1}|f(x)|^{p}\,dx\right\}^{1/p}. \]
   By \(W^{(r)}H_p^\omega\) (respectively \(W^{(r)}A_p^\omega\)), \(1\le p<\infty\), we denote the class of functions given on the interval \([-1,1]\) and having there an \(r\)-th derivative \(f^{(r)}(x)\), Lebesgue integrable to the \(p\)-th power, for which, for any \(h>0\), the inequality
   \[ \|f^{(r)}(x+h)-f^{(r)}(x)\|_{L_p(-1,\,1-h)}\le \omega(h) \]
   holds, respectively the inequality
   \[ \left\| \frac{f^{(r)}\!\left(x\sqrt{1-h^2}-h\sqrt{1-x^2}\right)-f^{(r)}(x)} {\omega\!\left(\sqrt{1-x^2}\,h+h^2\right)} \right\|_{L_p}\le C. \]

For \(\omega(t)=t^\alpha\) these classes will be denoted by \(H_p^{(r+\alpha)}\) (respectively \(A_p^{(r+\alpha)}\)). The classes \(W^{(r)}A_p^\omega\) were introduced by G. K. Lebed’ \((^5)\) and M. K. Potapov \((^{6,7})\). It is known that for \(p=\infty\) the classes \(W^{(r)}H_\infty^\omega\) and \(W^{(r)}A_\infty^\omega\) coincide (see, for example, \((^5)\)), while for \(1\le p<\infty\) the intersection of these classes is nonempty.

G. K. Lebed’ and M. K. Potapov gave the following constructive characterization of the class \(A_p^{(r+\alpha)}\) for \(0<\alpha<1\). In order that \(f(x)\in A_p^{(r+\alpha)}\), it is necessary and sufficient that for every natural number \(n\ge0\) there be an algebraic polynomial of degree not exceeding \(n\) such that the inequality
\[ \left\| \frac{f(x)-P_n(x)} {\left(\sqrt{1-x^2}+1/n\right)^{r+\alpha}} \right\|_{L_p} < \frac{C}{n^{r+\alpha}}, \]
holds, where the constant \(C\) does not depend on \(n\) and \(f(x)\). However, even after these works there remained open the question of a constructive characterization of the class \(W^{(r)}H_p^\omega\), and also the connection between the classes \(W^{(r)}H_p^\omega\) and \(W^{(r)}A_p^\omega\) had not been clarified.

1. In this section we formulate several theorems showing how the classes \(W^{(r)}H_p^\omega\) and \(W^{(r)}A_p^\omega\) are related. Obviously, it is enough for us to consider the case \(r=0\).

Theorem 1. For any function \(f(x)\in W^{(0)}H_p^\omega\) the inequality
\[ \left\| \frac{f\left(x\sqrt{1-h^2}-h\sqrt{1-x^2}\right)-f(x)} {\omega\left(\sqrt{1-x^2}\,h+h^2\right)} \right\|_{L_p} \leq C\ln^{1/p}\frac{1}{h}. \tag{1} \]

Theorem 2. For any function \(f(x)\in W^{(0)}A_p^\omega\) the inequality
\[ \|f(x+h)-f(x)\|_{L_p(-1,\,1-h)} \leq C\omega(h)\ln^{1/p}h\frac{1}{h}. \tag{2} \]

It can be shown by examples that, in the case \(\omega(t)=t^\alpha\) \((0<\alpha<1)\), there exist functions \(f_h(x)\) and \(\bar f_h(x)\), belonging respectively to the classes \(H_p^\alpha\) and \(A_p^\alpha\), and a constant \(C_0\) (\(C_0\), generally speaking, depends on \(\alpha\)) such that the sign of the inequality in (1) and (2) is reversed. In some cases the logarithms in the right-hand side of inequalities (1) and (2) can be omitted. For example, the classes of functions \(H_p^1\) and \(A_p^1\) coincide (consequently, the classes of functions \(H_p^{(r+1)}\) and \(A_p^{r+1}\) coincide for any integer \(r\)). Besides this case, it is interesting to note one more case of coincidence of the classes \(H_p^\alpha\) and \(A_p^\alpha\). A function \(f(x)\), defined on \([-1,1]\), belongs to the intersection of the classes \(H_p^\alpha\) and \(A_p^\alpha\) if it satisfies the condition
\[ \sum_i |f(x_i)-f(x_{i+1})|^p |x_i-x_{i+1}|^{-\alpha p+1}=O(1), \tag{3} \]
where \(x_i\) are arbitrary points of the interval \([-1,1]\) \((x_{i+1}>x_i)\); moreover, it is sufficient that condition (3) hold not on the whole interval, but only near the endpoints.

1. It follows from Theorem 1 that if \(f(x)\in H_p^{(r+\alpha)}\), then for each \(n=0,1,2,\ldots\) there exists an algebraic polynomial of degree not exceeding \(n\) such that the inequality
   \[ \left\| \frac{f(x)-P_n(x)} {\left(\sqrt{1-x^2}+1/n\right)^{r+\alpha}} \right\|_{L_p} \leq C\frac{\ln^{1/p} n}{n^{r+\alpha}}. \tag{4} \]
   holds.

This result is one of the variants of the strengthened Jackson theorem in the metric \(L_p\). However, inequality (4) does not characterize the class \(H_p^{(r+\alpha)}\), since functions of the class \(A_p^{(r+\alpha)}\) also satisfy this inequality. But, on the other hand, in the right-hand side of inequality (4) one cannot replace the quantity \(\frac{1}{n^\alpha}\ln^{1/p}n\) by a function \(\omega(1/n)\) such that
\[ \varlimsup_{h\to0} \left[ h\int_h^1 \frac{\omega(u)}{u^2}\,du \right]^{-1} h^\alpha \ln^{1/p}\frac{1}{h} =\infty. \tag{5} \]
The latter follows from the inverse theorems of G. K. Lebed’ (5) and the results of Sec. 3. As a consequence of inequality (4), we note Theorem 3.

Theorem 3. Any function \(f(x)\), belonging on the segment \([-1,1]\) to the class \(H_p^{(r+\alpha)}\) \((r+\alpha>1/4)\), is expanded in a Fourier–Legendre series converging in the metric \(L\) to the function \(f(x)\), and the estimate
\[ \int_{-1}^{1}\left|f(x)-\sum_{k=0}^{n} a_k P_k(x)\right|\,dx \leq C \begin{cases} \ln n/n^{2\alpha-1/2}, & r+\alpha<1/2,\\ \ln n/n^{r+\alpha}, & r+\alpha>1/2. \end{cases} \tag{6} \]
holds.

It was indicated above (see Sec. 2) that the class \(A_p^\alpha\), in terms of approximation by algebraic polynomials, can be defined as the class of functions for which the inequality
\[ \left\| \frac{f(x)-P_n(x)} {\rho_0^\alpha(1-x^2+1/n^2)} \right\|_{L_p} =O\left(\frac{1}{n^\alpha}\right), \tag{7} \]
holds.

where \(P_n(x)\) is an algebraic polynomial of degree not exceeding \(n\), \(\rho_0(t)=\sqrt t\). Naturally, the question arises: can \(\rho(t)\) be determined so that equality (7) would be equivalent to the assertion that \(f(x)\in H_p^\omega\)? The answer to this question is negative.

The following variant of an strengthened Jackson theorem in the integral metric is proposed.

Theorem 4. If a function \(f(x)\) is given on the interval \([-1,1]\) and belongs to the class \(W^{(0)}H_p^\omega\), then there exists a sequence of algebraic polynomials \(P_n(x)\) such that, for every \(n\), the inequality
\[ \left\{\sum_{i=-k-1}^{k}\int_{a_i}^{a_{i+1}} |f(x)-P_{n_i}(x)|^p\,dx\right\}^{1/p} = O\left[\omega\left(\frac{1}{n}\right)\right], \tag{8} \]
holds, where
\[ n_i=[n\sqrt{1-a_i}]; \]
\(a_i\) are points of the interval \([-1,1]\) such that
\[ a_{-i}=a_i,\qquad \sqrt{1-a_i}=\frac{1}{2^i},\qquad 2\sqrt n\ge 2^k\ge \sqrt n,\qquad a_{k+1}=1\quad (i=0,1,\ldots,k). \]

There is also a theorem converse to Theorem 4.

Theorem 5. If, for a function \(f(x)\) given on the interval \([-1,1]\), there exists a sequence of algebraic polynomials such that inequality (8) holds, where \(\omega(t)\) is some modulus of continuity, then
\[ \omega(f;t)_{L_p} = O\left[ t\int_t^1 \frac{\omega(u)}{u^2}\ln\frac{2u}{t}\,du \right]. \]

Thus, a theorem has been proved which gives a constructive characterization of the class \(W^{(0)}H_p^\omega\) under the additional assumption that \(\omega(t)\) satisfies the condition
\[ t\int_t^1 \frac{\omega(u)}{u^2}\,du = O[\omega(t)]. \tag{9} \]

Theorem 6. In order that a function \(f(x)\), given on the interval \([-1,1]\), belong to the class \(W^{(0)}H_p^\omega\), where \(\omega(t)\) satisfies condition (9), it is necessary and sufficient that inequality (8) hold.

Remark. The points \(a_i\) may be chosen arbitrarily, but so that
\[ \sqrt{1-a_i}<A\sqrt{1-a_{i+1}}, \]
where \(A\) is some positive number. In this case, of course, the constant in the right-hand side of inequality (8) depends on \(A\).

For classes of functions \(W^{(r)}H_p^\omega\) (\(r>0\)), under the condition that \(\omega(t)\) satisfies condition (9) and
\[ \int_0^t \frac{\omega(u)}{u}\,du = O[\omega(t)], \]
the following is valid.

Theorem 7. In order that a function, given on the interval \([-1,1]\), belong to the class \(W^{(r)}H_p^\omega\), it is necessary and sufficient that there exist a sequence of algebraic polynomials such that, for every \(n\), the inequality
\[ \left\{\sum_{i=-k-1}^{k}\int_{a_i}^{a_{i+1}} |f(x)-P_{n_i}(x)|\,dx\right\}^{1/p} = O\left[ \frac{1}{n^r}\omega\left(\frac{1}{n}\right) \right], \tag{10} \]
holds, where
\[ n_i=[n\sqrt{1-a_i}], \]
\(a_i\) are points of the interval \([-1,1]\) such that
\[ a_{-i}=a_i,\qquad \sqrt{1-a_i}=\frac{1}{2^i},\qquad a_{k+1}=1\quad (i=0,1,\ldots,k);\qquad 2\sqrt n\ge 2^k\ge \sqrt n. \]

5. In the works of M. K. Potapov \((^6,^7)\) and G. K. Lebed’ \((^5)\), other functional spaces \(L_p\) with norm \(\|\bar f\|_{L_p(-1,1)}=\) were also considered.

\[ = \left\{\int_{-1}^{1}|f(x)|^{p}\frac{dx}{\sqrt{1-x^{2}}}\right\}^{1/p}. \]

By definition, \(f(x)\in W^{(r)}\widetilde H_p^\alpha\), if \(f(x)\) has an \(r\)-th derivative \(f^{(r)}(x)\) for which, for any \(h>0\), the inequality
\[ \left\|\overline{f(x+h)-f(x)}\right\|_{L_p(-1,1-h)}<h^\alpha \]
holds, and \(f(x)\in W^{(r)}\widetilde A_p^\alpha\), if the \(r\)-th derivative \(f^{(r)}(x)\) exists and satisfies the condition
\[ \left\| \frac{ f^{(r)}\bigl(x\sqrt{1-h^{2}}-h\sqrt{1-x^{2}}\bigr)-f(x) }{ \bigl(\sqrt{1-x^{2}}\,h+h^{2}\bigr)^\alpha } \right\|_{L_p(-1,1)}=O(1). \]

As M. K. Potapov showed \((^7)\), the definition of the class \(W^{(r)}\widetilde A_p^\alpha\) in differential-difference terms is equivalent to the following definition of this class in terms of approximation by algebraic polynomials: the class \(W^{(r)}\widetilde A_p^\alpha\) consists of those and only those functions \(f(x)\), defined on the interval \([-1,1]\), for which

\[ \left\| \frac{f(x)-P_n(x)} {\bigl(\sqrt{1-x^{2}}+1/n\bigr)^{r+\alpha}} \right\|_{L_p(-1,1)} = O\left(\frac{1}{n^{r+\alpha}}\right). \tag{11} \]

In the same work M. K. Potapov posed the problem of a constructive characterization of the class \(W^{(r)}\widetilde H_p^\alpha\) and of the relation between the classes \(W^{(r)}\widetilde H_p^\alpha\) and \(W^{(r)}\widetilde A_p^\alpha\). Of all the theorems analogous in meaning to Theorems 1–7, we note the following one.

Theorem 8. In order that a function \(f(x)\), defined on the interval \([-1,1]\), belong to the class \(W^{(r)}\widetilde H_p^\alpha\), it is necessary and sufficient that there exist a sequence of polynomials \(P_n(x)\) such that

\[ \left\{ \sum_i \int_{a_i}^{a_{i+1}} \left|f(x)-P_{n_i}(x)\right| \frac{dx}{\sqrt{1-x^{2}}} \right\}^{1/p} = O\left(\frac{1}{n^{r+\alpha}}\right), \tag{12} \]

where \(n_i=[n\sqrt{1-a_i}]\) and \(\sqrt{1-a_i}<A\sqrt{1-a_{i+1}}\), \(A\) is some fixed number.

In conclusion I express my gratitude to Prof. S. M. Nikol’skii for posing the problem and for his attention to the work, and also to Yu. A. Brudnyi for valuable general considerations on the questions considered.

Received
17 III 1966

CITED LITERATURE

\(^1\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 10, No. 4 (1946).
\(^2\) A. F. Timan, DAN, 77, 969 (1951).
\(^3\) A. F. Timan, DAN, 78, 17 (1951).
\(^4\) V. K. Dzyadyk, Izv. AN SSSR, Ser. matem., 20, 623 (1956).
\(^5\) G. K. Lebed’, DAN, 118, 239 (1958).
\(^6\) M. K. Potapov, DAN, 111, 1185 (1956).
\(^7\) M. K. Potapov, Vestn. Mosk. univ., No. 4 (1960).