Yu. A. Brudnyi

On Local Best Approximation of Functions by Polynomials

(Presented by Academician S. N. Bernstein, 27 X 1964)

Mathematics

1. In the present note we give some new results on the connection between the properties of a function and the behavior of its local best approximations. The first works in this direction were carried out by S. N. Bernstein \((^1)\) and D. A. Raikov \((^4)\); in the time that has passed since then, comparatively few works have been done in which the above-mentioned connection was investigated explicitly (we know only the work \((^2)\)); however, some results in the theory of approximations (as a whole) and in the theory of functions of a real variable can be assigned to the domain under consideration (see, for example, \((^{3,5,6})\)). It seems to us, nevertheless, that the domain of investigation under consideration, which is a borderline one between approximation theory (as a whole) and the theory of functions of a real variable, is of undoubted interest.

2. Let \(B(I_0)\) be one of the functional spaces \(L^p(I_0)\), \(1 \le p \le \infty\), \(C(I_0)\), or \(M(I_0)\)*, where \(I_0\) is an interval of the real axis, not necessarily finite, and \(\|f; I\|\), where \(I \subset I_0\), is the corresponding local norm, i.e.
   \[ \|f; I\|=\left(\int_I |f(x)|^p dx\right)^{1/p}, \]
   if \(f \in L^p(I_0)\), and
   \[ \|f; I\|=\sup_{x\in I}|f(x)|, \]
   if \(f \in C(I_0)\) or \(f \in M(I_0)\). Everywhere in what follows \(I\) denotes a finite interval from \(I_0\), and \(|I|\) its length. Denote by \(E_r(f; I)\) the quantity \(\inf \|f-p; I\|\), where the infimum is taken over all polynomials \(p\) of degree \(\le r\); a polynomial on which the infimum is attained will be called a polynomial of best approximation for \(f\) on \(I\). Sometimes, instead of \(E_r(f; I)\), it is more convenient to consider the quantity \(E_T(f; I)\), defined as follows. Let \(\mathfrak M_r\) denote the class of linear operators acting from \(B(0,1)\) into the space of polynomials of degree \(r\) and leaving the polynomials of this space fixed. Let \(K_I\) be the isometric operator acting from \(B(I)\) into \(B(0,1)\) by the formula
   \[ K_I f=|I|^\gamma f(|I|x+a), \]
   where \(a\) is the left endpoint of \(I\); here and everywhere below \(\gamma\) is equal to \(1/p\) for \(L^p(I_0)\) and to \(0\) for the other spaces; then, if we put \(T_I=K_I^{-1}TK_I\), then \(E_T(f; I)=\|f-T_I f; I\|\). If \(T\in\mathfrak M_r\), then, as is known,
   \[ E_r(f; I)\le E_T(f; I)\le (1+\|T\|)E_r(f; I), \tag{1} \]
   therefore most of the results presented remain valid when \(E_r(f; I)\) is replaced by \(E_T(f; I)\).

3. By \(\Omega_r\) we denote the class of step functions \(\rho(t)\) with a finite number of jumps at the points \(\{a_j\}_{j=1}^N\) such that \(a_1=0\), and
   \[ \int_{-\infty}^{\infty} t^i\,d\rho(t)=0 \]
   for \(0\le i<r\) and \(\ne 0\) for \(i=r\). If \(\rho(t)\in\Omega_r\), then by the \(r\)-oscillation of a function \(f(x)\in B(I_0)\) on \(I\subset I_0\) we shall call the functional
   \[ \omega_\rho(f; I)=\sup_h \left\|\int_{-\infty}^{\infty} f(x+th)\,d\rho(t); I_{\rho,h}\right\|. \]

* \(M(I_0)\) is the space of bounded (not necessarily measurable) functions on \(I_0\).

Here \(I_{\rho,h}=\{x:x+\alpha_jh\in I,\ j=1,\ldots,N\}\). Let us note that if \(\rho(t)\in\Omega_1\), then in the uniform metric \(\omega_\rho(f;I)=\sup\limits_{x\in I} f(x)-\inf\limits_{x\in I} f(x)\), which also explains the name \(\omega_\rho(f;I)\). For the function
\(\rho_0(t)=\sum_{j\le t}(-1)^{r-j}\binom rj\;(\in\Omega_r)\) we shall write \(\omega_r(f;I)\) instead of \(\omega_\rho(f;I)\).* The following theorem, important for what follows, indicates the connection between \(\omega_\rho(f;I)\) and \(E_{r-1}(f;I)\).

Theorem 1. If \(f(x)\in B(I_0)\) and is measurable, then for every \(\rho(t)\in\Omega_r\) there exist constants \(A>0\) and \(B>0\), depending only on \(\rho(t)\), such that

\[ A\omega_\rho(f;I)\le E_{r-1}(f;I)\le B\omega_\rho(f;I). \tag{2} \]

For the case \(\omega_r(f;I)\) and the space \(M(I_0)\) the theorem was first proved by H. Whitney [5] (without the assumption of measurability). We note that our method of proof makes it possible, in the uniform metric, to prove (2) assuming only that \(f\) is measurable on \(I\), which answers a question of H. Whitney concerning \(\omega_r(f;I)\) (see [5]).

Let us record consequences of (2).

Corollary 1. If \(\rho_1(t)\) and \(\rho_2(t)\) belong to \(\Omega_r\), then there exist constants \(A>0\), \(B>0\), depending only on \(\rho_1(t)\) and \(\rho_2(t)\), such that

\[ A\omega_{\rho_1}(f;I)\le \omega_{\rho_2}(f;I)\le B\omega_{\rho_1}(f;I). \tag{3} \]

It follows from (3) that it is enough to use one of the \(r\)-oscillations. We shall use \(\omega_r(f;I)\).

Corollary 2. In order that \(f(x)\in B(I_0)\) be equivalent on \(I\subset I_0\) (in the case \(M(I_0)\), equal) to a polynomial of degree \(\le r-1\), it is necessary and sufficient that \(\omega_r(f;I)=0\).

1. In the present section we shall establish a connection between the existence of derivatives of the function \(f(x)\) (on the whole interval or at a point) and the behavior of local best approximations. We shall use the following definitions of derivatives (the first of them is due to Vallée-Poussin and Zygmund).

The function \(f(x)\in B(I_0)\) has at the point \(x_0\in I_0\) a strong derivative of order \(r\) (notation \(D^r f(x_0)\)) if there exists a polynomial \(p(x)=\sum_{k=0}^r a_kx^k\) such that
\[ \|f-p;I_h(x_0)\|=o(h^r) \]
as \(h\to +0\). Here \(I_h(x_0)=[x_0-h,x_0+h]\cap I_0\); moreover, by definition, \(D^r f(x_0)=r!a_r\). The function \(f(x)\in B(I_0)\) has at the point \(x_0\in I_0\) a weak derivative of order \(r\) (notation \(D_{\mathrm{sl}}^r f(x_0)\)) if
\[ \|h^{-r}\Delta^r(h)f-a;I_h(x_0)\|=o(h^r) \]
as \(h\to +0\); here \(a=D_{\mathrm{sl}}^r f(x_0)\). Let us note that if \(D^r f(x_0)\) exists at the point \(x_0\), then \(D_{\mathrm{sl}}^r f(x_0)\) also exists. The converse is false; however, if \(D_{\mathrm{sl}}^r f(x_0)\) exists for all \(x_0\in I\) and the relation
\[ \|h^{-r}\Delta^r(h)f-D_{\mathrm{sl}}^r f(x_0);I_h(x_0)\|=o(h^r) \]
holds uniformly with respect to \(x_0\in I\), then \(D^r f(x)\) exists everywhere on \(I\). Let us also note that for \(r=1\) and the space \(M(I_0)\) both definitions coincide with the classical definition of derivative. Finally, by the \(r\)-th derivative of a function \(f(x)\in B(I_0)\) on \(I_0\) (notation \(f^{(r)}(x)\)) we shall mean the Sobolev \(r\)-th derivative if \(B(I_0)=L^p(I_0)\), and the classical \(r\)-th derivative in the other cases.

Theorem 2. Let \(f(x)\in B(I_0)\), \(x_0\in I_0\), and let \(p(x;I)\) denote a polynomial of degree \(\le s\), where \(s\ge r\), least deviating from \(f(x)\) on \(I\). Then

\[ {}^* \ \text{Let us note that in this case } \int_{-\infty}^{\infty} f(x+th)\,d\rho_0(t) = \sum_{j=0}^r(-1)^{r-j}\binom rj f(x+jh) = \Delta^r(h)f(x). \]
A. F. Timan drew my attention to this generalization of the difference operation.

conditions

\[ \lim_{I\to x_0}\frac{E_r(f;I)}{|I|^{r+\gamma}}=0,\qquad \lim_{I\to x_0}\frac{\|f^{(r)}(x;I)-a;I\|}{|I|^\gamma}=0, \tag{4} \]

where \(a\) is some constant, are necessary for the existence of \(D^r f(x_0)\) (with \(a=D^r f(x_0)\)) and sufficient for the existence of \(D_{\mathrm{sl}}^r f(x_0)\) (in this case \(D_{\mathrm{sl}}^r f(x_0)=a\)).

Remark 1. \(\lim_{I\to x_0}\) denotes the limit over the partially ordered set of intervals \(I\) containing \(x_0\). Let us note that if the limits in (4) are taken along some decreasing sequence of intervals \(\{I_j\}\) containing \(x_0\) and such that \(|I_{j+1}|/|I_j|\ge \lambda>0,\ j=1,2,\ldots,\) then Theorem 2 remains valid.

Remark 2. It can be shown by examples that the fulfillment of only one of the conditions (4) is insufficient for the existence of \(D_{\mathrm{sl}}^r f(x_0)\).

The following theorem is a generalization of a result of S. N. Bernstein \((^1)\).

Theorem 3. Let \(f(x)\in B(I_0)\) and let \(D^r f(x_0)\) exist \((x_0\in I_0)\). Then

\[ \lim_{I\to x_0}\frac{E_{r-1}(f;I)}{|I|^{r+\gamma}} = c_r\,|D^r|\,f(x_0). \tag{5} \]

Here
\[ c_r=\frac{1}{r!}E_{r-1}(x^r;[0,1]) \]
(in the spaces \(M(I_0), C(I_0), L^1(I_0)\), and \(L^\infty(I_0)\), \(c_r\), as is known, is equal to \(1/2^{r-1}r!\); in the space \(L^2(I_0)\) it is equal to \((r!/(2r)!)\times (1/\sqrt{2r+1})\); for other \(B(I_0)\) the exact value of \(c_r\) is unknown).

Remark. For the space \(C(I_0)\), (5) is also true under the assumption of the existence of \(D_{\mathrm{sl}}^r f(x_0)\).

In the following theorems, \(\varphi(t)\) denotes a monotonically increasing function together with \(t\), such that for some constant \(C>0\),
\[ \varphi(2t)\le C\varphi(t) \]
for all \(t\).

Theorem 4. If \(E_r(f;I)\le \varphi(|I|)\) for all \(I\subset I_0\) and the integral
\[ \int_0 \frac{\varphi(t)}{t^{s+\gamma+1}}\,dt \]
converges for some natural \(s\le r\), then \(f^{(s)}(x)\) exists and

\[ E_{r-s}(f^{(s)};I) = O\left( |I|^\gamma \int_0^{|I|} \frac{\varphi(t)}{t^{s+\gamma+1}}\,dt \right), \qquad I\subset I_0. \tag{6} \]

Theorem 5. If \(E_r(f;I)\le \varphi(|I|)\) for all \(I\subset I_0\), then for an integer \(s\) \((0\le s<r)\) we have

\[ E_s(f;I) = O\left( |I|^{s+\gamma+1} \int_{|I|}^{|I_0|} \frac{\varphi(t)}{t^{s+\gamma+2}}\,dt \right), \qquad I\subset I_0. \tag{7} \]

Theorem 6. Let \(f(x)\in L^p(I_0)\) and \(E_r(f;I)_{L^p}\le \varphi(I)\) for all \(I\subset I_0\). Then, if the integral
\[ \int_0 \frac{\varphi(t)}{t^{1+\gamma}}\,dt \]
converges, then \(f(x)\in L^\infty(I_0)\) and

\[ E_r(f;I)_{L^\infty} = O\left( \int_0^{|I|} \frac{\varphi(t)}{t^{1+\gamma}}\,dt \right). \tag{8} \]

Among the numerous consequences of Theorems 4–6 we mention only the following:

Corollary. The condition \(E_r(f;I)=O(|I|^{s+\gamma+\alpha})\), \(I\subset I_0\), for \(0<\alpha<1\) is necessary and sufficient in order that \(f^{(s)}\in L^\infty(I_0)\) and
\[ \operatorname{vrai\,sup}_{x\in I_0}|\Delta^1(h)f^{(s)}(x)|=O(h^\alpha), \]
and for \(\alpha=0\), in order that \(f^{(s-1)}\in L^\infty(I_0)\) and
\[ \operatorname{vrai\,sup}_{x\in I_0}|\Delta^2(h)f^{s-1}(x)|=O(h). \]

1. In paper (1), S. N. Bernstein proved a criterion for the existence, for a function in \(C(I_0)\), of a continuous \(r\)-th derivative (in terms of local approximations). In the present section we shall give criteria for the membership of \(f^{(r)}(x)\) in \(L^p(I_0)\) or \(V(I_0)\) \(\bigl(f\in V(I_0)\), if \(f\) is equivalent to a function of bounded variation on \(I_0\bigr)\). For brevity put

\[ R_{p,q}^{(r)}(f;\{I_s\})= \left\{\sum_s \frac{E_{r-1}^{p}(f;I_s)_{L^q}\,L^q}{|I_s|^{rp+p/q-1}} \right\}^{1/p}, \]

where \(f(x)\in L^q(I_0)\), and \(\{I_s\}\) is a system of nonoverlapping intervals from \(I_0\).

Theorem 7. The condition
\[ \sup_{\{I_s\}} R_{p,q}^{(r)}(f;\{I_s\})<\infty \]
for \(1<p\le q\le\infty\) is necessary and sufficient for the existence of \(f^{(r)}(x)\in L^p(I_0)\), and for \(p=1\le q\le\infty\)—for the existence of \(f^{(r-1)}(x)\in V(I_0)\).

Remark. For \(p=q=\infty\) the result can be sharpened as follows: the condition
\[ E_{r-1}(f;I)_{L^\infty}\le M|I|^r/r!2^{\,2r-1} \]
for all \(I\subset I_0\) is necessary and sufficient for the existence of \(f^{(r)}(x)\in L^\infty(I_0)\), with
\[ \|f^{(r)};I_0\|\le M \]
(see (2)).

Theorem 8. In order that \(f^{(r)}(x)\in L^1(I_0)\) exist, it is necessary and sufficient that for every \(\varepsilon>0\) there be a \(\delta>0\) such that whenever
\[ \sum_s |I_s|<\delta, \]
where \(\{I_s\}\) is a system of nonoverlapping intervals from \(I_0\), one has
\[ R_{1,q}^{(r)}(f;\{I_s\})<\varepsilon. \]

1. Most of the results stated above are also valid for functions of \(n\) variables. In this case, to characterize “mixed” properties of a function (such as the existence of a mixed derivative), one uses approximation by quasipolynomials, i.e., by functions of the form
   \[ \sum_{i=1}^n\sum_{s=0}^{k_i} f_s(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)x_i^s, \]
   while to characterize properties of the function as a whole (such as the existence of a total differential), one uses approximation by algebraic polynomials.

Dnepropetrovsk
Agricultural Institute

Received
21 X 1964

REFERENCES

1. S. N. Bernstein, Collected Works, 2, 1954, p. 306.
2. Yu. A. Brudnyi, I. E. Gopengauz, Izv. AN SSSR, Ser. Matem., 27, 723 (1963).
3. F. John, L. Nirenberg, Comm. Pure and Appl. Math., 14, 415 (1961).
4. A. Raikov, DAN, 24, No. 7, 652 (1939).
5. H. Whitney, J. Math. Pures and Appl., (9), 36, 67 (1957).
6. H. Whitney, Proc. Am. Math. Soc., 10, No. 3, 480 (1959).