UDC 517.512.6

MATHEMATICS

Yu. A. BRUDNYI

APPROXIMATION OF FUNCTIONS DEFINED IN A CONVEX POLYHEDRON

(Presented by Academician L. V. Kantorovich, May 5, 1970)

1. Let \(\Gamma\) be a convex bounded polyhedron in \(s\)-dimensional space \(\mathbb{R}^s\); denote by \(\rho(x,\Gamma)\) the distance from the point \(x\) to the nearest vertex of \(\Gamma\). Further, let \(\omega(\tau)=\omega_k(f;\tau;\Gamma)\), \(k\) be the modulus of continuity of the function \(f\), defined on \(\Gamma\); this means that

\[ \omega(\tau)=\sup_{|y|\le \tau}\|\Delta_y^k f\|_C, \tag{1} \]

where the \(C\)-norm is taken over the domain of definition of the \(k\)-th difference of the function \(f\).

Denote by \(p_n(x)\) a polynomial of degree \(n\) in \(x=(x_1,\ldots,x_s)\), and let

\[ \lambda_n(x)=\lambda_n(x,\Gamma)=\sqrt{\rho(x,\Gamma)}+n^{-1}. \tag{2} \]

Theorem 1. If \(f\in C(\Gamma)\) and \(k\) is a fixed natural number, then there exists a sequence of polynomials \(p_n(x)\), \(n\ge k-1\), such that for \(x\in\Gamma\)

\[ |f(x)-p_n(x)|\le A\omega(\lambda_n(x)/n). \tag{3} \]

Here \(A=A(k,\Gamma)\), i.e., it depends only on \(k\) and \(\Gamma\).

Thus Theorem 1 asserts the possibility of improving the approximation at the vertices of \(\Gamma\). For functions of one variable, from (3) we obtain the known theorems of A. F. Timan \((^1)\) \((k=1)\) and V. K. Dzyadyk \((^2)\)—G. Freud \((^3)\) \((k=2)\); for arbitrary \(k\), Theorem 1 in the case of functions of one variable was proved by the author \((^4,^5)\). We note that the phenomenon of improvement of approximation at the endpoints of an interval was first discovered in \((^6)\).

1. Suppose that \(f\in C^l(\Gamma)\) and

\[ \sup_{|\alpha|=l}\omega_k(D^\alpha f;\tau;\Gamma)\le \omega(\tau). \tag{4} \]

Corollary 1. There exists a sequence of polynomials \(p_n(x)\), \(n\ge k+l-1\), such that for \(x\in\Gamma\)

\[ |f(x)-p_n(x)|\le A\{\lambda_n(x)/n\}^l\omega(\lambda_n(x)/n). \tag{5} \]

Here \(A=A(k+l,\Gamma)\).

Further, let \(U\) be a closed convex bounded subset of \(\mathbb{R}^s\), containing at least one interior point. We shall call a point \(x\in U\) conical if at this point there exist at least \(s\) linearly independent supporting hyperplanes. Suppose that \(U\) contains a finite number of conical points, and let \(\rho(x)\) be the distance from \(x\) to the nearest conical point of \(U\); define \(\lambda_n(x)\) by equality (2). Under these assumptions we have

Corollary 2. If \(f\in C^l(U)\) and for \(k=1\) condition (4) is satisfied, then there exists a sequence of polynomials \(p_n(x)\), \(n\ge l\), such that for \(x\in U\)

\[ |f(x)-p_n(x)|\le A\{\lambda_n(x)/n\}^l\omega(\lambda_n(x)/n). \tag{6} \]

Here \(A=A(l,U)\).

1. The question arises of the possibility of improving the approximation at nonconical points of a convex set. In this direction the following negative result holds.

Theorem 2. If \(x_0 \in U\) is not a conical point, then there exists a function \(f_0 \in C(U)\), satisfying a Lipschitz condition on \(U\), such that the approximation

\[ |f_0(x)-p_n(x)| \leq \frac{A}{n}\left(\sqrt{|x-x_0|}+n^{-1}\right) \]

is impossible for any sequence of polynomials \(p_n(x)\), \(n=n_0, n_0+1,\ldots\), and for any constant \(A=A(U)\).

The corresponding negative result for the one-dimensional case was obtained by I. E. Gopengauz \((^7)\).

1. Let us outline the proof of Theorem 1.

Lemma 1. For every natural number \(k\) and integer \(n \geq k-1\) there exists a linear operator \(P=P(n,k)\), mapping \(C(0,1)\) into the space of polynomials of degree \(n\), such that:

1) \(\|P\| \leq A=A(k)\);

2) for \(0 \leq x \leq 1\)

\[ |f(x)-P(f;x)| \leq B(k)\omega_k\left(f;\sqrt{x(1-x)}/n+1/n^2\right); \]

3) if \(0<a<a+\delta<1\), then there exists \(C=C(k,\delta)\) such that

\[ \max_{a+\delta \leq x \leq 1} |f(x)-P(f;x)| \leq C\left\{\max_{a\leq x\leq 1}|f(x)|+\omega_k(f;n^{-2})\right\}. \]

For the proof see \((^5)\).

Let now \(Q \subset \mathbb{R}^s\) be a cube and let \(U\) be an open convex body which contains only one vertex of \(Q\) and intersects only those faces of \(Q\) that are adjacent to this vertex. Denote further by \(U(\delta)\) the \(\delta\)-expansion of \(U\).

Lemma 2. Let \(f\in C(Q)\) and \(f=0\) outside \(Q\cap U\); let \(\omega(\tau)\) be (1) and \(\lambda_n(x)=\lambda_n(x;Q)\). Then there exists a sequence of polynomials \(p_n(x)\), \(n=sj\), \(j\geq k-1\), such that:

1) there exists a constant \(A=A(k,Q)\) for which

\[ |f(x)-p_n(x)|\leq A\omega(\lambda_n(x)/n); \]

2) for a given \(\delta>0\) there exists \(B=B(k,\delta,Q)\) such that

\[ \max_{Q\setminus U(\delta)} |p_n(x)| \leq B\omega(n^{-2}). \]

Proof. Without loss of generality we assume \(Q=\{x\mid 0\leq x_i\leq 1\}\); denote by \(P_i\) the operator \(P(n,k)\) of Lemma 1 applied to functions \(f(x)\in C(Q)\) with respect to the variable \(x_i\), and let \(P=P_1P_2\ldots P_n\). Then \(P(f;x)\) is a polynomial of degree \(sn\), and the application of properties 1)—3) of the operator \(P(n,k)\), established in Lemma 1, completes the proof.

Let now \(\Pi\) be a convex \(s\)-dimensional pyramid with vertex \(x_0\); let \(U\) be an open convex body containing \(x_0\) and not containing points of the base of \(\Pi\), and let \(U(\delta)\) be the \(\delta\)-expansion of \(U\).

Lemma 3. If \(f\in C(\Pi)\) and \(f=0\) outside \(\Pi\cap U\), then there exists a sequence of polynomials \(p_n(x)\), \(n=js\), \(j\geq k-1\), such that:

1) if \(\omega(\tau)\) is defined by formula (1), then for some \(A=A(k,\Pi)\)

\[ |f(x)-p_n(x)|\leq A\omega\left(\left(\sqrt{|x-x_0|}+n^{-1}\right)/n\right); \]

2) for a given \(\delta>0\) there exists \(B=B(k,\delta,\Pi)\) such that

\[ \max_{\Pi\setminus U(\delta)} |p_n(x)|\leq B\omega(n^{-2}). \]

Proof. Without loss of generality we assume \(x_0=0\) and let \(\Pi_+\) be the infinite pyramid generated by \(\Pi\), and let \(l_j(x)\), \(j=1,2,\ldots,N\), be the linear forms defining \(\Pi_+\), i.e.,
\[ \Pi_+ = \{x\in \mathbf{R}^s \mid l_j(x)\geq 0\}. \]
Define a mapping \(A:\Pi_+\to \mathbf{R}_+^N\) by the formula
\[ A(x)=\{l_1(x),\ldots,l_N(x)\}. \]
Then \(A(\Pi_+)\) is an \(s\)-dimensional section of \(\mathbf{R}_+^N\) by a plane passing through \(0\), and \(A(\Pi_+)\) is linearly isomorphic to \(\Pi_+\). Transferring \(f\) (which may be regarded as given on \(\Pi_+\)) by means of \(A\) to \(A(\Pi_+)\) and considering the transferred function on \(\mathbf{R}_+^N\), we can apply Lemma 2 to it. This implies everything.

For a given convex bounded polyhedron \(\Gamma\), consider a covering \(U_j\), \(1\leq j\leq r\), by open convex sets such that each \(U_j\) contains only one vertex of \(\Gamma\) and intersects only the faces adjacent to this vertex. Denote this vertex by \(x_j\), and let \(\Pi_j\) be the pyramid with vertex \(x_j\), generated by \(\Gamma\) and containing \(\Gamma\). Let \(\varphi_j\), \(1\leq j\leq r\), be an infinitely differentiable partition of unity subordinate to the covering \(\{U_j\}\), and let \(f_j=f\varphi_j\).

Lemma 4. We have: 1) \(\sum f_j=f\); 2) \(f_j\in C(\Pi_j)\) and is equal to \(0\) outside \(\Pi_j\cap U_j\); 3)
\[ \omega_k(f_j;\tau)\leq A(k,\Gamma)\omega_k(f;\tau),\qquad 1\leq j\leq r. \]

Now let \(p^j(x)\) be a polynomial of degree \(n\), constructed for the function \(f_j\) according to Lemma 3. Then from this lemma and Lemma 4 it follows that, for \(x\in\Gamma\),
\[ |f_j(x)-p^j(x)|\leq A\omega_k(f;\lambda_n(x,\Gamma)/n), \]
and, putting \(p(x)=\sum p^j(x)\), we prove the theorem for \(n\geq s(k-1)\).

It remains to prove Theorem 1 for \(n=k-1\); but in this case it follows from the multidimensional analogue of Whitney’s theorem, proved in (8).

Dnepropetrovsk Chemical-Technological
Institute

Received
23 IV 1970

CITED LITERATURE

1. A. F. Timan, DAN, 81, No. 4, 509 (1951).
2. V. K. Dzyadyk, Izv. AN SSSR, Ser. Matem., 22, No. 3, 337 (1958).
3. G. Freud, Math. Ann., 137, No. 1, 17 (1959).
4. Yu. A. Brudnyi, DAN, 148, No. 6, 1237 (1963).
5. Yu. A. Brudnyi, Izv. AN SSSR, Ser. Matem., 32, No. 4, 780 (1968).
6. S. M. Nikol’skii, Izv. AN SSSR, Ser. Matem., 10, 207 (1946).
7. I. E. Gopengauz, Matem. Zametki, 1, No. 2, 163 (1967).
8. Yu. A. Brudnyi, Matem. Sborn., 82 (124), No. 2, 175 (1970).