Abstract
Full Text
UDC 517.512
MATHEMATICS
V. S. VIDENSKII
AN EXISTENCE THEOREM FOR A POLYNOMIAL WITH A GIVEN SEQUENCE OF EXTREMA
(Presented by Academician S. N. Bernstein, 18 I 1966)
- Let the functions \(\{\varphi_k(x)\}_{k=0}^n\), continuous on \([a,b]\), form a Chebyshev system (a \(T_n\)-system), and let real numbers \(v_0, v_1, \ldots, v_n\) be given. Denote by \(V_n\) the set of all polynomials in the system \(\{\varphi_k(x)\}_{k=0}^n\) satisfying the conditions
\[ P(x_i)=v_i \qquad (i=0,1,\ldots,n), \tag{1} \]
where \(x_0=a,\ x_n=b\), and \(x_1<\cdots<x_{n-1}\) are arbitrary interpolation nodes from the interval \((a,b)\).
Theorem 1. Let points be given
\[ a=x_0<x_1<\cdots<x_{i-1}<x_{i+r}<\cdots<x_{n-1}<x_n=b, \tag{2} \]
where \(i\) and \(r\) are given natural numbers, \(i+r\le n\). If the inequalities
\[ v_{i+2j-1}>v_{i+2j}, \qquad v_{i+2j+1}>v_{i+2j} \]
\[ (j=0,1,\ldots,m-1);\qquad m=[(r+1)/2], \tag{3} \]
hold, and, for even \(r\), in addition, the inequality
\[ v_{i+2m-1}>v_{i+2m}, \tag{4} \]
holds, then in the interval \((x_{i-1},x_{i+r})\) there exist points
\[ x_i<x_{i+1}<\cdots<x_{i+r-1} \tag{5} \]
and a polynomial \(P(x)\in V_n\) with nodes (2) and (5) such that
\[ \min_{x_{i+2j-1}\le x\le x_{i+2j+1}} P(x) = P(x_{i+2j})=v_{i+2j}, \]
\[ \max_{x_{i+2j-2}\le x\le x_{i+2j}} P(x) = P(x_{i+2j-1})=v_{i+2j-1} \tag{6} \]
for \(j=0,1,\ldots,m-1\), and, for even \(r\), in addition,
\[ \max_{x_{i+2m-2}\le x\le x_{i+2m}} P(x) = P(x_{i+2m-1})=v_{i+2m-1}. \tag{7} \]
Moreover, the polynomial \(P(x)\in V_n\) is uniquely determined by the conditions (6), (7) in the sense that if in them the points (5) are replaced by other points \(y_i<\cdots<y_{i+r-1}\) from \((x_{i-1},x_{i+r})\), then there does not exist another polynomial \(Q(x)\in V_n\) with nodes (2) and \(y_s\), satisfying the conditions (6), (7).
Theorem 1 is a strengthening of the results of the papers \((^1,^2)\), in which it is assumed, first, that the functions \(\{\varphi_k(x)\}_{k=0}^n\) not only form a Chebyshev system, but are also continuously differentiable and such that the derivative of any polynomial in \(\{\varphi_k(x)\}_{k=0}^n\) can have \(\le n\) zeros, and, second, that the numbers \(v_0,v_1,\ldots,v_n\) have successively opposite signs. For \(r=n-1\) the question was first considered in the algebraic case \((\varphi_k(x)=x^k)\) in a paper of Davis \((^3)\).
A polynomial in the Chebyshev system \(\{\varphi_k(x)\}_{k=0}^n\) which at certain \(n+1\) points of the interval \([a,b]\) attains its greatest absolute value with successively opposite signs will, following S. N. Bernstein \((^4)\), be called an oscillating polynomial. If in Theorem 1 we put \(v_k=(-1)^k\) \((k=0,1,\ldots,n)\), then we may formulate the following
Corollary. For every \(T_n\)-system \(\{\varphi_k(x)\}_{k=0}^n\) on \([a,b]\) there exists a unique oscillating polynomial \(P(x)\), normalized by the condition \(P(a)=1\).
Moreover, this corollary can be derived directly from the works of S. N. Bernstein \((^4)\), § 12 and \((^5)\).
For the proof of Theorem 1 we apply the method of successive approximations \((^1,^2)\) and induction with respect to the number \(r\). Fix \(i\) and put \(r=1\); in this case inequalities (3) will be \(v_{i-1}>v_i,\ v_{i+1}>v_i\). Adjoin to the points (2) an arbitrary point \(y_i\) from \((x_{i-1},x_{i+1})\) and construct a polynomial \(A(x)\) from the conditions \(A(x_k)=v_k\ (k\ne i)\), \(A(y_i)=v_i\). If
\[ \min_{x_{i-1}\le x\le x_{i+1}} A(x)=A(y_i)=v_i, \tag{8} \]
then \(y_i\) is the desired point, and \(A(x)\) the desired polynomial. If, however,
\[ \min_{x_{i-1}\le x\le x_{i+1}} A(x)<A(y_i)=v_i, \tag{9} \]
then construct a polynomial \(B(x)\) from the conditions \(B(x_k)=0\ (k\ne i)\) and consider the polynomial \(C(x;\lambda)=A(x)+\lambda B(x)\). Since \(B(x)\ne0\) for \(x_{i-1}<x<x_{i+1}\), it is possible to choose such a value \(\lambda=\lambda_1\) that
\[ \min_{x_{i-1}\le x\le x_{i+1}} C(x;\lambda_1)>v_i. \tag{10} \]
Since \(\min C(x;\lambda)\) on \([x_{i-1},x_{i+1}]\) is a continuous function of \(\lambda\), and \(C(x;0)=A(x)\), it follows from (9) and (10) that there is a \(\lambda=\lambda_0\), \(0<\lambda_0<\lambda_1\), such that
\[
\min_{x_{i-1}\le x\le x_{i+1}} C(x;\lambda_0)=v_i.
\]
Denote the point at which the minimum of \(C(x;\lambda_0)\) is attained by \(x_i\), and \(P(x)=C(x;\lambda_0)\) is a polynomial satisfying the conditions of the theorem. Suppose that there exists another polynomial \(Q(x)\) such that \(Q(x_k)=v_k\ (k\ne i)\),
\[
\min_{x_{i-1}\le x\le x_{i+1}} Q(x)=Q(\xi_i)=v_i,\quad x_i<\xi_i<x_{i+1}.
\]
From \(P(x_i)=v_i<Q(x_i)\) and \(P(\xi_i)>v_i=Q(\xi_i)\) it follows that the polynomial \(P(x)-Q(x)\) has a zero in \((x_i,\xi_i)\); in addition, it has \(n\) other zeros \(x_k\ (k\ne i)\), which is impossible.
Assume, by induction, that Theorem 1 has been proved for some number \(r\) \((1\le r\le n-2)\) for any \(i\), and show that it is true for the number \(r+1\). For definiteness put \(r=2m-1\). Let the inequalities (3), (4) be satisfied and let the points be given
\[ a=x_0<x_1<\cdots<x_{i-1}<x_{i+2m}<\cdots<x_n=b. \tag{11} \]
Adjoin to them an arbitrary point \(x'_{i+2m-1}\), \(x_{i-1}<x'_{i+2m-1}<x_{i+2m}\). By the induction hypothesis, in \((x_{i-1},x'_{i+2m-1})\) there will be points
\[
x'_i<\cdots<x'_{i+2m-2}
\]
and a uniquely determined polynomial \(P_1(x)\in V_n\) with nodes \(x_k,\ x'_s\), which satisfies conditions (7) at the points \(x'_s\ (s=i,\ldots,i+2m-2)\). If
\[ \max_{x'_{i+2m-2}\le x\le x_{i+2m}} P_1(x)=P_1(x'_{i+2m-1})=v_{i+2m-1}, \tag{12} \]
then \(P_1(x)\) is the desired polynomial. If, however, (12) is not satisfied, then at certain points \(y\) of the interval \((x'_{i+2m-2},x_{i+2m})\) the equality
\[
P_1(y)=v_{i+2m-1}
\]
holds. Since \(P_1(x)\) is a polynomial in a \(T_n\)-system, the number of these points is finite; denote by \(y'_{i+2m-1}\) the nearest of them to the point \(x'_{i+2m-1}\), and put, for definiteness, \(x'_{i+2m-1}<y'_{i+2m-1}\). Choose
\(x''_{i+2m-1}=0.5\,(x'_{i+2m-1}+y'_{i+2m-1})\), adjoin it to the points (11) and repeat the preceding arguments. In the interval \((x_{i-1},x''_{i+2m-1})\) there will be points
\(x''_i<x''_{i+1}<\cdots <x''_{i+2m-2}\) and a polynomial \(P_2(x)\in V_n\) with nodes \(x_k,x''_s\), which satisfies conditions (7) at the points \(x''_s\) \((s=i,\ldots,i+2m-2)\). If \(P_2(x)\) does not satisfy the equality
\[ \max_{x''_{i+2m-2}\le x\le x_{i+2m}} P_2(x) = P_2(x''_{i+2m-1})=v_{i+2m-1}, \tag{13} \]
then one must continue the process of constructing the points \(x_s^{(k)}\) and the polynomials \(P_k(x)\in V_n\) according to the same scheme. Suppose that the process does not terminate at any finite step, and let us prove its convergence.
Thus, let \(P_2(x)\) fail to satisfy (13). We shall show that
\[ P_2(x)<P_1(x)\quad \text{for } x\in I=(x'_{i+2m-1},y'_{i+2m-1}). \tag{14} \]
Suppose that \(P_1(\xi_{i+2m-1})=P_2(\xi_{i+2m-1})=v^*_{i+2m-1}\), \(\xi_{i+2m-1}\in I\). Adjoin \(\xi_{i+2m-1}\) to (11) and denote by \(V_n^*\) the set of polynomials obtained from \(V_n\) by replacing \(v_{i+2m-1}\) by \(v^*_{i+2m-1}\). Then \(P_1(x)\in V_n^*\), \(P_2(x)\in V_n^*\), and both polynomials satisfy conditions (6) at \(x'_s\) and \(x''_s\) \((s=i,\ldots,i+2m-2)\), respectively. This contradicts the inductive assumption on the uniqueness of such a polynomial. Hence \(P_1(x)\ne P_2(x)\) for \(x\in I\). On the other hand, the point \(y'_{i+2m-1}\) was chosen so that \(P_1(x)>v_{i+2m-1}\) for \(x\in I\), and, in particular, \(P_1(x''_{i+2m-1})>P_2(x''_{i+2m-1})=v_{i+2m-1}\). Similarly to the preceding, we find in \((x''_{i+2m-2},x_{i+2m})\) such a point \(y''_{i+2m-1}\) that \(P_2(y''_{i+2m-1})=v_{i+2m-1}\). By (14), \(y''_{i+2m-1}\in I\) and
\[ |y''_{i+2m-1}-x''_{i+2m-1}|<0.5\,|y'_{i+2m-1}-x'_{i+2m-1}|. \tag{15} \]
We prove that the inequalities
\[ x'_i\le x''_i,\ldots,x'_{i+2m-2}\le x''_{i+2m-2}. \tag{16} \]
hold. Denote by \(z\) the leftmost of the points of \(I\) at which \(P_2(x)=v_{i+2m-1}\). Choose a point \(q_{i+2m-1}\) from \((x'_{i+2m-1},z)\), adjoin it to (11), and construct a polynomial \(Q(x)\in V_n\) which satisfies conditions (6) at the points
\(q_i<\cdots <q_{i+2m-2}\) from \((x_{i-1},q_{i+2m-1})\). Arguing as in the derivation of (14), we obtain \(P_2(x)<Q(x)<P_1(x)\) for \(q_{i+2m-1}<x<z\). It follows that, as \(q_{i+2m-1}\to x'_{i+2m-1}\), the polynomial \(Q(x)\) tends uniformly to \(P_1(x)\) on \([x'_{i+2m-1}+\varepsilon,z]\), \(\varepsilon>0\), and hence on the whole \([a,b]\). Clearly, \(Q(x)\) depends continuously on \(q_{i+2m-1}\). Choose \(\delta>0\) so small that the \(\delta\)-neighborhoods of the points \(x'_s\) \((s=i,\ldots,i+2m-2)\) do not intersect, and then choose \(q_{i+2m-1}\in I\) so close to \(x'_{i+2m-1}\) that the inequalities
\(|x'_i-q_i|<\delta,\ldots,|x'_{i+2m-2}-q_{i+2m-2}|<\delta\)
hold. In each \(\delta\)-neighborhood there lies at least one zero of the polynomial \(Q(x)-P_1(x)\); moreover, its zeros include the points (11). Consequently, in each \(\delta\)-neighborhood there lies exactly one zero. We show that \(x'_{i+2m-2}\le q_{i+2m-2}\). If \(x'_{i+2m-2}>q_{i+2m-2}\), then \(Q(x)-P_1(x)\) would have at least two zeros in \((q_{i+2m-2},x'_{i+2m-1})\), because of the inequalities
\(Q(q_{i+2m-2})-P_1(q_{i+2m-2})<0\),
\(Q(x'_{i+2m-2})-P_1(x'_{i+2m-2})>0\),
\(Q(x'_{i+2m-1})-P_1(x'_{i+2m-1})<0\).
Suppose that
\[ x'_{i+2m-2}=q_{i+2m-2},\ x'_{i+2m-3}=q_{i+2m-3},\ldots,\ x'_{i+k}=q_{i+k},\ x'_{i+k-1}\ne q_{i+k-1}. \tag{17} \]
If \(q_{i+k-1}<x'_{i+k-1}\), then
\((-1)^{2m-k}\{Q(q_{i+k-1})-P_1(q_{i+k-1})\}>0\),
\(Q(x'_{i+2m-1})-P_1(x'_{i+2m-1})>0\). Consequently, the number of zeros of the polynomial \(Q(x)-P_1(x)\) in \((q_{i+k-1},x'_{i+2m-1})\) exceeds the number of points (17) by an odd number, and hence the number of zeros in \((a,b)\) is greater than \(n\). Similarly one proves the inequalities
\(x'_{i+2m-3}\le q_{i+2m-3},\ldots,x'_i\le q_i\), from which (16) follows by continuity.
If \(\{P_k(x)\}\subset V_n\) is our sequence, then
\[ [x_{i+2m-1}^{(k+1)},\,y_{i+2m-1}^{(k+1)}]\subset [x_{i+2m-1}^{(k)},\,y_{i+2m-1}^{(k)}] \]
and inequalities similar to (15) and (16) hold:
\[ |y_{i+2m-1}^{(k+1)}-x_{i+2m-1}^{(k+1)}|<0.5\,|y_{i+2m-1}^{(k)}-x_{i+2m-1}^{(k)}|, \quad x_i^{(k)}\le x_i^{(k+1)},\ldots,x_{i+2m-2}^{(k)}\le x_{i+2m-2}^{(k+1)}. \]
It follows that the limits \(\lim_{k\to\infty}x_s^{(k)}=x_s\) exist for \(s=i,\ldots,i+2m-1\). Moreover, \(x_s\ne x_{s+1}\); if \(x_s=x_{s+1}\), then \(\{P_k(x)\}\) would converge to a discontinuous function, which is impossible, since \(\{P_k(x)\}\) is uniformly bounded on \(I\). Put \(P(x)=\lim_{k\to\infty}P_k(x)\). It is clear that
\[ P(x)\in V_n \]
and that it satisfies conditions (6) and (7).
It remains to prove the uniqueness of the polynomial \(P(x)\). Suppose that there exists another polynomial \(Q(x)\in V_n\) which satisfies conditions (6) and (7) at the points \(y_i<\cdots<y_{i+2m-1}\). If \(x_{i+2m-1}=y_{i+2m-1}\), then there would exist two polynomials \(P(x)\in V_n\) and \(Q(x)\in V_n\) satisfying conditions (6), which contradicts the inductive assumption on uniqueness. Let, for definiteness, \(x_{i+2m-1}<y_{i+2m-1}\). From the preceding it follows that the polynomial \(P(x)\) is uniquely determined by the initial choice of the point \(x'_{i+2m-1}\). Therefore we must assume that some points \(x'_{i+2m-1}\) from the interval \((x_{i+2m-1},y_{i+2m-1})\) lead to the system of points \(x_i<\cdots<x_{i+2m-1}\) and to the polynomial \(P(x)\), while others lead to the system of points \(y_i<\cdots<y_{i+2m-1}\) and to the polynomial \(Q(x)\). But this is impossible, since the points \(x_i<\cdots<x_{i+2m-2}\) are continuous monotone functions of \(x'_{i+2m-1}\). Indeed, if by \(\xi_{i+2m-1}\) we denote the upper bound of those \(x'_{i+2m-1}\) which generate the system of points \(x_i<\cdots<x_{i+2m-1}\), then, by continuity, both the point \(\xi_{i+2m-1}\) itself and the point \(\xi_{i+2m-1}+\Delta \xi\), for sufficiently small \(\Delta \xi>0\), will lead to the same system of points, which contradicts the definition of \(\xi_{i+2m-1}\). Theorem 1 is proved.
- A system \(\{\varphi_k(x)\}_{k=0}^{n}\) of continuous functions on \([a,b]\), such that \(\varphi_k(a)=\varphi_k(b)\), is called a \((^4)\) periodic \(T_n\)-system on \([a,b]\) if every polynomial with respect to this system has \(\le n\) zeros for \(a\le x<b\). We note that \(n\) must be an even number, since \(P(a)=P(b)\) for every polynomial, while there are polynomials that change sign \(n\) times. In this case Theorem 1 can be supplemented as follows.
Theorem 2. Let \(\{\varphi_k(x)\}_{k=0}^{2m}\) form a periodic \(T_{2m}\)-system on \([a,b]\), and let numbers \(v_0,v_1,\ldots,v_{2m-1}\) be given satisfying the inequalities
\[ v_0>v_1,\quad v_0>v_{2m-1},\quad v_{2j}>v_{2j-1},\quad v_{2j}>v_{2j+1} \quad (j=1,\ldots,m-1). \]
Then there exists a system of points
\[ a=x_0<x_1<\cdots<x_{2m-1}<x_{2m}=b \]
and a uniquely determined polynomial \(P(x)\) such that
\[ \begin{aligned} \max_{x_0\le x\le x_1} P(x)&=P(x_0)=v_0, & \max_{x_{2m-1}\le x\le x_{2m}} P(x)&=P(x_{2m})=v_0, \\ \max_{x_{2j-1}\le x\le x_{2j+1}} P(x)&=P(x_{2j})=v_{2j}, & \min_{x_{2j}\le x\le x_{2j+2}} P(x)&=P(x_{2j+1})=v_{2j+1} \end{aligned} \tag{18} \]
for \(j=1,\ldots,m-1\).
Put \(v_0=v_{2m}\) and consider an arbitrary sequence
\[ a<b_1<b_2<\cdots \]
converging to \(b\). On the interval \([a,b_k]\), by Theorem 1, one can uniquely construct a polynomial \(P_k(x)\) satisfying (18) at certain points
\[ a=x_0^{(k)}<x_1^{(k)}<\cdots<x_{2m}^{(k)}=b_k. \]
It can be proved that
\[ x_s^{(k)}\le x_s^{(k+1)}\quad (s=1,\ldots,2m), \]
whence it follows that \(\{P_k(x)\}\) converges uniformly to a polynomial \(P(x)\) satisfying conditions (18).
Leningrad Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich
Received
16 I 1966
CITED LITERATURE
- V. S. Videnskii, DAN, 162, No. 2, 251 (1965).
- V. S. Videnskii, Collection of Scientific Works of the Leningrad Mechanical Institute, No. 50, 1966.
- Ch. Davis, Am. Math. Monthly, 64, No. 9, 679 (1957).
- S. N. Bernstein, Extremal Properties of Polynomials, L.—M., 1937.
- S. N. Bernstein, Collection of Works, 2, Article No. 76, 1954, p. 287.
- Ch. Davis, Am. Math. Monthly, 64, No. 9, 679 (1957).