MATHEMATICS

E. V. VORONOVSKAYA

SOME CRITERIA FOR THE STABILITY OF FUNCTIONALS

(Presented by Academician S. L. Sobolev, 12 X 1964)

Suppose that a linear functional on the set of algebraic polynomials \(\{P_n(x)\}\) is given by the segment of numbers

\[ F(1),\ F(x),\ \ldots,\ F(x^n). \tag{*} \]

Let this segment be not absolutely monotone \((^1)\). We denote its true nodes by \((\sigma_k)_1^{s_0}\) and the weights by \((\delta_k)_1^{s_0}\). Here always \(2 \le s_0 \le n+1\); among the \((\sigma_k)\) there are no equal ones, and among the \((\delta_k)\) there are no zeros.

Put

\[ R_{s_0}(x)=\prod_1^{s_0}(x-\sigma_k) \]

—the resolvent of the functional \((*)\), and

\[ R_{s_0,m}(x)=\prod_{k\ne m}(x-\sigma_k). \]

The weights \((\delta_k)\) are determined from the compatible system of \(n+1\) equations

\[ \sum_{k=0}^{s_0}\delta_k\sigma_k^m=F(x^m)\qquad (m=0,1,\ldots,n) \tag{1} \]

by the formulas

\[ \delta_m=\frac{F(R_{s_0,m})}{\prod_{i\ne m}(\sigma_m-\sigma_i)}. \tag{2} \]

Choose on \([0,1]\) arbitrarily \(s\) mutually distinct points; call them conditional nodes, and

\[ \overline{R}_s(x)=\prod_1^s(x-\rho_i) \]

the conditional resolvent; let

\[ \overline{R}_{s,m}(x)=\prod_{i\ne m}(x-\rho_i). \]

From the first \(s\) equations of the system \((s\le n+1)\)

\[ \sum_{i=0}^s \overline{\delta}_i\rho_i^m=F(x^m)\qquad (m=0,1,\ldots,n) \tag{3} \]

we have the conditional weights

\[ \overline{\delta}_m=F(\overline{R}_{s,m})\Big/\prod_{i\ne m}(\rho_m-\rho_i). \tag{4} \]

In contrast to system (1), system (3) is in general incompatible when \(s<n+1\).

In the particular case where \(s=s_0\) and all \((\rho_i)=(\sigma_k)\), respectively, then \((\overline{\delta}_i)=(\delta_k)\) and the entire system (3) is compatible. Conversely, if for \(s=s_0\) all \(\overline{\delta}_i=\delta_i\) and system (3) is compatible, then the system of nodes \((\rho_i)\) coincides with \((\sigma_k)\).

We note some consequences of the formulas written above.

Corollary 1. If \(s_0<s\) \((\leq n+1)\), then one can always choose the conditional nodes \((\rho_i)_1^s\) so that exactly \(l=s-s_0\) of the nodes have zero loads in formulas (4). For this it is sufficient that \((\sigma_k)_1^{s_0}\subset(\rho_i)_1^s\).

Corollary 2. If, for no \((\rho_i)_1^s\), it is possible to turn in system (4), for \(m=s\), at least one load \(\delta\) into zero, then certainly \(s_0\geq s\).

Corollary 3. If, although it is possible under the conditions of the corollary to turn some \(\delta\) into zero, this is possible only for such \(\rho\) that certainly cannot be a system of true nodes of the given functional, then even in that case \(s_0\geq s\).

Theorem. A sufficient condition for the number \(s_0\) of true nodes (*) to satisfy the condition \(s_0\geq s\), where \(s\) is chosen in advance \((s\leq n+1)\), consists in the inconsistency, for all \((\rho_i)_1^s\), of the system of conditions, numbering \(n+2-s\),

\[ F(\overline R_s)=0,\quad F(x\overline R_s)=0,\ldots,\quad F(x^{n-s}\overline R_s)=0. \tag{a} \]

\[ F(\overline R_{s,m})=0, \tag{б} \]

where \(m\) is equal to one of the numbers \(1,2,\ldots,s\).

Indeed, if \(s_0<s\) and \((\rho_i)\) are chosen so that they contain \((\sigma_k)\), then \(\overline R_s(x)=R_{s_0}(x)r(x)\), and the conditions (a) are necessarily fulfilled. As for condition (б), it is the requirement, following from (4), that at least one of the loads \(\delta=0\). The impossibility of satisfying conditions (a) and (б) simultaneously indicates that \(s_0\geq s\).

Remark. According to Corollary 3, the theorem is also valid under smaller restrictions: if conditions (a) and (б) are satisfied jointly only for such sets \((\rho_i)_1^s\) that certainly cannot be a set of true nodes of degree \(n\) \((^1)\), then in this case too \(s_0\geq s\).

We shall call a functional \(F\), depending on some parameter \(\xi\), stable in the interval \(\alpha<\xi<\beta\), if in this interval the number of its true nodes \(s_0(\xi)\geq n\).

We give examples of functionals whose stability is investigated with the aid of the criterion proved above.

Example 1. \(F=0_0,0_1,\ldots,0_{n-1},\xi\) \((\xi\neq0)\). Put \(s=n+1\). We have one condition (б): \(F(R_{n+1,m})=0\), i.e. \(\xi=0\). Thus, \(F\) has \(n+1\) true nodes for all \(\xi\neq0\).

Example 2. \(F=0_0,0_1,\ldots,0_{n-2},1_{n-1},\xi\); let \(s=n+1\). Owing to the arbitrariness of \((\rho_i)\) on \([0,1]\), for \(\xi\leq0\) and for \(\xi\geq n\) the condition is clearly not fulfilled; consequently, \(s_0=n+1\). Put \(s=n\); then we have two conditions: \(F(\overline R_n)=0\), \(F(\overline R_{n,m})=0\). They give \(\xi=s_1\) and \(1=0\). Thus, the given functional has \(s_0\geq n\) for all real \(\xi\).

Example 3. \(F_\xi(P_n)=P_n^{(k)}(\xi)\) \((1\leq k\leq n)\) \((^2)\). Put \(s=n+1\). Condition (б) gives

\[ R_{n+1,m}^{(k)}(\xi)=0. \tag{5} \]

The impossibility of this requirement for \(\xi\geq1\) or \(\xi\leq0\) is obvious, since all roots of equation (5) lie inside \([0,1]\).

Put \(s=n\). We have two conditions: \(R_n^{(k)}(\xi)=0\), \(\overline R_{n,m}(\xi)=0\). Denote \(\overline R_{n,m}(\xi)=Q_{n-1}(\xi)\); then \(\overline R_n(\xi)=Q_{n-1}(\xi)(\xi-\rho_m)\), and the system of conditions takes the form \(Q_{n-1}^{(k)}(\xi)=0;\ Q_{n-1}^{(k-1)}(\xi)=0\), which is inconsistent for simple roots. Thus, \(s_0\geq n\) for all \(\xi\).

Example 4. 1) \(F_{\rho,\xi}(P_n)=\operatorname{Re}P_n(z)\); 2) \(F_{\rho,\xi}(P_n)=\operatorname{Im}P_n(z)\), i.e. 1) \(F_{\rho,\xi}(x^k)=\rho^k\cos k\xi\); 2) \(F_{\rho,\xi}(x^k)=\rho^k\sin k\xi\) \((k=0,1,\ldots,n)\). Let \(s=n\); then we have: 1) \(\operatorname{Re}\overline R_n(z)=0\), \(\operatorname{Re}\overline R_{n,m}(z)=0\), or, putting \(\overline R_{n,m}(z)=Q_{n-1}(z)\), \(\operatorname{Re}Q_{n-1}(z)=0\), \(\operatorname{Re}[Q_{n-1}(z)(z-\rho)]=0\). These requirements are compatible only for real \(z\).

The same result is obtained in case 2).

Thus, off the axis \(Ox\) one has \(s_0\geq n\) \((^3)\).

Example 5. \(F_x=1,\ \dfrac{x}{2},\ \dfrac{x^2}{3},\ldots,\dfrac{x^n}{n+1}\) is an integral functional on \(\{P_n(x)\}\), since

\[ F_x(P_n)=\frac1x\int_0^x P_n(\xi)\,d\xi . \]

It is obvious that for \(x<0\) the number of true nodes is \(s_0=n+1\), while for \(0<x\leq 1\) the functional is absolutely monotone (amorphous) (1).

Before applying the criterion to the case \(x>1\), we shall have to introduce some auxiliary estimates.

Lemma. Let \(f(x)\) be continuous on \([0,1]\) and \(0<\alpha<1\). Then, after replacing \(\alpha\) by one of the numbers \(0\) or \(1\) in

\[ I(\alpha)=\int_0^1 (x-\alpha) f(x)\,dx \]

the modulus of the integral will not decrease, i.e. \(I(\alpha)\) is majorized either by

\[ \left|\int_0^1 x f(x)\,dx\right|, \]

or by

\[ \left|\int_0^1 (1-x) f(x)\,dx\right|. \]

Indeed,

\[ I(\alpha)=\int_0^1 x f(x)\,dx-\alpha\int_0^1 f(x)\,dx . \]

1) Let \(I(\alpha)>0\). Then, if \(\int_0^1 f(x)\,dx>0\), then

\[ 0<I(\alpha)<\int_0^1 x f(x)\,dx, \]

and if \(\int_0^1 f(x)\,dx<0\), then

\[ 0<I(\alpha)<\int_0^1 (x-1) f(x)\,dx . \]

2) Let \(I(\alpha)<0\); if \(\int_0^1 f(x)\,dx>0\), then

\[ \int_0^1 (x-1)Q(x)\,dx<I(\alpha)<0; \]

whereas if \(\int_0^1 f(x)\,dx<0\), then

\[ \int_0^1 xQ(x)\,dx<I(\alpha)<0 . \]

Thus, the lemma is proved.

Corollary. For \(0\leq \alpha_1\leq \alpha_2\leq\cdots\leq \alpha_n\leq 1\), in

\[ \int_0^1 \prod_1^n (x-\alpha_i)\,dx \]

each of the numbers \(\alpha_i\) may be replaced by one of the numbers \(0\) or \(1\) in such a way that

\[ \left|\int_0^1 \prod_1^n (x-\alpha_i)\,dx\right| \leq \int_0^1 x^k(1-x)^{\,n-k}\,dk . \]

Equality in this estimate occurs only if \(\alpha_1=\alpha_2=\cdots=\alpha_k=0\) and \(\alpha_{k+1}=\cdots=\alpha_n=1\).

Remark 1. The beta integral

\[ \int_0^1 x^k(1-x)^{\,n-k}\,dx=B_n(k)\qquad (0\leq k\leq n) \]

has a maximum at \(k=0\) and \(k=n\) and a minimum at \(k=E(n/2)\). Thus,

\[ B_n(k)\leq \frac1{n+1}. \]

Remark 2. Denote

\[ \int_\alpha^\beta \xi^k(\xi-1)^{\,n-k}\,d\xi=I_{k,n-k}(\alpha,\beta); \]

we have the directly verified recurrence formula

\[ I_{k+1,n-k-1}(\alpha-\beta)-I_{k,n-k}(\alpha,\beta)=I_{k,n-1-k}(\alpha,\beta). \tag{6} \]

We return to the integral functional, putting in it \(x>1\). The general form of conditions (a) and (b) here reduces to the conditions

\[ \int_0^x \xi^k \overline{R}_s(\xi)\,d\xi=0\quad (k=0,1,\ldots,s-k); \tag{a′} \]

\[ \int_0^x \overline{R}_{s,m}(\xi)\,d\xi=0. \tag{b′} \]

Let \(s=n+1\). We have one condition

\[ \int_0^x \overline{R}_{n+1,m}(\xi)\,d\xi=0 \]

or

\[ \int_0^1 \prod_1^n(\xi-\rho_i)\,d\xi+\int_1^x \prod_1^n(\xi-\rho_i)\,d\xi=0, \]

briefly,

\[ I_1+I_2(x)=0; \]

but

\[ |I_1|\leqslant \int_0^1 x(1-x)^{n-1}\,dx=\frac{1}{n(n+1)} \]

(see Corollary 3);

\[ I_2(x)>\int_1^x(\xi-1)^n\,d\xi=\frac{(x-1)^{n+1}}{n+1}. \]

Thus, the condition is certainly impossible if

\[ (x-1)^{n+1}>\frac{1}{n}. \]

For all \(n=2,3,\ldots\) and \(x\geqslant 2\) we have \(s_0=n+1\).

Let \(s=n\). We have two conditions which, if we put \(R_{n,m}(\xi)=Q_{n-1}(\xi)\), may be written in the form

\[ \int_0^x Q_{n-1}(\xi)\,d\xi=0,\qquad \int_0^x \xi Q_{n-1}(\xi)\,d\xi=0. \]

We find conditions for the certain impossibility of the second, as the stronger one:

\[ \int_0^1 \xi\prod_1^{n-1}(\xi-\rho_i)\,d\xi+ \int_1^x \xi\prod_1^{n-1}(\xi-\rho_i)\,d\xi =I_1+I_2(x), \]

\[ |I_1|\leqslant \int_0^1 \xi(1-\xi)^{n-1}\,d\xi=\frac{1}{n(n+1)}, \]

\[ I_2(x)>\int_1^x \xi(\xi-1)^{n-1}\,d\xi =(x-1)^n\frac{nx+1}{n(n+1)}. \]

Thus, for violation of the condition it is sufficient that the inequality

\[ (x-1)^n>\frac{1}{nx+1}\qquad (1<x<2). \]

be satisfied.

Suppose

\[ x=2-\frac{p}{n}\quad (p<n\ \text{and const}). \]

We have

\[ \left(1-\frac{p}{n}\right)^n>\frac{1}{2n+1-p}; \]

then for \(p=1\) and \(n>2\) the inequality is satisfied and \(s_0\geqslant n\) for

\[ x\geqslant 2-\frac{1}{n}. \]

For \(p=2\) and \(n\geqslant 6\), also \(s_0\geqslant n\) for

\[ x\geqslant 2-\frac{2}{n}, \]

and so on.

In conclusion, we note that the integral functional in any interval \((2-\varepsilon,2)\), where \(\varepsilon>0\) is an arbitrarily small constant, is not stable for sufficiently large \(n\).

Leningrad Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich

Received
12 IX 1964

REFERENCES

1. E. V. Voronovskaya, The method of functionals and its applications. L., 1963.
2. V. A. Gusev, Izv. Akad. Nauk SSSR, Ser. Mat., 25, No. 3, 367 (1961).
3. E. V. Voronovskaya, M. Ya. Zinger, Dokl. Akad. Nauk SSSR, 143, No. 5 (1962).