UDC 517.512.6

MATHEMATICS

V. A. GUSEV

THE INTEGRAL FUNCTIONAL AND ITS NORM

(Presented by Academician S. N. Bernstein on 25 VIII 1965)

In the monograph by E. V. Voronovskaya \((^1)\), on p. 177, a number of still unsolved problems were formulated. Among them is the question of investigating the integral functional

\[ 1,\quad \frac{\xi}{2},\quad \frac{\xi^2}{3},\ldots,\frac{\xi^n}{n+1} \quad \text{for } -\infty<\xi<+\infty . \tag{1} \]

In the present paper a functional of a more general form is investigated:

\[ a_0,\ a_1\xi,\ a_2\xi^2,\ldots,\ a_n\xi^n \quad \text{for } -\infty<\xi<+\infty , \tag{2} \]

where \((a_i)_0^n\) is an absolutely monotone segment \(\bigl((^1),\ \text{p. }25\bigr)\). The functional (2), which we shall denote by \(F_\xi\), may also be called an integral functional, since the numbers \((a_i)_0^n\) are the moments of some nondecreasing function \(g(t)\) defined on the segment \([0,1]\). Indeed,

\[ F_\xi(x^i)=a_i\xi^i =\xi^i\int_0^1 t^i\,dg(t) =\int_0^\xi x^i\,dg\!\left(\frac{x}{\xi}\right) \quad (i=0,1,2,\ldots,n), \]

where \(\xi t=x\) has been put, and if

\[ P_n(x)=\sum_{i=1}^{n}p_i x^i \]

is an algebraic polynomial of degree not exceeding \(n\), then

\[ F_\xi[P_n(x)] = \int_0^\xi P_n(x)\,dg\!\left(\frac{x}{\xi}\right). \tag{3} \]

It is obvious that the functional (1) is a particular case of the functional (2), when \(g(t)\equiv t\).

Theorem 1. If the segment \((a_i)_0^n\) is amorphous \(\bigl((^1),\ \text{p. }27\bigr)\), then the integral functional (2) is an absolutely monotone segment for \(0\le \xi\le \xi_0\), where

\[ \xi_0= \begin{cases} 1/x_m, & \text{if } n=2m-1,\\ 1/y_m, & \text{if } n=2m. \end{cases} \]

Here \(x_m\) is the largest root of the orthogonal polynomial \(X_m(x)\) corresponding to the integral weight \(g(t)\), and \(y_m\) is the largest root of the orthogonal polynomial \(Y_m(x)\), corresponding to the integral weight

\[ \int_0^t \tau\,dg(\tau) \]

\[ \bigl((^2),\ \text{p. }39\bigr). \]

Proof. 1) Let \(n=2m-1\). We compute determinants of the form

\[ \Delta(\alpha_i\xi,\ \alpha_{2k+1}\xi^{2k+1}) = \left|\alpha_{i+j+1}\xi^{i+j+1}\right|_{i,j=0}^{k} \quad (k=0,1,\ldots,m-1), \]

\[ \Delta(\alpha_0-\alpha_1\xi,\ \alpha_{2k}\xi^{2k}-\alpha_{2k+1}\xi^{2k+1}) = \left|\alpha_{i+j}\xi^{i+j}-\alpha_{i+j+1}\xi^{i+j+1}\right|_{i,j=0}^{k} \]

\[ (k=0,1,\ldots,m-1). \]

Obviously,

\[ \Delta(\alpha_1 \xi,\alpha_{2k+1}\xi^{2k+1}) = \Delta(\alpha_1,\alpha_{2k+1})\xi^{(k+1)^2} \qquad (k=0,1,\ldots,m-1), \]

\[ \Delta(\alpha_0-\alpha_1\xi,\alpha_{2k}\xi^{2k}-\alpha_{2k+1}\xi^{2k+1}) = \xi^{(k+1)^2}X_{k+1}\!\left(\frac1\xi\right) \qquad (k=0,1,\ldots,m-1), \]

where

\[ X_{k+1}(x)=\left|\alpha_{i+j}x-\alpha_{i+j+1}\right|_{i,j=0}^{k} \qquad (k=0,1,2,\ldots) \]

is a system of orthogonal polynomials on the segment \([0,1]\), corresponding to the integral weight \(g(t)\) \(((^2), p. 40)\). Since \(\Delta(\alpha_1,\Delta_{2k+1})>0\) \((k=0,1,\ldots,m-1)\) \(((^1), p. 31)\), and the largest root \(x_m\) of the polynomial \(X_m(x)\) lies on the segment \([0,1]\) to the right of all roots of the polynomials \(X_{k+1}(x)\) \((k=0,1,\ldots,m-2)\) \(((^2), p. 58)\), it follows that for \(0<\xi<1/x_m\) all determinants \(\Delta(\alpha_1\xi,\alpha_{2k+1}\xi^{2k+1})>0\) and \(\Delta(\alpha_0-\alpha_1\xi,\alpha_{2k}\xi^{2k}-\alpha_{2k+1}\xi^{2k+1})>0\) \((k=0,1,\ldots,m-1)\), and the segment (2) is an absolutely monotone amorphous segment \(((^1), p. 31)\). For \(\xi=1/x_m=\xi_0\) only one determinant

\[ \Delta(\alpha_0-\alpha_1\xi_0,\alpha_{2m-2}\xi_0^{2m-2}-\alpha_{2m-1}\xi_0^{2m-1})=0, \]

and, consequently, the segment (2) for \(\xi=\xi_0\) is an absolutely monotone segment with a nodal structure \(((^1), pp. 22 and 31)\). For \(\xi=0\), the segment (2), obviously, is absolutely monotone with a nodal structure \(((^1), p. 28)\).

2) Let \(n=2m\). We compute determinants of the form

\[ \Delta(\alpha_0,\alpha_{2k}\xi^{2k}) = \left|\alpha_{i+j}\xi^{i+j}\right|_{i,j=0}^{k} \qquad (k=0,1,\ldots,m), \]

\[ \Delta(\alpha_1\xi-\alpha_2\xi^2,\alpha_{2k-1}\xi^{2k-1}-\alpha_{2k}\xi^{2k}) = \left|\alpha_{i+j-1}\xi^{i+j-1}-\alpha_{i+j}\xi^{i+j}\right|_{i,j=1}^{k} \]

\[ (k=1,2,\ldots,m). \]

Obviously,

\[ \Delta(\alpha_0,\alpha_{2k}\xi^{2k}) = \Delta(\alpha_0,\alpha_{2k})\xi^{k(k+1)} \qquad (k=0,1,\ldots,m), \]

\[ \Delta(\alpha_1\xi-\alpha_2\xi^2,\alpha_{2k-1}\xi^{2k-1}-\alpha_{2k}\xi^{2k}) = \xi^{k(k+1)}Y_k(1/\xi) \]

\[ (k=1,2,\ldots,m), \]

where

\[ Y_k(x)=\left|\alpha_{i+j-1}x-\alpha_{i+j}\right|_{i,j=1}^{k} \qquad (k=1,2,\ldots) \]

is a system of orthogonal polynomials on the segment \([0,1]\), corresponding to the integral weight

\[ \int \tau\,dg(\tau) \]

\(((^2), p. 40)\). The rest of the proof is carried out analogously to case 1).

Corollary 1. The extremal polynomial \(Q_n(x)\) \(((^1), p. 24)\) of the functional \(F_{\xi_0}\) has the form

\[ Q_n(x)= \begin{cases} \displaystyle 1+\frac{c\xi_0^{\,n+1}X_m^2(x/\xi_0)} {\Delta^2(\alpha_0,\alpha_{n-1})(x-1)}, & \text{if } n=2m-1,\\[1.2em] \displaystyle 1+\frac{c\xi_0^{\,n}xY_m^2(x/\xi_0)} {\Delta^2(\alpha_1,\alpha_{n-1})(x-1)}, & \text{if } n=2m, \end{cases} \]

where the multiplier \(c>0\) is chosen so that

\[ \max_{[0,1]} |Q_n(x)|=1. \]

1) Let \(n=2m-1\). Since

\[ \Delta(\alpha_0-\alpha_1\xi_0,\alpha_{2m-2}\xi_0^{2m-2}-\alpha_{2m-1}\xi_0^{2m-1})=0, \]

then, according to the corollary on p. 31 \((^1)\), the functional \(F_{\xi_0}\) has an extremal polynomial \(Q_n(x)\), whose resolvent

\[ R_m(x)=\prod_{i=1}^{m}(x-\sigma_i), \]

where

\[ 0<\sigma_1<\sigma_2<\cdots<\sigma_m=1, \]

satisfies the conditions

\[ F_{\xi_0}\!\left[x^kR_m(x)\right] = \int_0^{\xi_0} [[\unclear: integrand]](x)\,dg\!\left(\frac{x}{\xi_0}\right) = 0 \qquad (k=0,1,\ldots,m-1). \]

Consequently,
\[ R_m(x)=\frac{\xi_0^m}{\Delta(\alpha_0,\alpha_{n-1})}X_m\left(\frac{x}{\xi_0}\right) \]
\(((^2),\ p. 41)\). Next we construct the extremal polynomial \(Q_n(x)\) according to \((^1)\), p. 19, case III.

2) Let \(n=2m\). Since
\[ \Delta(\alpha_1\xi_0-\alpha_2\xi_0^2,\ \alpha_{2m-1}\xi_0^{2m-1}-\alpha_{2m}\xi_0^{2m})=0, \]
the functional \(F_{\xi_0}\) has an extremal polynomial \(Q_n(x)\), whose resolvent
\[ R_{m+1}(x)=\prod_{i=1}^{m+1}(x-\sigma_i), \]
where \(0=\sigma_1<\sigma_2<\cdots<\sigma_{m+1}=1\), satisfies the conditions
\[ F_{\xi_0}[x^kR_{m+1}(x)]=\int_0^{\xi_0}x^kR_{m+1}(x)\,dg\left(\frac{x}{\xi_0}\right)=0 \quad (k=0,1,\ldots,m-1). \]
Consequently,
\[ R_{m+1}(x)=\frac{\xi_0^m x}{\Delta(\alpha_1,\alpha_{n-1})}Y_m\left(\frac{x}{\xi_0}\right), \]
and the extremal polynomial \(Q_n(x)\) is constructed according to \((^1)\), p. 19, case IV.

Corollary 2. If \(\max_{[0,1]}|P_n(x)|=1\), then for \(0\le \xi\le \xi_0\) the inequality
\[ \left|\int_0^\xi P_n(x)\,dg\left(\frac{x}{\xi}\right)\right|\le \alpha_0 \tag{4} \]
holds.

Indeed, for \(0\le \xi\le \xi_0\) the norm \(N(\xi)\) of segment (2) is \(\alpha_0\) (\((^1)\), p. 25), and therefore
\[ |F_\xi(P_n(x))|\le N(\xi)=\alpha_0, \]
which coincides with inequality (4) by virtue of (3).

Theorem 2. If the segment \((a_i)_0^n\) is amorphous, then the functional \(F_\xi\) for \(\xi>\xi_0\) has the greatest possible number \(s\) of nodes with positive loads, namely: \(s=m\), if \(n=2m-1\), and \(s=m+1\), if \(n=2m\).

This theorem is a special case of Theorem 3 of E. V. Voronovskaya \((^3)\).

Corollary 1. The extremal polynomial \(Q_n(x,\xi)\) of the functional \(F_\xi(\xi>\xi_0)\) has the form
\[ Q_n(x,\xi)= \begin{cases} 1-q_n(\xi)(1-x)\displaystyle\prod_{i=1}^{m-1}[x-\sigma_i(\xi)]^2, & \text{if } n=2m-1,\\[1.2em] 1-q_n(\xi)x(1-x)\displaystyle\prod_{i=2}^{m}[x-\sigma_i(\xi)]^2, & \text{if } n=2m, \end{cases} \]
where the leading coefficient \(q_n(\xi)\) is determined by the conditions
\[ \max_{[0,1]}|Q_n(x,\xi)|=1 \]
and
\[ Q_n(x_0,\xi)=-1 \]
for some \(x_0\in[0,1]\), while \(\sigma_1(\xi),\sigma_2(\xi),\ldots,\sigma_m(\xi)\) are the nodes with positive loads of the polynomial \(Q_n(x,\xi)\).

Corollary 2. Segment (2), for \(\xi>\xi_0\), is of class II \((^1)\), p. 42, and, consequently, has a unique extremal polynomial. According to the theorem on continuous deformation \((^1)\), p. 70, the coefficients and nodes of the extremal polynomial \(Q_n(x,\xi)\) are continuous functions of the parameter \(\xi\).

Corollary 3. All roots of the derivative \(\dfrac{\partial}{\partial x}Q_n(x,\xi)\) are real, distinct, and lie inside the interval \((0,1)\), i.e. the polynomial \(Q_n(x,\xi)\) has an internal deformation \((^1)\), p. 120).

Remark. The norm \(N(\xi)\) of the integral functional \(F_\xi\) satisfies the inequality
\[ N(\xi)\le \sum_{k=0}^{n} \left| \frac{1}{R'_{n+1}(\tau_k)} \int_0^\xi \frac{R_{n+1}(x)}{x-\tau_k}\, dg\left(\frac{x}{\xi}\right) \right|, \]

where \(R_{n+1}(x)=\prod_{i=0}^{n}(x-\tau_i)\) is the resolvent of the Chebyshev polynomial
\(T_n(x)=\cos n\arccos(2x-1)\), and \((\tau_i)_0^n\) are its nodes. This remark is a special case of Theorem 13 (\({}^1\), p. 39), when \(\mu_k=\alpha_k \xi^k\) \((k=0,1,\ldots,n)\) and \(c_i=\tau_{i-1}\) \((i=1,2,\ldots,n+1)\). From this remark there follows the existence of a number \(\xi_1\ge \xi_0\) such that for all \(\xi\ge \xi_1\), \(Q_n(x,\xi)=T_n(x)\). The number \(\xi_1\) may be defined as the least positive root of the equation

\[ \int_0^\xi T_n(x)\,dg\!\left(\frac{x}{\xi}\right) = \sum_{k=0}^{n} \left| \frac{1}{R'_{n+1}(\tau_k)} \int_0^\xi \frac{R_{n+1}(x)}{x-\tau_k}\, dg\!\left(\frac{x}{\xi}\right) \right|. \]

We also note that \(Q_n(x,\xi)=(-1)^n T_n(x)\) for \(\xi<0\), since the segment (2) in this case is alternating.

Theorem 3. The norm \(N(\xi)\) of the segment

\[ \xi,\ \frac{\xi^2}{2},\ldots,\frac{\xi^{n+1}}{n+1} \tag{5} \]

for \(\xi_0\le \xi\le \xi_1\) is a function convex downward.

Proof.

\[ N(\xi+\Delta\xi)-2N(\xi)+N(\xi-\Delta\xi)= \]

\[ = \int_0^{\xi+\Delta\xi} Q_n(x,\xi+\Delta\xi)\,dx - 2\int_0^\xi Q_n(x,\xi)\,dx + \int_0^{\xi-\Delta\xi} Q_n(x,\xi-\Delta\xi)\,dx > \]

\[ > \int_0^{\xi+\Delta\xi} Q_n(x,\xi)\,dx - 2\int_0^\xi Q_n(x,\xi)\,dx + \int_0^{\xi-\Delta\xi} Q_n(x,\xi)\,dx = \]

\[ = \int_\xi^{\xi+\Delta\xi} Q_n(x,\xi)\,dx - \int_{\xi-\Delta\xi}^{\xi} Q_n(x,\xi)\,dx = \bigl[ Q_n(\xi+\theta_1\Delta\xi,\xi) - Q_n(\xi-\theta_2\Delta\xi,\xi) \bigr]\Delta\xi>0. \]

\[ (0<\theta_1<1,\quad 0<\theta_2<1), \]

since, according to Corollary 3 of Theorem 2, the polynomial \(Q_n(x,\xi)\) for \(x\ge 1\) is an increasing function of \(x\). Thus, the inequality
\[ N(\xi+\Delta\xi)-2N(\xi)+N(\xi-\Delta\xi)>0 \]
has been proved, which is sufficient for downward convexity of the norm \(N(\xi)\).

Corollary. The norm \(N(\xi)\) of the segment (5), for \(\xi_0<\xi<\xi_1\), can be estimated by means of a linear function, namely

\[ N(\xi)<\xi_0+\frac{\xi-\xi_0}{\xi_1-\xi_0} \left\{ \int_0^{\xi_1} T_n(x)\,dx-\xi_0 \right\}. \]

In conclusion, we give estimates of integrals of algebraic polynomials \(P_n(x)\) satisfying the condition \(\max_{[0,1]}|P_n(x)|=1\):

\[ \left|\int_0^\xi P_n(x)\,dx\right| \le \begin{cases} \left|\displaystyle\int_0^\xi T_n(x)\,dx\right|, & \text{for } \xi<0,\\[1.2em] \xi, & \text{for } 0\le \xi\le \xi_0,\\[1.2em] \xi_0+\dfrac{\xi-\xi_0}{\xi_1-\xi_0} \left\{\displaystyle\int_0^{\xi_1}T_n(x)\,dx-\xi_0\right\}, & \text{for } \xi_0<\xi<\xi_1,\\[1.2em] \displaystyle\int_0^\xi T_n(x)\,dx, & \text{for } \xi\ge \xi_1. \end{cases} \]

Leningrad Institute
of Aviation Instrument Engineering

Received
25 VIII 1965

References

\({}^1\) E. V. Voronovskaya, The Method of Functionals and Its Applications, 1963.
\({}^2\) G. Szegő, Orthogonal Polynomials, 1962.
\({}^3\) E. V. Voronovskaya, DAN, 166, No. 6 (1966).