V. S. VIDENSKII

ON THE ZEROS OF ORTHOGONAL POLYNOMIALS

(Presented by Academician S. N. Bernstein, 4 V 1963)

1. Let \(\alpha(x)\) be a given nondecreasing function with an infinite number of points of increase on the interval \([a,b]\), and let the algebraic polynomials \(\{p_n(x)\}\) form an orthogonal system on \([a,b]\) corresponding to the distribution \(d\alpha(x)\). It is known that all zeros of the polynomial \(p_n(x)\) are real and simple and lie inside the interval \((a,b)\); moreover, the zeros of the polynomial \(p_n(x)\) interlace with the zeros of the polynomial \(p_{n-1}(x)\). We shall generalize this theorem to the case where a sequence of continuously differentiable orthogonal functions \(\{\varphi_j(x)\}_{j=0}^{\infty}\) is such that \(\{\varphi_j(x)\}_{j=0}^{n}\) forms a Chebyshev system for every natural \(n\). Recall that, by definition \((^1)\), a sequence of functions \(\{\psi_j(x)\}_{j=0}^{n}\) continuous on \([a,b]\) forms a Chebyshev system of order \(n\) (a \(T_n\)-system) on \([a,b]\) if every nonidentically zero polynomial

\[ P_n(x)=\sum_{j=0}^{n} a_j\psi_j(x) \tag{1} \]

can have on \([a,b]\) no more than \(n\) zeros.

We shall adopt the following convention for counting the multiplicity of zeros of functions (cf. \((^1)\), p. 8). If a continuously differentiable function \(f(x)\) is such that it has on the interval \([a,b]\) under consideration a finite number of zeros, then a point \(x_0\) is regarded as a simple zero of the function \(f(x)\) if \(f(x_0)=0\), \(f'(x_0)\ne 0\); but if \(f(x_0)=f'(x_0)=0\), then the point \(x_0\) is regarded as a double zero if, in passing through \(x_0\), the function \(f(x)\) does not change sign, and if the function \(f(x)\) changes sign in passing through \(x_0\), then the point \(x_0\) is regarded as a triple zero. We note, however, that all the arguments and conclusions given below remain valid also in the case where the functions considered are differentiable a sufficient number of times and the multiplicity of zeros is counted in the usual way.

Theorem 1. Let a sequence of functions \(\{\varphi_j(x)\}_{j=0}^{\infty}\), continuously differentiable on the interval \([a,b]\), form an orthogonal system corresponding to the given distribution \(d\alpha(x)\), and let, moreover, \(\{\varphi_j(x)\}_{j=0}^{n}\) form a \(T_n\)-system for every natural \(n\). Then, whatever real number \(\lambda\) may be, the function

\[ \Phi(x;\lambda)=\varphi_n(x)+\lambda\varphi_{n-1}(x) \tag{2} \]

has on \([a,b]\) either \(n\) or \(n-1\) zeros, and all of them are simple. The function \(\varphi_n(x)\) has inside the interval \((a,b)\) \(n\) simple zeros, which interlace with the zeros of \(\varphi_{n-1}(x)\).

An analogous theorem on the interlacing of the zeros of two consecutive polynomials of least deviation from zero in the metric of the space \(C\) was proved in my note \((^2)\). In the case of algebraic polynomials, the proof of Theorem 1 relies essentially on the possibility of factoring these polynomials into factors (see \((^3,^4)\)). In the general case, the following theorem will serve as our tool.

Theorem 2. Let \(U(x)\) and \(V(x)\) be continuously differentiable functions on the interval \([a,b]\) such that the function \(U(x)\) has \(n\) simple zeros on \([a,b]\), while the function \(V(x)\) has either \(n\) or \(n-1\) simple zeros on \([a,b]\), and, for every real \(\lambda\), the function

\[ F(x;\lambda)=U(x)+\lambda V(x) \]

has \(\leqslant n\) zeros on \([a,b]\). In order that the zeros of the functions \(U(x)\) and \(V(x)\) alternate, it is necessary and sufficient that, for every real \(\lambda\), the function \(F(x;\lambda)\) have only simple zeros.

Denote by \(x_1<x_2<\cdots<x_n\) the zeros of \(U(x)\), and by \(\xi_1<\xi_2<\cdots<\xi_n\) the zeros of \(V(x)\). Suppose that the zeros of \(U(x)\) and \(V(x)\) alternate, i.e.

\[ a\leqslant x_1<\xi_1<x_2<\xi_2<\cdots<x_{n-1}<\xi_{n-1}<x_n<\xi_n\leqslant b. \tag{3} \]

By virtue of the inequalities (3), the function \(V(x)\) changes sign in the interval \((x_k,x_{k+1})\), so that for every \(\lambda\ne0\) we have

\[ F(x_k;\lambda)F(x_{k+1};\lambda)=\lambda^2 V(x_k)V(x_{k+1})<0. \]

We see that the function \(F(x;\lambda)\) changes sign in \((x_k,x_{k+1})\); consequently, the function \(F(x;\lambda)\) has an odd number of zeros in \((x_k,x_{k+1})\). Since the number of such intervals is \(n-1\), and \(F(x;\lambda)\) can have \(\leqslant n\) zeros on \([a,b]\), in each interval \((x_k,x_{k+1})\) there lies one and only one simple zero of the function \(F(x;\lambda)\). The function \(F(x;\lambda)\) can have one more simple zero in \([a,x_1)\) or in \((x_n,b]\).

Suppose now that, for every \(\lambda\), \(F(x;\lambda)\) has on \([a,b]\) only simple zeros. This means that for no value of \(x\), \(a\leqslant x\leqslant b\), is the system of homogeneous equations

\[ F(x;\lambda)=U(x)+\lambda V(x)=0, \]
\[ F'(x;\lambda)=U'(x)+\lambda V'(x)=0, \]

consistent; consequently, the determinant of this system does not vanish,

\[ \Delta(x)=U(x)V'(x)-U'(x)V(x),\qquad a\leqslant x\leqslant b. \]

For definiteness, suppose that \(\Delta(x)<0\). We have \(U(x_k)=U(x_{k+1})=0\), and, since these zeros are simple, it follows that

\[ U'(x_k)U'(x_{k+1})<0. \tag{4} \]

On the other hand, since \(\Delta(x)>0\), we have

\[ \Delta(x_k)=-U'(x_k)V(x_k)>0,\qquad \Delta(x_{k+1})=-U'(x_{k+1})V(x)_{k+1}>0. \tag{5} \]

From (4) and (5) it follows that

\[ V(x_k)V(x_{k+1})<0. \tag{6} \]

Consequently, the function \(V(x)\) has an odd number of zeros in \((x_k,x_{k+1})\), and since the number of these intervals is \(n-1\), each of them contains exactly one zero of the function \(V(x)\), which means that the zeros of the functions \(U(x)\) and \(V(x)\) alternate.

To prove Theorem 1, let us first establish that the function \(\varphi_n(x)\) \((n=1,2,\ldots)\) has \(n\) simple zeros on \([a,b]\). It is clear that the function \(\varphi_n(x)\) must change sign at least once inside \((a,b)\), since

\[ \int_a^b \varphi_n(x)\varphi_0(x)\,d\alpha(x)=0, \]

and the function \(\varphi_0(x)\) does not vanish on \([a,b]\), since it constitutes a \(T_0\)-system. Suppose that the function \(\varphi_n(x)\) changes sign on \([a,b]\) \(k\) times, \(k<n\). Denote the corresponding zeros of the function \(\varphi_n(x)\) by \(x_1,x_2,\ldots,x_k\). Construct the polynomial

\[ P_k(x)= \begin{vmatrix} \varphi_0(x_1) & \varphi_1(x_1) & \ldots & \varphi_k(x_1)\\ \varphi_0(x_2) & \varphi_1(x_2) & \ldots & \varphi_k(x_2)\\ \ldots & \ldots & \ldots & \ldots\\ \varphi_0(x_k) & \varphi_1(x_k) & \ldots & \varphi_k(x_k)\\ \varphi_0(x) & \varphi_1(x) & \ldots & \varphi_k(x) \end{vmatrix}, \tag{7} \]

which has \(k\) simple zeros at the points \(x_1,x_2,\ldots,x_k\). From the orthogonality conditions it follows that

\[ \int_a^b \varphi_n(x)P_k(x)\,d\alpha(x)=0, \]

which is impossible, since the product \(\varphi_n(x)P_k(x)\) does not change sign on \([a,b]\). Thus, the function \(\varphi_n(x)\) has \(n\) simple zeros on \([a,b]\).

It is proved similarly that for any \(\lambda\) the function \(\Phi(x;\lambda)\) has only simple zeros. Indeed, if \(\Phi(x;\lambda)\) had at least one double zero, then the number of zeros of the function \(\Phi(x;\lambda)\) at which it changes sign would be \(k\le n-2\), since \(\Phi(x;\lambda)\) is a polynomial in the \(T_n\)-system and, consequently, the total number of its zeros is \(\le n\). On the other hand, the function \(\Phi(x;\lambda)\) is orthogonal to any polynomial of order \(k\le n-2\), so that, in particular,

\[ \int_a^b \Phi(x;\lambda)P_k(x)\,d\alpha(x)=0, \tag{8} \]

where \(P_k(x)\) is the polynomial defined by (7), and \(x_1,x_2,\ldots,x_k\) are those zeros of \(\Phi(x;\lambda)\) at which this function changes sign. But for the polynomial \(P_k(x)\) equality (8) is impossible, since the product \(\Phi(x;\lambda)P_k(x)\) does not change sign on \([a,b]\). Consequently, the function \(\Phi(x;\lambda)\) has \(\ge n-1\) simple zeros on \([a,b]\).

On the basis of Theorem 2 we conclude from this that the zeros of the functions \(\varphi_n(x)\) and \(\varphi_{n-1}(x)\) mutually interlace.

1. We indicate another application of Theorem 2, namely we establish the following theorem.

Theorem 3. Let

\[ P(x)=\prod_{\nu=1}^{m}(x-\alpha_\nu)=\sum_{k=0}^{m}a_kx^k,\qquad Q(x)=\prod_{\nu=1}^{m}(x-\beta_\nu)=\sum_{k=0}^{m}b_kx^k \tag{9} \]

be algebraic polynomials all of whose zeros are real and mutually interlace; let \(u(x)\) and \(v(x)\) be continuously differentiable functions on the interval \([a,b]\), having \(n\) simple zeros on \([a,b]\) which mutually interlace, and, moreover, let the function \(f(x;\lambda)=u(x)+\lambda v(x)\) have \(\le n\) zeros on \([a,b]\) for every real \(\lambda\). Then all zeros of the functions

\[ p(x)=\sum_{k=0}^{m}a_ku^k(x)v^{m-k}(x),\qquad q(x)=\sum_{k=0}^{m}b_ku^k(x)v^{m-k}(x), \tag{10} \]

lying on \([a,b]\), are simple and mutually interlace.

This result generalizes a theorem of P. Montel \((^5)\), in which algebraic polynomials of degree \(n\) are considered instead of the functions \(u(x)\) and \(v(x)\).

For the proof, consider the algebraic polynomial \(R(x;\lambda)=P(x)+\lambda Q(x)\). By Theorem 2 it has either \(m\) or \(m-1\) simple real ...

real zeros, so that

\[ R(x;\lambda)=\prod_{\nu=1}^{t}(x-\gamma_\nu), \tag{11} \]

where all \(\gamma_\nu\) are real and distinct, and \(t=m\) or \(m-1\). Taking into account (9), (10), and (11), we can write

\[ p(x)=\prod_{\nu=1}^{m}[u(x)-\alpha_\nu v(x)],\qquad q(x)=\prod_{\nu=1}^{m}[u(x)-\beta_\nu v(x)]; \]

\[ r(x;\lambda)=p(x)+\lambda q(x)=\prod_{\nu=1}^{t}[u(x)-\gamma_\nu v(x)]. \tag{12} \]

Since each of the factors occurring on the right-hand sides of (12) is a function of the form \(f(x;\mu)\), it has only simple zeros on \([a,b]\), their number being either \(n\) or \(n-1\). Moreover, no two factors have common zeros. Consequently, each of the functions \(p(x)\), \(q(x)\), \(r(x;\lambda)\) has \(\le mn\) zeros on \([a,b]\), all of them simple. By Theorem 2 it follows from this that the zeros of \(p(x)\) and \(q(x)\) interlace on \([a,b]\).

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

Received
17 IV 1963

CITED LITERATURE

\(^{1}\) S. N. Bernstein, Extremal Properties of Polynomials and the Best Approximation of Functions of One Real Variable, Moscow–Leningrad, 1937.
\(^{2}\) V. S. Videnskii, DAN, 116, No. 5, 723 (1957).
\(^{3}\) G. Szegő, Orthogonal Polynomials, Moscow, 1962.
\(^{4}\) N. I. Akhiezer, The Classical Moment Problem, Moscow, 1961.
\(^{5}\) P. Montel, Mathematica (Cluj), 5, 110 (1931).