Mathematics

V. KABAILA

ON SOME INTERPOLATION PROBLEMS IN THE CLASS \(H_p\) FOR \(p<1\)

(Presented by Academician V. I. Smirnov on 18 II 1960)

1. In papers \((^{1-5})\), among other questions, the following problem is considered: given \(n\) points \(\lambda_1,\lambda_2,\ldots,\lambda_n\) \((|\lambda_k|<1)\) and \(n\) numbers \(c_1,c_2,\ldots,c_n\), it is required to find a function of the class \(H_p\), interpolating at the points \(\lambda_k\) the values \(c_k\), with the least norm* in comparison with other functions of the same class interpolating the same values. The function sought in the problem will henceforth be called minimal. In the cited papers the existence, general form, and uniqueness of the minimal function are proved in the case \(p\geq 1\). In particular, in \((^4)\) it is proved that, in order that the function \(f(z)\) be minimal in the class \(H_p\), \(p\geq 1\), with respect to the given \(\lambda_k\) and \(c_k\), it is necessary and sufficient that \(f(z)\) have the form

\[ f(z)=A\prod' \frac{z-\alpha_k}{1-\overline{\alpha}_k z} \prod_{k=1}^{n-1}(1-\overline{\alpha}_k z)^{2/p} \prod_{k=1}^{n}(1-\overline{\lambda}_k z)^{-2/p}, \tag{1} \]

where \(A,\alpha_1,\alpha_2,\ldots,\alpha_{n-1}\) \((|\alpha_k|\leq 1)\) are certain constants, and the product \(\prod'\) extends over some or all \(\alpha_k\) for which \(|\alpha_k|<1\). The existence of a minimal function in the class \(H_p\), \(p>0\), follows trivially from the compactness principle.

The author of the present article considers the form of the minimal function and the question of its uniqueness in the case \(p<1\).

2. Theorem 1. If the function \(f(z)\) is minimal in the class \(H_p\), \(p>0\), with respect to the given \(\lambda_k\) and \(c_k\) \((k=1,2,\ldots,n)\); \(b(z)\) is the Blaschke product formed for the zeros of \(f(z)\), then the function

\[ g(z)=\left[\frac{f(z)}{b(z)}\right]^{p'/p} b(z),\qquad \text{where } p'>p, \tag{2} \]

is minimal and, moreover, the unique minimal function in the class \(H_{p'}\) with respect to \(\lambda_k\) and to its values (for multiple \(\lambda_k\), with respect to the values of the corresponding derivatives) at these points.

* The norm of a function \(f(z)\in H_p\), \(p>0\), is defined by the equality

\[ \|f\|_{H_p}= \left\{ \frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^p\,d\theta \right\}^{1/p}, \qquad \text{where } f(e^{i\theta})=\lim_{r\to 1} f(re^{i\theta}). \]

Obviously, for \(p<1\) the name “norm” can be retained only conditionally, since in this case the triangle inequality is not satisfied in general.

The scheme of the proof is as follows. Suppose that there exists a minimal function \(g^*(z)\in H_{p'}\), \(g^*(z)\ne g(z)\), interpolating the same values as \(g(z)\) at the points \(\lambda_k\) (i.e. \(\|g^*\|_{H_{p'}}\leqslant \|g\|_{H_{p'}}\)); then we would obtain:

1) if \(\|g^*\|_{H_{p'}}<\|g\|_{H_{p'}}\), then it follows that \(\|f^*\|_{H_p}<\|f\|_{H_p}\), where

\[ f^*(z)=\left[\frac{g(z)}{b(z)}\right]^{p'/p-1} g^*(z) \]

interpolates the same values as \(f(z)\) at the points \(\lambda_k\), which is impossible, since \(f(z)\) is minimal;

2) if \(\|g^*\|_{H_{p'}}=\|g\|_{H_{p'}}\), then it follows that almost everywhere \(\left|g^*(e^{i\theta})\right|=\left|g(e^{i\theta})\right|\); hence it follows that \(g^*(z)\equiv g(z)\), which is impossible by assumption.

Theorem 2. A minimal function in the class \(H_p\), \(0<p<1\), has the form (1).

For the proof it is enough to apply Theorem 1 for \(p'\geqslant 1\). Then the minimal function (2), as is known, must have the form (1) (where \(p=p'\)), and therefore

\[ f(z)=\left[\frac{g(z)}{b(z)}\right]^{p'/p} b(z) \tag{3} \]

will have the form (1), since the transformation (3) changes only \(p\) in the expression (1).

Definition. We shall call the index of minimality \((p_0)\) the exact lower bound of those \(p\) for which the expression (1) (with fixed \(\alpha_k,\lambda_k\) and zeros) is a minimal function with respect to the points \(\lambda_k\) and its values at these points, and the index of uniqueness \((p_0^*)\) the exact lower bound of those \(p\) for which the indicated expression is the unique minimal function.

Obviously, by definition \(p_0\leqslant p_0^*\).

From Theorem 1 and from the investigation of the interpolation problem in the case \(\lambda_1=\lambda_2=0;\ c_1=1,\ c_2>0\), it follows:

Theorem 3. The index of minimality \(p_0\) is equal to the index of uniqueness and, with a suitable choice of \(\lambda_k\) and \(c_k\) \((k=1,2,\ldots,n)\), can assume any values from the interval \([0,1]\); for \(p=p_0\) the function (1) is minimal, but may be a nonunique minimal function.

Example. The function \(f_1(z)=(1+z)^2\) is minimal in the class \(H_1\) in the case \(\lambda_1=\lambda_2=0\) with respect to the values \(c_1=1,\ c_2=2\) (i.e. \(f_1(0)=1,\ f_1'(0)=2\)), but the function \(f_p(z)=(1+z)^{2/p}\) will not be minimal in any class \(H_p\) for \(p<1\).

Remark. For \(\lambda_1=\lambda_2=0\) and any \(p<1\), there exists a unique value of the ratio \(|c_2/c_1|\) for which in the class \(H_p\) there exist two distinct minimal functions with respect to \(\lambda_1,\lambda_2;\ c_1,c_2\) (i.e. interpolating the same values).

1. From the investigation of the interpolation problem in the case \(\lambda_1=\lambda_2=0\), one obtains an exact estimate for \(|f'(z)|\). In particular, for \(f(z)\in H_p,\ p<1\), the estimate

\[ |f'(z)|\leqslant \frac{h(z)\|f\|_{H_p}}{(1-|z|^2)^{1/p+1}}, \tag{4} \]

holds, where \(h(z)\) is a bounded real function defined by the equality

\[ h(z)=\max\left\{ 2\left(1-\frac{p}{2}\right)^{1/p} \left(1+\frac{|z|}{\sqrt{p(2-p)}}\right), \right. \]

\[ \left. \frac{2}{p}\, \frac{ \rho(z)-\dfrac{|z|}{2-p} +\sqrt{\left(\dfrac{|z|}{2-p}\right)^2+\dfrac{p}{2-p}} }{ \left[ 1+2\left(\dfrac{|z|}{2-p}\right)^2+\dfrac{p}{2-p} -\dfrac{2|z|}{2-p}\sqrt{\left(\dfrac{|z|}{2-p}\right)^2+\dfrac{p}{2-p}} \right]^{1/p} } \right\}. \]

Vilnius State University
named after V. Kapsukas

Received
16 II 1960

REFERENCES

¹ S. Ya. Khavinson, Uch. zap. MGU, mathematics, 6, 148, 133 (1955).
² S. Ya. Khavinson, Matem. sborn., 36 (78), 3, 445 (1955).
³ S. Kakeya, Trans. Am. Math. Soc., 22, 489 (1921).
⁴ A. J. Macintyre, W. W. Rogosinski, Acta Math., 82, 275 (1950).
⁵ J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, N. Y., 1935.