UDC 512.52

MATHEMATICS

Yu. A. KAZMIN

ON A CLASS OF INTERPOLATION PROBLEMS

(Presented by Academician A. N. Kolmogorov on 8 IV 1966)

In what follows we shall use the following notation. The space of functions analytic in the disk \(|z|<R\), \(0<R<\infty\), will be denoted by the symbol \(A(|z|<R)\), and the space of functions analytic for \(|z|\ge R\) and tending to zero at infinity will be denoted by \(A(|z|\ge R)\). It is known that the spaces \(A(|z|<R)\) and \(A(|z|\ge R)\) are mutually conjugate. By the sign \([\rho,\sigma]\) below we denote everywhere the class of entire functions of growth not exceeding order \(\rho\) and type \(\sigma\).

Let us first consider one interpolation problem of a particular nature, whose method of solution makes it possible to give an exhaustive answer for a certain class of interpolation problems.

Thus, let the following be given: 1) an integer \(p\), \(p\ge 1\); 2) \(p\) points \(\eta\delta^k\), \(\eta\ne 0\), \(\delta=e^{2\pi i/p}\), \(k=0,1,\ldots,p-1\); 3) a point \(\omega=e^{i\alpha}\) lying on the unit circle \(|z|=1\), where, for definiteness, \(\alpha\in[0;2\pi]\); and, finally, 4) \(p\) sequences of complex numbers

\[ \{a_{ks}\},\quad s=0,1,2,\ldots;\quad k=0,1,\ldots,p-1. \]

The following questions are posed.

I. Does there exist an entire function \(F(z)\in[1;\sigma]\) for which the relations

\[ F^{(ps)}(\eta\delta^k\omega^s)=a_{ks},\quad s=0,1,2,\ldots;\quad k=0,1,\ldots,p-1. \tag{1} \]

hold?

II. If there exists a function \(F(z)\in[1;\sigma]\) satisfying conditions (1), then how can \(F(z)\) be reconstructed from the given values of its derivatives (1)?

III. What is the set of all functions of the space \(A(|z|<R)\), \(R>|\eta|>0\), for which the equalities

\[ F^{(ps)}(\eta\delta^k\omega^s)=0,\quad s=0,1,2,\ldots;\quad k=0,1,\ldots,p-1. \tag{2} \]

hold?

Questions of an analogous type are studied for the space \(A(|z|\ge R)\).

IV. What is the set of all functions \(F(z)\in A(|z|\ge R)\), \(0<R<|\eta|\), satisfying conditions (2)?

V. What properties must the numbers \(\{a_{ks}\}\) have in order that there exist a function \(F(z)\in A(|z|\ge R)\) corresponding to relations (1)?

VI. Suppose there exists a function \(F(z)\in A(|z|\ge R)\) for which equalities (1) hold. It is asked how to reconstruct it from the known values of the derivatives (1).

Lemma 1. I. If, for the function

\[ F(z)=\sum_{k=0}^{\infty}\frac{x_k}{k!}z^k\in A(|z|<R),\quad R>|\eta|>0, \]

relations (1) are satisfied, then each of the auxiliary functions generated by it,

\[ F_j(z)=\sum_{s=1}^{\infty}\frac{x_{ps-j}}{(ps-j)!}\omega^{(ps-j)(ps-j-1)/2p}z^{ps-j}\in A(|z|<R),\quad j=1,2,\ldots,p, \]

in some neighborhood of the origin (in any case, for \(|z| < R-|\eta|\)) satisfies the corresponding linear differential equation of infinite order with constant coefficients

\[ L_j[F_j]\equiv \sum_{s=1}^{\infty} \frac{\eta^{ps-j}}{(ps-j)!}\, \omega^{-(ps-j)(ps-j-1)/2p} F_j^{(ps-j)}(z)=\Phi_j(z),\qquad j=1,2,\ldots,p, \tag{3} \]

where in (3) the right-hand side, for a given \(j\), is the function

\[ \Phi_j(z)=\sum_{s=0}^{\infty}\frac{B_{js}}{(ps)!}\, \omega^{s(ps-1)/2}z^{ps}\in A(|z|<r),\qquad r=R-|\eta|>0, \tag{4} \]

whose coefficients \(B_{js}\) are defined by the equalities

\[ B_{js}=\frac{1}{p}\sum_{q=0}^{p-1}\delta^{qj}a_{qs},\qquad s=0,1,2,\ldots \tag{5} \]

II. If, for each \(j,\ j=1,2,\ldots,p\), there exists a solution \(F_j(z)\in A(|z|<R)\), \(R>|\eta|>0\), of the \(j\)-th equation (3), whose right-hand side is determined by the series (4), with coefficients \(B_{js}\) computed by formulas (5), and the series (4) converges uniformly for \(|z|<R-|\eta|\), while the \(j\)-th function \(F_j(z)\), \(j=1,2,\ldots,p\), has a lacunary Taylor series in powers of \(z\), containing only the powers \(z^{ps-j}\), then for the function

\[ F(z)=\Omega(z)*\left[\sum_{j=1}^{p}F_j(z)\right]\in A(|z|<R), \]

where

\[ \Omega(z)=\sum_{s=0}^{\infty}\omega^{-s(s-1)/2p}z^s\in A(|z|<1), \]

and the asterisk denotes the Hadamard product of the functions \(\Omega\) and \(\sum_{j=1}^{p}F_j\), the relations (1) are satisfied.

Recall that the Hadamard product of the functions \(a(z)=\sum_{k=0}^{\infty}a_kz^k\in A(|z|<R_1)\) and \(b(z)=\sum_{k=0}^{\infty}b_kz^k\in A(|z|<R_2)\) is the function

\[ c(z)=a*b=\sum_{k=0}^{\infty}a_kb_kz^k\in A(|z|<R_1R_2). \]

Renumber all zeros \(\beta_{nj}\) of the characteristic function

\[ \varphi_j(\eta t)=\sum_{s=1}^{\infty} \frac{\eta^{ps-j}}{(ps-j)!}\, \omega^{-(ps-j)(ps-j-1)/2p}t^{ps-j}\in[1;\eta] \]

of the \(j\)-th equation (3) \((j=1,2,\ldots,p)\) in the order of nonincreasing moduli:

\[ 0=|\beta_{0j}|<|\beta_{1j}|\le\cdots\le|\beta_{nj}|\le|\beta_{n+1,j}|\le\cdots; \]

moreover, for the function \(\varphi_p(\eta t)\), which does not vanish at the origin, the list of roots in the displayed string begins with \(|\beta_{1p}|>0\). From the lacunary character of the expansion of the function \(\varphi_j(\eta t)\) in powers of \(t\) it follows that if \(\beta_{nj}\), \(n=1,2,\ldots\), is its zero of multiplicity \(m_{nj}\), then each of the \(p\) points

\[ \{\beta_{nj}\delta^l\},\qquad l=0,1,\ldots,p-1\quad(\delta=e^{2\pi i/p}) \tag{6} \]

is also a zero of the function \(\varphi_j(\eta t)\) of the same multiplicity \(m_{nj}\). For each group of roots of the form (6) we choose arbitrarily one representative \(\hat\beta_{nj}\) and fix it.

Theorem A. I. In order that there exist an entire function \(F(z)\in[1;\sigma]\) satisfying conditions (1), it is necessary and sufficient that the inequalities

\[ \varlimsup_{s\to\infty}\sqrt[ps]{|a_{ks}|}\le \sigma,\qquad k=0,1,\ldots,p-1. \]

II. The set of all entire functions of class \([1;\sigma]\) (\(\sigma\) fixed) for which the equalities (1) hold is the family of the form

\[ F(z)=\sum_{j=1}^{p}\left[\frac{1}{2\pi i}\int_{|t|=\sigma+\varepsilon} \frac{\gamma_j(t)}{\varphi_j(\eta t)}\varphi_j(zt)\,dt+ \sum_{0<|\widetilde{\beta}_{nj}|\leq\sigma}\sum_{l=0}^{m_{nj}-1} C_{njl}\frac{\partial^l}{\partial\widetilde{\beta}_{nj}^{\,l}} \varphi_j(\widetilde{\beta}_{nj}z)\right], \tag{7} \]

where the functions

\[ \gamma_j(t)=\sum_{s=0}^{\infty}\frac{B_{js}\omega^{s(ps-1)/2}}{t^{ps+1}}, \quad j=1,2,\ldots,p, \]

are regular for \(|t|>\sigma\)*; the functions \(\varphi_j(\eta t)\) and the numbers \(\widetilde{\beta}_{nj}, m_{nj}\) were defined above; the contour of integration \(|t|=\sigma+\varepsilon\), \(\varepsilon>0\), is chosen so that in the annulus \(\sigma<|t|\leq\sigma+\varepsilon\) the functions \(\varphi_j(\eta t)\) have no zeros; the sum in square brackets in representation (7) is taken over the zeros \(\widetilde{\beta}_{nj}\) of all groups of the form (6) chosen by us that fall in the circle \(|t|\leq\sigma\), and \(C_{njl}\) are arbitrary constants.

If, however, in the circle \(|t|\leq\sigma\) there are no zeros of the functions \(\varphi_j(\eta t)\), \(j=1,2,\ldots,p\), then the second term on the right-hand side of equality (7) is identically equal to zero, and interpolation problem II has in the class \([1;\sigma]\) a unique solution.

III. Every function \(F(z)\in A(|z|<R)\), \(0<|\eta|<R\), satisfying (2), is uniquely representable in the form of the sum of \(p\) series

\[ \sum_{j=1}^{p}\sum_{n=1}^{\infty} \left(\sum_{l=0}^{m_{nj}-1} C_{njl}\frac{\partial^l}{\partial\widetilde{\beta}_{nj}^{\,l}} \varphi_j(\widetilde{\beta}_{nj}z)\right). \tag{8} \]

Moreover, a certain subsequence of partial sums of the sum of the series (8) converges uniformly in the circle \(|z|<R-|\eta|\) to the function \(F(z)\) \((1^{-4})\).

In what follows, for brevity, we shall say that the circle of questions I–III constitutes interpolation problem A, and questions IV–VI constitute problem B. Theorem A gives a complete solution of problem A. Note that for \(p=2\) and \(\omega=1\), item II of Theorem A solves a problem to which the well-known Lidstone problem on the reconstruction of an entire function \(F(z)\in[1;\sigma]\) from the prescribed values of its even-order derivatives at two distinct points is reduced by a simple substitution:
\(F^{(2n)}(\alpha)=a_n,\ F^{(2n)}(\beta)=b_n,\ \alpha\ne\beta,\ n=0,1,2,\ldots\) (see, for example, \((5^{-8})\)).

By means of Lemma 1, problem A is reduced to a whole class of interpolation problems defined by prescribing relations of the form

\[ F^{(ps-j)}(0)=a_s^{(j)},\quad j=j_1,j_2,\ldots,j_n;\quad j_m\ne j_l,\ m\ne l;\quad 0\leq j_m\leq p-1; \]

\[ F^{ps}(\eta\delta_k\omega^s)=a_{ks},\quad k=k_1,k_2,\ldots,k_{p-n};\quad k_m\ne k_l,\ m\ne l;\quad 0\leq k_m\leq p-1 \]

\[ (s=0,1,2,\ldots). \]

Problem A is solved in an analogous way when \(|\omega|<1\) and \(|\omega|>1\).

Theorem B. I. Whatever may be: 1) an integer \(p,\ p\geq1\); 2) a complex \(\eta,\ 0<R<|\eta|<+\infty\); 3) a complex \(\omega,\ |\omega|=1\), there does not exist any function \(F(z)\in A(|z|\geq R)\), different from the identically zero function, satisfying conditions (2).

II. In order that there exist a function \(F(z)\in A(|z|\geq R)\) satisfying equalities (1), it is necessary and sufficient that the aggregate of complex numbers \(\{a_{ks}\}\) generate functions

\[ f_j^+(z)=\sum_{s=0}^{\infty}(-1)^{ps} \frac{\eta^{ps}\omega^{s(ps+j+2)}}{(ps)!} \left[\frac{1}{p}\sum_{k=0}^{p-1}a_{ks}\delta^{-k(j-1)}\right]z^{ps} \in A\left(|z|<1-\frac{R}{|\eta|}+\varepsilon\right), \]

\[ \varepsilon>0,\quad j=1,2,\ldots,p, \tag{9} \]

represented respectively by the series

\[ f_j^+(z)=\sum_{s=1}^{\infty}C_{ps-j} \left[\frac{d^{ps-j}}{dt^{ps-j}} \left(\frac{t^{p-j}}{1-t^p}\right)\right]_{t=z/\omega^s}, \tag{10} \]

\[ \text{* The function } \gamma_j(t) \text{ is usually called the Borel-associated function of } \Phi_j(t)\in[1;\sigma]\ (4). \]

\[ \varlimsup_{s\to\infty}\sqrt[ps-j]{(ps-j)!\,|C_{ps-j}|}<\frac{R}{|\eta|},\qquad j=1,2,\ldots,p, \tag{10} \]

uniformly convergent for \(|z|<1-R/|\eta|+\varepsilon\) \((\varepsilon>0)\).

The representation (10), if it exists, is unique.

III. The Laurent coefficients \(x_n\) of the function
\[ F(z)=\sum_{n=0}^{\infty}\frac{x_n}{z^{n+1}}\in A(|z|\ge R), \]
satisfying the relations (1), are found from (10) by the formulas
\[ x_{ps-j}=(ps-j)!\,\eta^{ps-j+1}C_{ps-j},\qquad s=1,2,\ldots;\quad j=1,2,\ldots,p. \]

For each of the systems of functions
\[ \left\{ \left. \frac{d^{ps-j}}{dt^{ps-j}}\left(\frac{t^{p-j}}{1-t^p}\right) \right|_{t=z/\omega^s} \right\}_{s=1}^{\infty}, \]
considered in the space \(A(|z|\ge 1+R/|\eta|)\), a system of polynomials \(\{p_{ps-j}(z)\}_{s=1}^{\infty}\) is constructed, containing only powers \(z^{ps-1}\) and forming, with the functions
\[ \left. \frac{d^{ps-j}}{dt^{ps-j}}\left(\frac{t^{p-j}}{1-t^p}\right) \right|_{t=z/\omega^s}, \]
a biorthogonal sequence. Therefore, if the functions \(f_j^{-}(z)\), \(j=1,2,\ldots,p\), are known, to which, for \(|z|>1+R/|\eta|-\varepsilon\), the series (10) respectively converge uniformly, then the coefficients \(C_{ps-j}\) (and then also \(x_{ps-j}\)) can be computed by the formulas
\[ C_{ps-j}=\frac{1}{2\pi i}\int_{|t|=r>1+R/|\eta|} f_j^{-}(t)\,p_{ps-j}(t)\,dt,\qquad s=1,2,\ldots;\quad j=1,2,\ldots,p. \]

In connection with this, the following is of interest.

Theorem C. Let there exist a function
\[ F(z)=\sum_{n=0}^{\infty}\frac{x_n}{z^{n+1}}\in A(|z|\ge R), \]
satisfying the equalities (1), and let its elements \(\{a_{ks}\}\) generate the functions (9)
\[ f_j^{+}(z)\in A(|z|<1-R/|\eta|-\varepsilon),\qquad j=1,2,\ldots,p, \]
analytically continuable from the disk \(|z|\le 1-R/|\eta|\), respectively, along continuous curves \(\Gamma_j\), \(j=1,2,\ldots,p\), without self-intersections, each of which connects the circles \(|z|=1-R/|\eta|\) and \(|z|=1+R/|\eta|\) and is located in the angle \(\alpha_j\) (each in its own) with vertex at the origin and aperture not greater than \(2(\pi-\arcsin R/|\eta|)\).

Then each of the functions (9)
\[ f_j^{+}(z)\in A(|z|<1-R/|\eta|+\varepsilon),\qquad j=1,\ldots,p, \]
can be analytically continued to the whole domain \(|z|\ge 1+R/|\eta|\), and the result of continuing the function \(f_j^{+}(z)\) along \(\Gamma_j\) into the domain \(|z|\ge 1+R/|\eta|\) coincides respectively with
\[ f_j^{-}(z)\in A(|z|\ge 1+R/|\eta|),\qquad j=1,2,\ldots,p. \]

To the interpolation problem B considered in the space \(A(|z|\ge R)\) there reduces any of the interpolation problems of the following type: construct a function
\[ F(z)=\sum_{n=0}^{\infty}\frac{x_n}{z^{n+1}}\in A(|z|\ge R), \]
if the following are known: 1) its Laurent coefficients \(x_{ps-j}\), \(s=1,2,\ldots;\ j=j_1,j_2,\ldots,j_m,\ j_n\ne j_l,\ n\ne l;\ 1\le j_n\le p,\ 1\le n\le p\); 2) the values of the derivatives
\[ F^{(ps)}(\eta\delta^k\omega^s)=a_{ks},\qquad s=0,1,\ldots;\quad k=k_1,k_2,\ldots,k_{p-m},\ k_n\ne k_l,\ n\ne l,\ 0\le k_n\le p-1 \]
\((|\eta|>R)\). With the corresponding modifications, the method proposed by us makes it possible to solve problems B of the indicated type also in the case when \(|\omega|>1\).

Moscow State University
named after M. V. Lomonosov

Received
8 IV 1966

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