MATHEMATICS

Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashian

ON THE COMPLETION AND CLOSURE OF AN INCOMPLETE SYSTEM OF FUNCTIONS ({e^{-\mu_k x}x^{s_k-1}})

1°. Let ({\mu_k}), (\operatorname{Re}\mu_k>0) ((k=1,2,\ldots)), be an arbitrary sequence of complex numbers, among which numbers may occur (but not necessarily consecutively) with finite or even infinite multiplicity. Denoting by (s_k\ge 1) the multiplicity of the number (\mu_k) among the system ({\mu_1,\ldots,\mu_k}), we associate with the sequence of numbers ({\mu_k}) ((k=1,2,\ldots)) the sequence of functions ({e^{-\mu_k x}x^{s_k-1}}) ((k=1,2,\ldots)) from (L_2(0,+\infty)).

In the space of functions (L_2(0,+\infty)) the well-known approximation theorem of Müntz can be formulated as follows*:

For completeness of the system of functions

[
{e^{-\mu_k x}x^{s_k-1}}\qquad (k=1,2,\ldots)
\tag{1}
]

in (L_2(0,\infty)) it is necessary and sufficient that the condition

[
\sum_{k=1}^{\infty}\frac{\operatorname{Re}\mu_k}{1+|\mu_k|^2}=+\infty
\tag{2}
]

be satisfied.

Let us agree that ({\mu_k}\subset S), according as the series (2) diverges or converges. We shall denote further by (M_2^{(n)}{\mu_k}) the linear span of the system of functions ({e^{-\mu_k x}x^{s_k-1}}) ((k=1,2,\ldots,n\le \infty)) in the metric of (L_2(0,\infty)). By Müntz’s theorem, (M_2^{(\infty)}{\mu_k}\equiv L_2(0,+\infty)) only if ({\mu_k}\subset R). In this connection, apparently long ago the problem was posed of a complete characterization of the whole class (M_2^{(\infty)}{\mu_k}) (or of the class (C^{(\infty)}{\mu_k}) when taking the closure of the same system in the metric of uniform approximation) in the case when ({\mu_k}\subset S).

We note that some necessary, but by no means sufficient, conditions which the functions of these classes must satisfy, in the particular assumptions (\operatorname{Im}\mu_k=0;\ s_k=1) ((k=1,2,\ldots)) on the sequence ({\mu_k}\subset S), were established by Laurent Schwartz ((^6)) and independently by A. F. Leont'ev ((^7)).

In the present article we give formulations of several new results on the completion and biorthogonalization of the incomplete system (1) and on the complete characterization of the class (M_2^{(\infty)}{\mu_k}) for ({\mu_k}\subset S). In obtaining these results we rely essentially on the concept of a unitary pair of operators and on its analytic characteristic, given in the note ((^8)).

2°. Putting, for (n=1,2,\ldots),

[
B_n(z)=\prod_{k=1}^{n}\frac{z+\mu_k}{z-\overline{\mu}_k}\,\varepsilon_k,\qquad
\varepsilon_k=\frac{|(1-\mu_k)(1+\mu_k)|}{(1-\mu_k)(1+\mu_k)}
]

((k=1,2,\ldots)), in the case ({\mu_k}\subset S), we shall also include here the Blaschke products

[
B_\infty(z)\equiv B(z)\equiv \prod_{k=1}^{\infty}\frac{z+\mu_k}{z-\overline{\mu}_k}\,\varepsilon_k,
]

which converge everywhere except at the points of the imaginary axis.

* In the case (s_k=1) ((k=1,2,\ldots)) see ((^1,^3)), and also the book ((^3)). In the case (s_k\ge 1), but with (\operatorname{Im}\mu_k=0), see ((^4,^5)). The proof of the theorem in the general formulation given here is in fact contained in the moment method of proof used in the book ((^3)), under the condition (s_k=1) ((k=1,2,\ldots)), which is completely superfluous.

(\operatorname{Re} z=0) and the points of the sequence ({\bar u_k}). Observing further that for any measurable function (\varphi(x)), bounded on the whole axis ((-\infty,+\infty)), for each fixed value (\xi\in(-\infty,+\infty)) there exists the mean limit on (-\infty<x<+\infty)

[
L(\xi;\varphi(x))\equiv {1\over 2\pi}\,\text{l.i.m.}{\sigma\to\infty}
\int\,d\tau}^{\sigma}\varphi(\tau)\,{e^{-i\xi\tau}-1\over -i\tau}\,e^{ix\tau
\in L_2(-\infty,+\infty),
\tag{3}
]

we introduce for consideration the functions (K_n(\xi,x)\equiv L(\xi; B_n(ix))), (K_n^(\xi,x)\equiv L(\xi;\overline{B_n(ix)})), and, under the condition ({\mu_k}\subset S), also the functions (K_\infty(\xi,x)\equiv K(\xi,x)\equiv L(\xi,B(ix))), (K_n^(\xi,x)\equiv K^*(\xi,x)\equiv L(\xi,\overline{B(ix)})), where by (B(ix)) is meant the boundary value of the Blaschke function on the imaginary axis, which exists and is of modulus one almost everywhere for all (-\infty<x<+\infty).

Denote by ({\gamma_k(x)}) ((k=1,2,\ldots)) the orthogonalization of the sequence of functions ({e^{-\mu_k x}x^{k-1}}) ((k=1,2,\ldots)) on the half-axis ((0,+\infty)). By a direct calculation one easily verifies the validity of the integral formula

[
{1\over \sqrt{2\pi}}\int_{-\infty}^{+\infty}
\sqrt{{\operatorname{Re}\mu_k\,B_{k-1}(i\tau)\over \pi}}\,
{i\tau+\mu_k}\,e^{ix\tau}\,d\tau
=
\begin{cases}
\gamma_k(x), & 0<x<\infty,\
0, & -\infty<x<0,
\end{cases}
\tag{4}
]

(where one must put (B_0(i\tau)\equiv 1)), obtained in another way by G. V. Badalyan ((^5)) for the case when (\operatorname{Im}\mu_k=0). Finally, putting

[
R_n(\xi,x)\equiv
\sum_{k=1}^{n}\left(\int_0^\xi \gamma_k(t)\,dt\right)\gamma_k(x)
\quad (n=1,2,\ldots),
]

we note that for each fixed value (\xi\in(0,+\infty)) on the half-axis (0<x<\infty) there exists the mean limit

[
R_\infty(\xi,x)\equiv R(\xi,x)=\text{l.i.m.}_{n\to\infty} R_n(\xi,x).
\tag{5}
]

The following important auxiliary theorem is valid.

Theorem 1. For any (1\le n<\infty), and also for (n=\infty), if ({\mu_k}\subset S), the following equations hold among the functions (K_n(\xi,x)), (K_n^(\xi,x)), (R_n(\xi,x)):*

[
\left.
\begin{aligned}
\text{a)}\quad&
\int_0^\infty \overline{K_n(\xi,x)}\,K_n(\eta,x)\,dx
+
\int_0^\infty \overline{R_n(\xi,x)}\,R_n(\eta,x)\,dx
\[2mm]
\text{b)}\quad&
\int_0^\infty \overline{K_n^(\xi,x)}\,K_n^(\eta,x)\,dx
\end{aligned}
\right}
= \min(\xi,\eta);
]

[
\text{c)}\quad
\int_0^\eta K_n(\xi,x)\,dx
=
\int_0^\xi \overline{K_n^*(\eta,x)}\,dx;
]

[
\text{d)}\quad
\int_0^\eta R_n(\xi,x)\,dx
=
\int_0^\xi \overline{R_n(\eta,x)}\,dx,
]

where (\xi>0,\ \eta>0) are arbitrary.

Hence, using Theorem 1 of our preceding note ((^8)), we may assert that the quadruple of functions (K_n(\xi,x)), (K_n^(\xi,x)), (R_n(\xi,x)\equiv R_n^(\xi,x)) generates a certain unitary pair of operators ({U_1,U_2}), acting in the space (L_2(0,+\infty)), in accordance with the following main theorem:

Theorem 2. Let (f(x)\in L_2(0,+\infty)) be arbitrary; then for any (1\le n<\infty), and also for (n=\infty), if ({\mu_k}\subset S), the representation holds

[
\int_0^\xi f(x)\,dx
=
\sum_{k=1}^{n} c_k(f)\int_0^\xi \gamma_k(x)\,dx
+
\int_0^\infty \overline{K_n(\xi,x)}\,g_n(x)\,dx,
\qquad
\xi\in(0,+\infty),
\tag{6}
]

where (g_n(x)\in L_2(0,+\infty)),

[
\int_0^\xi g_n(x)\,dx=\int_0^\infty \overline{K_n^*(\xi,x)}\,f(x)\,dx,\quad \xi\in(0,\infty)
]

and

[
c_k(f)=\int_0^\infty f(x)\overline{\gamma_k(x)}\,dx\quad (k=1,2,\ldots);
]

moreover always

[
\int_0^\infty |f(x)|^2\,dx\equiv |f|^2=\sum_{k=1}^n |c_k(f)|^2+|g_n|^2.
\tag{7}
]

Thus, the representation (6) gives an effective completion of each finite system
({e^{-\mu_k x}x^{s_k-1}}) ((k=1,2,\ldots,n)), as well as of the whole system (1) in the case when ({\mu_k}\subset S), i.e. when this system is incomplete.

(3^\circ). Let us now note that, in general, under the condition ({\mu_k}\subset S), Theorem 2 implies:

Corollary. The class (M_2^{(n)}{\mu_k}) ((1\le n\le\infty)) coincides with the set of functions in (L_2(0,+\infty)) satisfying the additional condition

[
\int_0^\infty \overline{K_n^*(\xi,x)}\,f(x)\,dx=0,\quad \xi\in(0,+\infty).
\tag{8}
]

To put condition (8) into a more transparent and equivalent form, we introduce the definition: a function (F(w)\in H_2^{(+)}) (respectively (F(w)\in H_2^{(-)})) if it is holomorphic in the half-plane (\operatorname{Re}w>0) ((\operatorname{Re}w<0)) and

[
\int_{-\infty}^{+\infty} |F(u+iv)|^2\,dv\le M_0<+\infty
]

for all (0<u<\infty) ((-\infty<u<0)), where (M_0) does not depend on (u). From Theorem 2 it follows:

Theorem 3. If ({\mu_k}\subset S), then the class (M_2^{(n)}{\mu_k}) ((1\le n\le\infty)) coincides with the subset of those (f(x)\in L_2(0,+\infty)) for which the expression

[
F(iv)=\operatorname{l.i.m.}_{\sigma\to\infty}\int_0^\sigma e^{-ivx}f(x)\,dx\in L_2(-\infty,+\infty)
\tag{9}
]

(being, as is known, the boundary function for some (F(w)\in H_2^{(+)})) is such that the product (F(iv)B_n(iv)) is the boundary function for a certain function (F_0(w)\in H_2^{(-)}).

From this it follows immediately:

Corollary*. If ({\mu_k}\subset S), and moreover (\lim |\mu_k|=\infty), then the class (M_2^{(n)}{\mu_k}) ((1\le n\le\infty)) coincides with the set of functions (f(x)) represented in the form

[
f(x)=\operatorname{l.i.m.}{\sigma\to\infty}\frac{1}{2\pi}\int\,dv,\quad 0<x<\infty,}^{\sigma} F(iv)e^{ivx
\tag{10}
]

where (F(w)) is an arbitrary meromorphic function in the entire (w)-plane satisfying the conditions
(F(w)\in H_2^{(+)}), (F(w)B_n(w)\in H_2^{(-)}).

When additional restrictions are imposed on the sequence ({\mu_k}), one can indicate necessary conditions for a function to belong to the class (M_2^{(\infty)}{\mu_k}).

We shall say that ({\mu_k}\subset S(\alpha)) ((0\le\alpha<1)), if ({\mu_k}\subset S), and, in addition,
(|\arg\mu_k|\le \frac12\pi\alpha) ((0\le\alpha<1)), (k=1,2,\ldots). Then the following holds:

* In the statement of Theorem 3 and its corollary we have included also the cases when (1\le n<+\infty), in order to emphasize that, in general, for all (1\le n\le+\infty) the result has a unified formulation, although for finite (n) it is established very simply.

Theorem 4. a) If ({\mu_k}\subset S(\alpha)) ((0\le \alpha<1)), then every function (f(x)\in M_2^{(\infty)}{\mu_k}) is holomorphic in the angle (\Delta_\alpha: |\arg z|<\frac12\pi(1-\alpha)) and, moreover,

[
\int_0^\infty |f(re^{i\psi})|^2\,dr<+\infty,\qquad |\psi|<\frac12\pi(1-\alpha).
\tag{11}
]

b) If ({\mu_k}\subset S(\alpha)) ((0\le \alpha<1)), and (\lim |\mu_k|=\infty), then for every function (f(x)\in M_2^\infty{\mu_k}) the expansion

[
f(z)=\sum_{k=1}^{\infty} c_k(f)\gamma_k(z),\qquad z\in \Delta_\alpha,
\tag{12}
]

holds, converging uniformly and absolutely in every subdomain of (\Delta_\alpha).

(4^\circ). Let us now note that under the condition ({\mu_k}\subset S), for every (n\ge 1) the multiplicity (p_n) of the number (\mu_n) in the entire sequence ({\mu_k}) is always finite, and it is obvious that (1\le s_n\le p_n) ((n=1,2,\ldots)). Finally denoting

[
\omega_n(x)=\frac{1}{2\pi}\operatorname*{l.i.m.}{\sigma\to\infty}
\int}^{\sigma
\left{
\frac{p!}{(s_n-1)!\,B^{(p_n)}(-\mu_n)}
\frac{B(it)}{(it+\mu_n)^{p_n-s_n+1}}
\right}e^{itx}\,dt,
\tag{13}
]

we have:

Theorem 5. a) If ({\mu_k}\subset S), then the systems of functions ({e^{-\mu_k x}x^{s_k-1}}_1^\infty), ({\omega_k(x)}_1^\infty) are biorthogonal on the half-axis ((0,+\infty)). Moreover, if (f(x)\in M_2^{(\infty)}{\mu_k}) and

[
S_n(x,f)=\sum_{k=1}^{n} c_k(f)\gamma_k(x)
=\sum_{k=1}^{n} a_k^{(n)}(f)e^{-\mu_k x}x^{s_k-1},
\tag{14}
]

then all the limits exist

[
a_k(f)=\lim_{n\to\infty} a_k^{(n)}(f)
=\int_0^\infty f(x)\overline{\omega_k(x)}\,dx
\qquad (k=1,2,\ldots).
\tag{15}
]

b) If ({\mu_k}\subset S(\alpha)) ((0\le \alpha<1)), (\lim|\mu_k|=\infty), then for every function (f(x)\in M_2^{(\infty)}{\mu_k}) the formal expansion

[
f(z)\sim \sum_{k=1}^{\infty} a_k(f)e^{-\mu_k z}z^{s_k-1},
\qquad
a_k(f)=\int_0^\infty f(x)\overline{\omega_k(x)}\,dx,
\tag{16}
]

at least after a suitable rearrangement of its terms, will converge uniformly to (f(z)) inside the domain (\Delta_\alpha).

In conclusion, we note that in the works of L. Schwartz and A. F. Leont’ev({}^{6,7}), by entirely different methods, results were established which in essence differ little from assertion a) of Theorem 4 and from Theorem 5 (without the fact of the existence of a biorthogonal system), but only for the case when ({\mu_k}\subset S(0)) (i.e. when (\operatorname{Im}\mu_k=0), (\sum |\mu_k|^{-1}<+\infty)) and (s_k=1) ((k=1,2,\ldots)).

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
21 VIII 1961

REFERENCES

({}^{1}) S. N. Müntz, Schwarz’s Festschrift, Berlin, 1914, S. 303—312.
({}^{2}) O. Szasz, Math. Ann., 77, 482 (1916).
({}^{3}) R. Paley, N. Wiener, Fourier Transforms, N. Y., 1934, p. 26—36.
({}^{4}) A. O. Gel’fond, Izv. AN SSSR, ser. matem., 14, 413 (1950).
({}^{5}) G. V. Badalyan, Izv. AN ArmSSR, ser. fiz.-matem. nauk, 8, No. 5 (1955); 9, No. 1 (1956).
({}^{6}) L. Schwartz, Ann. Fac. Sci. Univ. Toulouse, Sci. Math. et Sci. Phys., (1943).
({}^{7}) A. F. Leont’ev, DAN, 72, No. 4 (1950).
({}^{8}) M. M. Dzhrbashyan, DAN, 141, No. 2 (1961).