MATHEMATICS

M. L. RASULOV

ON ONE FORMULA FOR THE EXPANSION OF AN ARBITRARY FUNCTION

(Presented by Academician S. L. Sobolev, 20 XI 1957)

In the present note a new formula is given for the expansion of an arbitrary vector-function (see formula (8)), connected with a boundary-value problem with a complex parameter for a system of linear differential equations with piecewise-smooth coefficients. To such a problem there leads the more general problem of investigating the formula for the expansion of an arbitrary vector-function in a series in fundamental* functions of a boundary-value problem with a parameter for a system of linear differential equations of higher order. In addition, the result of this note (see formula (8)) is characterized by the fact that it pertains to a boundary-value problem with a parameter for a system of equations whose coefficients also contain negative powers of (\lambda). As far as we know, such a case has not yet been considered by anyone.

We consider the system of differential equations

[
\frac{d y_k^{(i)}}{dx}
-
\sum_{j=1}^{n} a_{kj}^{(i)}(x,\lambda)y_j^{(i)}
=
f_k^{(i)}(x)
\quad \text{for } x\in(a_i,b_i)
\tag{1}
]

[
(i=1,\ldots,m;\ k=1,\ldots,n)
]

with piecewise-smooth coefficients (a_{kj}^{(i)}(x,\lambda)) under the boundary conditions

[
\sum_{i=1}^{m}\sum_{j=1}^{n}
\left{
\alpha_{kj}^{(i)}(\lambda)y_j^{(i)}(a_i)
+
\beta_{kj}^{(i)}(\lambda)y_j^{(i)}(b_i)
\right}
=0,
\tag{2}
]

where

[
a_{kj}^{(i)}(x,\lambda)
=
\lambda a_{kj}^{(i)}(x)
+
\sum_{\nu=0}^{N}\lambda^{-\nu}a_{kj\nu}^{(i)}(x);
]

(\alpha_{kj}^{(i)}(\lambda)), (\beta_{kj}^{(i)}(\lambda)) are polynomials in (\lambda); ((a_i,b_i)) are mutually non-overlapping intervals having common ends.

Let the following conditions be satisfied:

(1^\circ). On the interval ([a_i,b_i]) the functions (a_{kj}^{(i)}(x)) are twice continuously differentiable, (a_{kj0}^{(i)}(x)) are continuously differentiable once, and the remaining (a_{kj\nu}^{(i)}(x)) are merely continuous.

(2^\circ). For (x\in[a_i,b_i]) the roots (\varphi_1^{(i)}(x),\ldots,\varphi_n^{(i)}(x)) of the characteristic equations**

[
\Phi^{(i)}(\theta)=\det\left(A^{(i)}(x)-\theta E\right)=0
\quad (i=1,\ldots,m)
]

* Speaking of fundamental functions, we have in mind the eigenfunctions and associated functions.

** (A^{(i)}(x)) is the matrix of the functions (a_{kj}^{(i)}(x)); (E) is the identity matrix.

are distinct and different from zero; their arguments and the arguments of their differences do not depend on (x). For large (\lambda) the sign of the differences (\operatorname{Re}\varphi_R^{(i)}(x)-\operatorname{Re}\varphi_j^{(i)}(x)) does not depend on (x).

(3^\circ). For (|\lambda|>R) ((R) a sufficiently large number) the rank of the matrix of the coefficients (\alpha_{kj}^{(i)}(\lambda), \beta_{kj}^{(i)}(\lambda)) of the boundary conditions (2) is equal to (mn).

Let (y_{k1}^{(i)}(x,\lambda),\ldots,y_{kn}^{(i)}(x,\lambda)) be a fundamental system of particular solutions of the homogeneous system corresponding to system (1). For all (\lambda), with the exception of a countable set of values, the solution (y_k^{(i)}(x,\lambda)) of problem (1)—(2) exists and is found by the usual method in the form

[
y_k^{(i)}(x,\lambda)
=
\sum_{p=1}^{n}\sum_{q=1}^{m}
\int_{a_q}^{b_q}
G_{kp}^{(i,q)}(x,\xi,\lambda) f_p^{(q)}(\xi)\,d\xi,
]

where (G_{kp}^{(i,q)}(x,\xi,\lambda)) is the Green’s function of problem (1)—(2), which has the form
[
G_{kp}^{(i,q)}(x,\xi,\lambda)=
\Delta_{kp}^{(i,q)}(x,\xi,\lambda)/\Delta(\lambda),
]
(\Delta_{kp}^{(i,q)}(x,\xi,\lambda)), (\Delta(\lambda)) are analytic functions of (\lambda) for (\lambda\ne 0) (the point at infinity may also be a singular point);

[
\Delta(\lambda)=
\left|
\begin{array}{cccccc}
u_{1,1}^{(1)} & \cdots & u_{1n}^{(1)} & \cdots & u_{11}^{(m)} & \cdots \ u_{1n}^{(m)}\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot\
u_{mn,1}^{(1)} & \cdots & u_{mn,n}^{(1)} & \cdots & u_{mn,1}^{(m)} & \cdots \ u_{mn,n}^{(m)}
\end{array}
\right|,
]

[
u_{kj}^{(i)}
=
\sum_{p=1}^{n}
\left{
\alpha_{kp}^{(i)}(\lambda)y_{pj}^{(i)}(a_i)
+
\beta_{kp}^{(i)}(\lambda)y_{pj}^{(i)}(b_i,\lambda)
\right}.
]

According to conditions (1^\circ)—(3^\circ), the equations

[
\operatorname{Re}\lambda\varphi_p^{(i)}(x)
=
\operatorname{Re}\lambda\varphi_q^{(i)}(x)
\qquad
(p,q=1,\ldots,m)
]

for (p\ne q) determine straight lines passing through the origin of the (\lambda)-plane. By these straight lines the (\lambda)-plane is divided into sectors ((\Sigma_s)), in each of which, under a certain numbering of the roots of the characteristic equations, the inequalities

[
\operatorname{Re}\lambda\varphi_1^{(i)}(x)
\le
\operatorname{Re}\lambda\varphi_2^{(i)}(x)
\le
\cdots
\le
\operatorname{Re}\lambda\varphi_n^{(i)}(x).
\tag{3}
]

hold. Consequently, according to the theorem of J. D. Tamarkin ((^1)), under conditions (1^\circ)—(3^\circ) the homogeneous system corresponding to system (1) has a fundamental system of particular solutions (y_{pq}^{(i)}(x,\lambda)), admitting in the sectors ((\Sigma_s)) the asymptotic representation

[
y_{pq}^{(i)}(x,\lambda)
=
\exp\left{
\lambda\int_{a_i}^{x}\varphi_q^{(i)}(\xi)\,d\xi
\right}
[\eta_{pq}^{(i)}(x)],
\tag{4}
]

where ([\eta_{pq}^{(i)}(x)]) is G. D. Birkhoff’s notation ((^2)): the sum
[
\eta_{pq}^{(i)}(x)+E_{pq}^{(i)}(x,\lambda)/\lambda;
]
(\eta_{pq}^{(i)}(x)), (E_{pq}^{(i)}(x,\lambda)) are continuous in (x) on ([a_i,b_i]); (E_{pq}^{(i)}(x,\lambda)) is bounded for large (|\lambda|).

Substituting (4) into the expression (u_{kj}^{(i)}), we find

[
u_{kj}^{(i)}(\lambda)
=
\lambda^{l_k}
\left{
[A_{kj}^{(i)}]+[B_{kj}^{(i)}]e^{\lambda w_j^{(i)}}
\right},
\tag{5}
]

where (A_{kj}^{(i)}), (B_{kj}^{(i)}) are constants;
[
w_j^{(i)}=\int_{a_i}^{b_i}\varphi_j^{(i)}(x)\,dx;
]
(l_k) is the greatest exponent-

of the powers (\lambda) occurring in the (k)-th row of the matrix of boundary conditions (2) with a nonzero coefficient.

Through the origin and the points (w_1^{(i)}, \ldots, w_n^{(i)}) draw straight lines. Arrange the azimuths (\alpha_j) of these lines in increasing order. Next define the straight lines (d_j) by the equations

[
\text{azimuth } d_j =
\begin{cases}
\dfrac{\pi}{2}-\alpha_j, & \text{if } 0 \leqslant \alpha_j \leqslant \dfrac{\pi}{2};\[6pt]
\dfrac{3\pi}{2}-\alpha_j, & \text{if } \dfrac{\pi}{2}<\alpha_j<\pi .
\end{cases}
]

After this we divide the (\lambda)-plane into sectors ((T_j)), not overlapping and having common boundaries (not coinciding with the lines (d_j)), containing these lines (d_j) (each such sector contains a half of only one line (d_j)). Denote by (w_{k_j}^{(i)}) ((k=1,\ldots,\gamma_j)) those of the numbers (w_1^{(i)}, \ldots, w_n^{(i)}) which lie on the line with azimuth (\alpha_j). For these numbers, on one half of the line (d_j) contained in ((T_j)), (\operatorname{Re}\lambda w_{k_j}^{(i)}) vanishes.

Let (w_{k_j}^{(i)}=\mu_{k_j}^{(i)}\exp{\alpha_j\sqrt{-1}}), where (\mu_{k_j}^{(i)}) are real numbers arranged in increasing order. After excluding the numbers (w_{k_j}^{(i)}) from the set ({w_1^{(i)}, \ldots, w_n^{(i)}}), the remaining numbers can be divided into two categories, for which in the sector ((T_j)) we have, respectively: (\operatorname{Re}\lambda w_k^{(i,1)}\to-\infty), (\operatorname{Re}\lambda w_k^{(i,2)}\to+\infty) as (\lambda\to\infty).

Consequently, using the asymptotic formulas (5), we find

[
\Delta(\lambda)=\lambda^l \exp\left{\lambda\sum w_k^{(i,2)}\right} H_j(z),
\tag{6}
]

where (l=l_1+\cdots+l_{mn}), the sum extends over all numbers (w) of the second category for all (i=1,\ldots,m); (z=\exp{\alpha_j\sqrt{-1}}\lambda),

[
H_j(z)=[M_{1j}]\exp{m_{1j}z}+\cdots+[M_{\sigma_j j}]\exp{m_{\sigma_j j}z};
]

(M_{kj}, m_{kj}) are constants; (m_{1j}<m_{2j}<\cdots<m_{\sigma_j j}).

With the aid of the asymptotic representation (6), using the method of J. D. Wilder—J. D. Tamarkin (\left({}^{3,4}\right)), it is easy to show that (\Delta(\lambda)) has a countable set of zeros, not accumulating at infinity, contained in a finite number of strips of bounded width parallel to the lines (d_j) and containing them. Further, if these zeros and the point (\lambda=0) are isolated by disks of radius (\delta) with centers at the isolated points, then in the remaining part of the (\lambda)-plane the inequality holds

[
|H_j(z)|>N_\delta>0,
\tag{7}
]

where (N_\delta) is a constant depending only on (\delta).

Using the estimate (7), by estimating (\Delta_{pq}^{(i,q)}(x,\xi,\lambda)) in the sectors ((T_j)) (with the aid of the asymptotic formulas (4)), by the method of J. D. Tamarkin (\left({}^{4}\right)) it is not difficult to prove the following theorem.

Theorem. Under conditions (1^\circ)—(3^\circ), if all the numbers (M_{1j}, M_{\sigma_j j}) are different from zero, then there exists a sequence of closed expanding contours (\Gamma_\nu) ((\nu=1,2,\ldots)) such that for every vector-function (f^{(i)}(x))

* These contours are chosen so that they do not intersect the above-mentioned disks of radius (\delta) with centers at the isolated points.

with components (f_k^{(i)}(x) \in L_2(a_i,b_i)) ((i=1,\ldots,m;\ k=1,\ldots,n)), the integral

[
-\frac{1}{2\pi\sqrt{-1}}\int_{\Gamma_\lambda^\nu} y^{(i)}(x,\lambda)\,d\lambda
\Rightarrow (A^{(i)}(x))^{-1}f^{(i)}(x)
\quad \text{as } \nu \to \infty
\tag{8}
]

in the sense of (L_2(a_i,b_i)).*

We note that the formulation of this theorem for (m=1) is contained in ((5)).

Lviv State University
named after Ivan Franko

Received
18 XI 1957

REFERENCES

({}^{1}) Ya. D. Tamarkin, On some general problems in the theory of ordinary linear differential equations and on expansions of arbitrary functions in series, Petrograd, 1917.
({}^{2}) G. D. Birkhoff, Trans. Am. Math. Soc., 9 (1908).
({}^{3}) Ch. E. Wilder, Trans. Am. Math. Soc., 18 (1917); 19 (1918).
({}^{4}) J. Tamarkin, Math. Zs., 27 (1928).
({}^{5}) M. L. Rasulov, Matem. sborn., 30 (72), 3 (1952).

* By imposing stronger restrictions on (f^{(i)}(x)), one can obtain convergence in the pointwise sense.