Abstract
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Reports of the Academy of Sciences of the USSR
1967, Volume 177, No. 3
UDC 517.916:517.94:517.944(045) MATHEMATICS
V. G. Khryshtun
CLASSES OF FUNCTIONS QUASIANALYTIC WITH RESPECT TO AN ORDINARY LINEAR DIFFERENTIAL OPERATOR, AND THEIR APPLICATION
(Presented by Academician S. L. Sobolev, February 4, 1967)
1. Let the operator \(L\) be given by the formula
\[ L=D_x^n+\sum_{k=0}^{n-1}p_k(x)D_x^k \left(D_x=\frac{d}{dx},\ D_x^0=I\right), \tag{1} \]
where the coefficients \(p_k(x)\) \((k=0,1,\ldots,n-1)\) are continuous functions on an arbitrary fixed interval \(\Delta\) of the real axis, and \(I\) is the identity operator. By \(M\) we denote another operator
\[ M=D_y^{n_1}+\sum_{k=0}^{n_1-1}q_k(y)D_y^k \left(D_y=\frac{d}{dy}\right), \tag{2} \]
of the same type, with coefficients \(q_k(y)\) continuous on an arbitrary fixed interval \(\Delta_1\) of the real axis.
Denote by \(\{k_i\}_{i=0}^{\infty}\) and \(\{\nu_j\}_{j=0}^{\infty}\) arbitrary fixed monotonically increasing sequences of positive integers, and by \(\{M_{k_i}\}_{i=0}^{\infty}\) and \(\{N_{\nu_j}\}_{j=0}^{\infty}\) arbitrary fixed sequences of positive numbers.
An infinitely \(L\)-differentiable (see \((^5)\)) function \(f(x)\) on the interval \(\Delta\) will be said to belong to the class \(C^L\{M_{k_i}\}\) if, for every closed interval \(\delta \subset \Delta\), there exist constants \(N\) and \(C\) (depending on \(L\), \(f\), and \(\delta\)) such that
\[ \left|D_x^{r_i}L^{q_i}f(x)\right|_{\delta} \leq NC^{k_i}M_{k_i},\qquad k_i=nq_i+r_i,\quad 0\leq r_i\leq n-1,\quad q_i\geq 0, \tag{3} \]
where \(q_i\) and \(r_i\) are integers.
The class \(C^L\{M_{k_i}\}\) of functions \(f(x)\) defined on the interval \(\Delta\) will be called a quasi-\(L\)-analytic class if, from the equalities \(D_x^rL^qf(a)=0\), holding at some point \(a\in\Delta\) for all \(r=0,1,\ldots,n-1\) and \(q=0,1,2,\ldots\), it follows that \(f(x)\equiv 0\) on \(\Delta\).
A function \(g(x,y)\), defined in the domain \(x\in\Delta,\ y\in\Delta_1\), infinitely \(L\)-differentiable with respect to \(x\) and infinitely \(M\)-differentiable with respect to \(y\), will be said to belong to the class
\[ C^{L,M}\{M_{k_i};N_{\nu_j}\}, \]
if for every closed interval \(\delta\subset\Delta\) and every closed interval \(\delta_1\subset\Delta_1\) there exist constants \(A\) and \(B\) (depending on \(L\), \(M\), \(g\), \(\delta\), and \(\delta_1\)) such that
\[ \left|D_x^{r_i}D_y^{\sigma_j}L^{q_i}M^{\mu_j}g(x,y)\right|_{\delta\times\delta_1} \leq AB^{k_i+\nu_j}M_{k_i}N_{\nu_j}, \tag{4} \]
\[ k_i=nq_i+r_i,\quad 0\leq r_i\leq n-1,\quad q_i\geq 0; \]
\[ \nu_j=n_1\mu_j+\sigma_j,\quad 0\leq \sigma_j\leq n_1-1,\quad \mu_j\geq 0, \]
where \(q_i,r_i,\mu_j,\sigma_j\) are integers.
Suppose that the function \(\sigma(x,y)\) in the domain \(x \in \Delta,\ y \in \Delta_1\) is the sum of the series
\[ \sigma(x,y)=\sum_{p=0}^{\infty} b_p(x,y)(x-x_1)^p \qquad (x_1 \in \Delta) \tag{5} \]
and the functions \(b_p(x,y)\) are continuous in the domain \(\Delta \times \Delta_1\). Then the function
\[ \varphi(x)=\sum_{p=0}^{\infty} a_p(x-x_1)^p, \tag{6} \]
whose coefficients \(a_p\) are given by the formulas
\[ a_p=\max_{\delta \times \delta_1}|b_p(x,y)|,\qquad p=0,1,2,\ldots \]
will be called an exact Cauchy majorant of the function (5) in the closed domain \(\delta \times \delta_1 \subset \Delta \times \Delta_1\). If for every closed domain \(\delta \times \delta_1 \subset \Delta \times \Delta_1\) the function (6) is an entire function of order of growth \(\rho\), then this number \(\rho\) will be called the order of conditional growth with respect to the variable \(x\) of the function (5) in the domain \(\Delta \times \Delta_1\).
We shall assume that
\[ \lim_{i\to\infty}\sqrt[k_i]{M_{k_i}}=\infty, \]
and then, for the sequence of points \((k_i,\ln M_{k_i})\), one can construct the Newton polygon (see (2)). The sequence \(\{M^c_{k_i}\}_{i=0}^{\infty}\) will be called the convex regularization by means of the logarithms of the sequence \(\{M_{k_i}\}_{i=0}^{\infty}\), if the number \(\ln M^c_{k_i}\) is equal to the ordinate of the Newton polygon at the point \(k_i\).
If \(L=D_x\) and \(k_i=i\), then the classes of functions \(C^L\{M_i\}\) coincide with the classes of functions \(C\{M_i\}\), whose properties are set forth in \((^2)\).
In the present note we establish that the conditions
\[ \lim_{i\to\infty}\sqrt[k_i]{M_{k_i}}=\infty,\qquad \sum_{i=1}^{\infty}(k_i-k_{i-1})(M_{k_{i-1}}/M^c_{k_i})^{1/(k_i-k_{i-1})}=\infty \tag{7} \]
are sufficient for the quasi-\(L\)-analyticity of the classes of functions \(C^L\{M_{k_i}\}\), if the coefficients of the operator \(L\) of the form (1) are continuous. Under additional restrictions on the coefficients \(p_k(x)\) we single out those operators \(L\) for which it has been possible to establish that the conditions (7) are necessary for the quasi-\(L\)-analyticity of the classes of functions \(C^L\{M_{k_i}\}\). We use these results in investigating the properties of solutions of the equation
\[ L^s f(x,y)=M^m f(x,y), \tag{8} \]
satisfying the initial data
\[ D_x^r L^q f(x,y)\big|_{x=x_1}=\varphi_{nq+r}(y),\qquad r=0,1,\ldots,n-1;\ q=0,1,\ldots,s-1, \tag{9} \]
where \(s\) and \(m\) are natural numbers. The solution of problem (8)—(9) is sought in the classes of functions \(C^{L,M}\{M_{k_i};N_{v_j}\}\), the initial data (9) being chosen in such a way that these classes are quasi-analytic with respect to one of the operators \(L\) or \(M\). The direction of the investigation was chosen under the influence of A. N. Tikhonov’s work \((^3)\) and M. K. Fage’s work \((^6)\).
We also note that the sufficient conditions for the quasi-\(L\)-analyticity of the note (7) are special cases of the conditions (7), and that A. A. Tyanovskii \((^4)\) also arrived at the conditions (7) as sufficient conditions for quasi-\(L\)-analyticity.
- Theorem 1. Conditions (7) are sufficient for the quasi-\(L\)-analyticity of the classes of functions \(C^L\{M_{ki}\}\), if \(L\) is an operator (1) of arbitrary order \(n \ge 1\) with continuous coefficients.
The proof is carried out in the same way as in note (7), with the aid of the generalized Bang formula.
Theorem 2. Conditions (7) are necessary for the quasi-\(L\)-analyticity of the classes of functions \(C^L\{M_k\}\) for the following operators \(L\) of the form (1):
1) \(L\) is a first-order operator with coefficient \(p_0(x)\) continuous on the interval \(\Delta\);
2) \(L\) is a second-order operator with coefficients continuous on the interval \(\Delta\), and the coefficient \(p_1(x)\) has a second derivative, while \(p_0(x)\) has a first derivative to the right (or to the left) of some point \(x_1 \in \Delta\);
3) \(L\) is an operator of arbitrary order, all coefficients of which are continuous on the interval \(\Delta\) and are analytic functions in an arbitrarily small neighborhood of one of the endpoints of the interval \(\Delta\);
4) \(L\) is an operator of arbitrary order, all coefficients of which are continuous on the interval \(\Delta\) and, in an arbitrarily small neighborhood of some point \(x_1 \in \Delta\), are constant.
For the first three types of operators \(L\), the proof is carried out by means of a transformation operator (see (1)).
In the next four theorems we establish some properties of solutions of problem (8)—(9).
Theorem 3. Let \(n_1m \le ns\), and let on the interval \(\Delta_1\) the functions (9) belong to the class \(C^M\{N_j\}\), where \(N_j = j!\). Then in a neighborhood of each point \((x_1, y_1) \in \Delta \times \Delta_1\) there exists a unique solution of problem (8)—(9), belonging to the class \(C^{L,M}\{M_{nsi}; N_j\}\), where \(M_{nsi} = (nsi)!\).
If \(n_1m < ns\), then the order of conditional growth with respect to the variable \(x\) of the solution of problem (8)—(9) does not exceed the number \(ns/(ns - n_1m)\).
Theorem 4. Let \(n_1m < ns\), let \(\varepsilon\) be an arbitrary positive number, and let \(\beta\) and \(\gamma\) be defined by the equalities
\[ ns/(ns-\beta)=ns/(ns-n_1m)+\varepsilon,\qquad \gamma=\beta/n_1m. \]
If on the interval \(\Delta_1\) the functions (9) belong to the class \(C^M\{N_j\}\), where \(N_j=(\gamma j)^{\gamma j}\), then in the domain \(\Delta \times \Delta_1\) there exists a unique solution of problem (8)—(9), belonging to the class \(C^{L,M}\{M_{nsi}; N_j\}\), where \(M_{nsi}=(nsi)!\). The order of conditional growth with respect to the variable \(x\) of the solution of problem (8)—(9) is not greater than the number \(ns/(ns-n_1m)+\varepsilon\).
Theorem 5. Let \(n_1m > ns\) and \(\alpha=ns/n_1m\). If on the interval \(\Delta_1\) the functions (9) belong to the class \(C^M\{N_j\}\), where \(N_j \le \Gamma(\alpha j+1)\), then in a neighborhood of each point \((x_1,y_1)\in \Delta \times \Delta_1\) there exists a unique solution of problem (8)—(9), belonging to the class \(C^{L,M}\{M_{nsi}; N_j\}\), where \(M_{nsi}=(nsi)!\) and \(N_j \le \Gamma(\alpha j+1)\). The order of conditional growth with respect to the variable \(y\) of the solution of problem (8)—(9) is not greater than \(n_1m/(n_1m-ns)\);
Theorem 6. If the operator \(L\) satisfies the conditions of Theorem 2, \(n_1m > ns\), and the order of conditional growth with respect to the variable \(y\) of the solution of problem (8)—(9) is greater than \(n_1m/(n_1m-ns)\), then the solution is not unique.
If in Theorem 6 we put \(s=m=1\), \(L=D_x\), and \(M=D_y^2\), then we obtain the well-known result of A. N. Tikhonov (3).
Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR
Received
27 I 1967
CITED LITERATURE
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- S. Mandelbrojt, Attached series. Regularization of sequences, IL, 1955.
- A. N. Tikhonov, Mat. sborn., 42, No. 2, 199 (1935).
- A. A. Tynovskii, Operator Taylor—Bang formulas and some of their applications, Candidate’s dissertation, Kiev, 1966.
- M. K. Fage, Tr. Moscow Math. Soc., 7, 227 (1958).
- M. K. Fage, Operator-analytic functions of one independent variable, Lviv, 1959.
- V. G. Khryshtun, Siberian Math. Journal, 6, No. 6, 1395 (1965).