MATHEMATICS

B. A. VOSTRETSOV

ON CONDITIONS FOR THE REPRESENTABILITY OF A FUNCTION OF MANY VARIABLES IN THE FORM OF A SUM OF A FINITE NUMBER OF PLANE WAVES OF GIVEN DIRECTIONS

(Presented by Academician I. G. Petrovskii on May 9, 1963)

Let in Euclidean space \(R_n=\{\mathbf{x}\}\), \(\mathbf{x}=(x_1,\ldots,x_n)\), there be given a system of pairwise noncollinear vectors \(\mathbf{a}_i=(a_{i1},\ldots,a_{in})\), \(i=1,\ldots,k\), and some domain \(D\).

Denote by \(F_m[D]\) the totality of all functions having in the domain \(D\) continuous partial derivatives up to order \(m\) inclusive, and by \(C_m[\mathbf{a}_i,D]\) the totality of all functions \(\varphi_i(t_i)\), \(i=1,\ldots,k\), that are \(m\) times continuously differentiable on the interval

\[ \inf_{\mathbf{x}\in D}(\mathbf{a}_i\mathbf{x})<t_i<\sup_{\mathbf{x}\in D}(\mathbf{a}_i\mathbf{x}),\qquad \mathbf{a}_i\mathbf{x}=a_{i1}x_1+\ldots+a_{in}x_n, \tag{1} \]

In the present paper we derive necessary, and under certain restrictions imposed on the domain \(D\), sufficient conditions for there to exist, for a function \(f(\mathbf{x})\), \(f(\mathbf{x})\in F_m[D]\), where \(m\) is a sufficiently large natural number, functions \(\varphi_i(t_i)\in C_m[\mathbf{a}_i,D]\), \(i=1,\ldots,k\), such that in the domain \(D\)

\[ f(\mathbf{x})=\sum_{i=1}^{k}\varphi_i(\mathbf{a}_i\mathbf{x}). \tag{2} \]

The indicated necessary and sufficient conditions are obtained in the form of a system of partial differential equations with one unknown function. In what follows in the paper, starting from the set \(\{\mathbf{a}_1,\ldots,\mathbf{a}_k\}\), a special system of differential operators is constructed. The constructed operators are applied to a function \(f(\mathbf{x})\) representable in the form (2). Specifying part of these operators on a segment \([\mathbf{x}_0,\mathbf{x}_1]\) of a straight line not orthogonal to any of the directions \(\mathbf{a}_1,\ldots,\mathbf{a}_k\), and the remaining operators at some point of this segment, determines the function \(f(\mathbf{x})\) uniquely. The domain of its definition is the intersection of the strips

\[ \mathbf{a}_i\mathbf{x}_0\leq \mathbf{a}_i\mathbf{x}\leq \mathbf{a}_i\mathbf{x}_1,\qquad i=1,\ldots,k,\qquad \mathbf{x}\in R_n. \tag{3} \]

Let us first note that the coordinates of the vectors \(\mathbf{a}_i=(a_{i1},\ldots,a_{in})\), \(i=1,\ldots,k\), may be regarded as homogeneous coordinates of points of the \((n-1)\)-dimensional projective space \(\Pi_{n-1}\). Denote by \(\mathbf{a}\) an arbitrary point of this space, determined by homogeneous coordinates \(a_1,\ldots,a_n\). Let some set of points \(M\) be given in the space \(\Pi_{n-1}\). Consider, in the ring of polynomials \(K[\mathbf{y}]=K[y_1,\ldots,y_n]\) over the field of real numbers, the totality \(\{P(\mathbf{y})\}\) of all forms (homogeneous polynomials) \(P(\mathbf{y})\), each of which contains the set \(M\) (vanishes at every point of the set \(M\)). Among these forms (see (1)) there exists a finite number of forms

\[ P_{iM}(\mathbf{y}),\qquad i=1,\ldots,s, \tag{4} \]

such that for every homogeneous polynomial \(P(\mathbf{y})\) containing the set \(M\), the equality

\[ P(\mathbf{y})=\sum_{i=1}^{s} R_i(\mathbf{y})P_{iM}(\mathbf{y}), \tag{5} \]

will hold,

where \(R_i(y)\) are certain forms from the ring \(K[y]\). In what follows we assume that the system (4) is minimal: none of the polynomials of this system is expressed in terms of the others by formula (5). Any such system of polynomials will be called a basis belonging to the set \(M\). Replacing in each term of the polynomial \(P(y)\) the power \(y_p^{\nu_p}\) by the symbolic power \(\partial^{\nu_p}/\partial x_p^{\nu_p}\), \(p=1,\ldots,n\), we obtain the differential operator \(P(\partial/\partial x)\) corresponding to the form \(P(y)\). For this operator the representation

\[ P\left(\frac{\partial}{\partial x}\right) = \sum_{i=1}^{s} R_i\left(\frac{\partial}{\partial x}\right) P_{iM}\left(\frac{\partial}{\partial x}\right). \tag{5'} \]

is valid.

Using the basis (4), we construct a system of differential equations with one unknown function

\[ P_{i,M}(\partial/\partial x)u = P_{iM}(\partial/\partial x_1,\ldots,\partial/\partial x_n)u = 0, \quad i=1,\ldots,s. \tag{6} \]

Lemma 1. If the set \(\{P(y)\}\) consists only of the identically zero form, then for any homogeneous polynomial \(Q(x)\) of arbitrary degree \(m\) there exist real numbers \(\lambda_i\) and, in the set \(M\), points \(a_i\), \(i=1,\ldots,r\), such that for all \(x\in R_n\)

\[ Q(x)=\sum_{i=1}^{r}\lambda_i(a_i x)^m. \tag{7} \]

If, however, the set \(\{P(y)\}\) contains at least one nontrivial form, then in order that a homogeneous polynomial \(Q(x)\) be representable in the form (7), it is necessary and sufficient that this polynomial satisfy the system (6).

The validity of the first assertion of the lemma was proved in the note \(\left({}^{2}\right)\). The necessity of the second is obvious. Its sufficiency is proved analogously to how this was done in the paper \(\left({}^{3}\right)\) for the special case \(Q(x)=(ax)^m\).

Suppose now that \(M\) is a finite set of points of \(P_{n-1}\). The index of a point \(a\) with respect to the set \(M\) will mean the least of the degrees of forms of the ring \(K[y]\) which contain the set \(M\) but do not contain the point \(a\). A form containing the set \(M\) and not containing the point \(a\), whose degree is equal to the index of the point \(a\) with respect to the set \(M\), will be called the index form for the point \(a\) and the set \(M\) and will be denoted by \(\Gamma_{aM}(y)\). The corresponding differential operator \(\Gamma_{aM}(\partial/\partial x)\) will be called the index operator for the indicated point and set.

Theorem 1. Let \(M=\{a_1,\ldots,a_k\}\). If every section of the domain \(D\) by a hyperplane orthogonal to the vector \(a_i\) is connected, \(i=1,\ldots,k\), then for the given set \(M\) there exists a natural number \(l=l(M)\) such that every function \(f(x)\), \(f(x)\in F_m[D]\), \(m\ge l\), satisfying in the domain \(D\) the system (6), is represented in this domain in the form (2), moreover \(\varphi_i(t_i)\in C_m[a_i,D]\).

Conversely, if for a function \(f(x)\) in some domain the equality (2) is valid, where \(\varphi_i(t_i)\in C_m[a_i,D]\), \(m\ge l\), then in this domain \(f(x)\) satisfies the system (6).

We shall prove the direct assertion by induction on the number of points \(k\). For \(k=1\), \(M=\{a_1\}\), we put \(P_{i\{a_1\}}(y)=e_i y\), \(i=1,\ldots,n-1\), where \(\{e_1,\ldots,e_{n-1}\}\) is an arbitrary collection of linearly independent vectors orthogonal to the vector \(a_1\). The system (6) is written in the form

\[ P_{i\{a_1\}}(\partial/\partial x)u = (e_i,\partial/\partial x)u = 0, \quad i=1,\ldots,n-1, \]

and in this case the theorem is easily proved. Assuming it true for a set consisting of \(k-1\) points, we construct, for the set \(M_1=\{a_2,\ldots,a_k\}\), the forms \(P_{iM_1}(y)\), \(i=1,\ldots,r\), and consider the products \(P_{iM_1}(y)\cdot P_{j\{a_1\}}(y)\), \(i=1,\ldots,r\); \(j=1,\ldots,n-1\) (we assume here that the forms

\(P_{j\{a_1\}}(\mathbf y),\ j=1,\ldots,n-1,\) do not contain the points \(M_1\). Each such product contains the set \(M=\{\mathbf a_1,\ldots,\mathbf a_k\}\), and hence is expressed in terms of the polynomials \(P_{\sigma M}(\mathbf y),\ \sigma=1,\ldots,s,\) by formula (5). The latter means that any product
\[ P_{iM_1}(\partial/\partial \mathbf x)\cdot P_{j\{a_1\}}(\partial/\partial \mathbf x) \]
can be expressed in terms of the forms \(P_{\sigma M}(\partial/\partial \mathbf x),\ \sigma=1,\ldots,s,\) by means of equality \((5')\). Therefore every solution \(u=f(\mathbf x)\) of system (6), defined in the domain \(D\), \(f(\mathbf x)\in F_m[D]\), \(m\ge l(M)\ge l(M_1)+1\)*, is in this domain a solution of the system
\[ P_{iM_1}(\partial/\partial \mathbf x)\cdot P_{j\{a_1\}}(\partial/\partial \mathbf x)u=0,\qquad i=1,\ldots,r;\ j=1,\ldots,n-1. \]

Hence, and from the induction assumptions, it follows that this solution \(u=f(\mathbf x)\) will satisfy in the domain \(D\) the system:
\[ P_{j\{a_1\}}(\partial/\partial \mathbf x)u =\sum_{i=2}^{k}\varphi_{ij}(\mathbf a_i\mathbf x),\qquad j=1,\ldots,n-1, \tag{8} \]
where the functions \(\varphi_{ij}(t_i)\in C_{m-1}[\mathbf a_i,D]\), \(j=1,\ldots,n-1;\ i=2,\ldots,k\), and in the indicated domain are subject to the conditions
\[ \sum_{i=2}^{k}\varphi'_{it}(\mathbf a_i\mathbf x)P_{q\{a_1\}}(\mathbf a_i) = \sum_{i=2}^{k}\varphi'_{iq}(\mathbf a_i\mathbf x)P_{t\{a_1\}}(\mathbf a_i), \qquad q,t=1,\ldots,n-1. \tag{9} \]

Using the relations (9), let us express the functions \(\varphi_{ih}(t_i)\), \(h=1,\ldots,n-2\), for example, in terms of the function \(\varphi_{i\,n-1}(t_i)\), \(i=2,\ldots,k\). To this end, putting \(t=h\) and \(q=n-1\), we act on both sides of (9) with the operator
\(\Gamma_{\alpha_i M-\{a_i\}}(\partial/\partial \mathbf x)\). Then, integrating the result obtained \(\alpha_i+1\) times (\(\alpha_i\) is the index of the point \(\mathbf a_i\) with respect to the set \(M-\{a_i\}\)), we shall have
\[ \varphi_{ih}(t_i)= \frac{P_{h\{a_1\}}(\mathbf a_i)} {P_{n-1\{a_1\}}(\mathbf a_i)} \,\varphi_{i\,n-1}(t_i)+Q_{ih}(t_i), \]
\[ h=1,\ldots,n-2;\qquad i=2,\ldots,k, \]
where \(Q_{ih}(t_i)\) is a certain polynomial in \(t_i\) of degree not exceeding \(\alpha_i\).

It can be shown that, when conditions (9) and (10) are fulfilled, every solution of system (8) of the class \(F_m[D]\), \(m\ge l\), has the form
\[ u(\mathbf x)=\varphi_1(\mathbf a_1\mathbf x) +\sum_{i=2}^{k} \frac{1}{P_{n-1\{a_1\}}(\mathbf a_i)} \int_{t_i^0}^{\mathbf a_i\mathbf x}\varphi_{i\,n-1}(\tau)\,d\tau +T(\mathbf x), \]
where \(\varphi_1(t_1)\in C_m[\mathbf a_1,D]\), \(m\ge l\), \(T(\mathbf x)\) is a certain polynomial, and \(t_i^0\) is a number from the interval (1). The function \(f(\mathbf x)\), being a solution of system (8), must be of the same form. It follows from this that the corresponding polynomial \(T(\mathbf x)\) is a solution of system (6). By virtue of Lemma 1 it must be representable by formula (2). The direct theorem is proved. The converse theorem is obvious.

Let \(M\in \Pi_{n-1}\) be any set of points for which there exists a nontrivial form \(P(\mathbf y)\) containing it. Denote by \(\gamma_1<\cdots<\gamma_\mu\) the sequence of all distinct degrees of the polynomials of some basis belonging to the set \(M\), and by \(M_{\gamma_i}\) the algebraic variety consisting of the collection of all common zeros of those basis polynomials whose degrees are less than or equal to \(\gamma_i\), \(i=0,1,\ldots,\mu\) (\(\gamma_0=0,\ M_0=\Pi_{n-1}\)). The following lemmas are easily proved:

\[ \text{* The number } l(M) \text{ is chosen, in addition, so that all derivatives of the functions under consideration which enter the argument are continuous.} \]

Lemma 2. Let \(\gamma_i \leq m < \gamma_{i+1}\), \(i=0,1,\ldots,\mu\); \(\gamma_{\mu+1}=\infty\). Then and only then does the degree \((\mathbf a\mathbf x)^m\) decompose according to formula (7), when \(\mathbf a\in M_{\gamma_i}\).

We shall call the index of a finite set of points \(M\in \Pi_{n-1}\) the greatest of the indices of the points of this set relative to their complements in the set \(M\).

Lemma 3. Suppose that the set of points \(M\) has index \(\gamma\), and let a point \(\mathbf a\), \(\mathbf a\in M\), have index \(\gamma\) relative to the set \(M-\{\mathbf a\}\). Then the index of the set \(M-\{\mathbf a\}\) is either \(\gamma\) or \(\gamma-1\). If some point \(\mathbf b\in M\) has, relative to \(M-\{\mathbf b\}\), index \(\beta<\gamma\), then its index relative to the set \(M-\{\mathbf a\}-\{\mathbf b\}\) is also equal to \(\beta\).

Let again \(M=\{\mathbf a_1,\ldots,\mathbf a_k\}\), and let \([\mathbf x_0,\mathbf x_1]\) be some segment of the line of the space \(R_n\), not orthogonal to any of the directions \(\mathbf a_i\), \(i=1,\ldots,k\).

We now pose the problem of determining a sufficiently smooth function \(f(\mathbf x)\), representable in the form (2) in the domain which is the intersection of the strips (3), from the following data.

Take arbitrary continuous functions \(\varphi_i(t_i)\), \(\mathbf a_i\mathbf x_0\leq t_i\leq \mathbf a_i\mathbf x_1\), \(i=1,\ldots,k\). We first require that at each point \(\mathbf z\) of the segment \([\mathbf x_0,\mathbf x_1]\) the equalities

\[ \left. \Gamma_{\mathbf a_i M-\{\mathbf a_i\}}\left(\frac{\partial}{\partial \mathbf x}\right) f(\mathbf x) \right|_{\mathbf x=\mathbf z} = \varphi_i(\mathbf a_i\mathbf z), \qquad i=1,\ldots,k, \tag{11} \]

hold, where \(\Gamma_{\mathbf a_i M-\{\mathbf a_i\}}(\partial/\partial \mathbf x)\) is the index operator for the point \(\mathbf a_i\) and the set \(M-\{\mathbf a_i\}\).

Assume further that the index of the set \(M\) is equal to \(\gamma\). In addition to the data (11), specify at some point of the segment \([\mathbf x_0,\mathbf x_1]\), for instance at the point \(\mathbf x_0\), the following set of index operators from the function \(f(\mathbf x)\) of orders lower than \(\gamma\).

Let \(M^1\) be such a subset of \(M\) of index \(\gamma-1\) that adjoining to it any point of the set \(M-M^1\) turns it into a set of index \(\gamma\). In the set \(M^1\) we single out all points \(\mathbf a_i\) of index \(\gamma\) relative to the set \(M-\{\mathbf a_i\}\). For each such point and its complement to \(M^1\), consider the index operator from \(f(\mathbf x)\) and prescribe its value at the point \(\mathbf x_0\). We obtain a system \(S_1\) of homogeneous differential operators of order \(\gamma-1\) from the function \(f(\mathbf x)\), specified at the point \(\mathbf x_0\).

By \(M^2\) denote a subset of \(M^1\) of index \(\gamma-2\) such that adjoining to it any point of the set \(M^1-M^2\) turns it into a set of index \(\gamma-1\). In the set \(M^2\) we single out all points \(\mathbf a_i\) of index \(\gamma-1\) relative to the set \(M^1-\{\mathbf a_i\}\). For these points and their complements to the set \(M^2\) we consider the corresponding index operators from the function \(f(\mathbf x)\) and prescribe their values at the point \(\mathbf x_0\). We obtain a system \(S_2\) of homogeneous differential operators from the function \(f(\mathbf x)\) of order \(\gamma-2\), the values of which are specified at the point \(\mathbf x_0\). Continuing in the same way, we specify systems of differential operators \(S_3,\ldots,S_\gamma\) of orders respectively \(\gamma-3,\ldots,1\) from the function \(f(\mathbf x)\) at the point \(\mathbf x_0\). Finally, specify the value \(f(\mathbf x_0)\). One can prove the theorem:

Theorem 2. The equalities (11) and the values of all operators from the set \(S=S_1+\cdots+S_\gamma\) from the function \(f(\mathbf x)\) at the point \(\mathbf x_0\), as well as the value \(f(\mathbf x_0)\), together uniquely determine a sufficiently smooth function \(f(\mathbf x)\), representable by formula (2), in the domain which is the intersection of the strips (3).

Received
9 V 1963

REFERENCES

1. Van der Waerden, Modern Algebra, Part II, 1931, p. 57.
2. B. A. Vostretsov, M. A. Kreines, DAN, 140, No. 6 (1961).
3. B. A. Vostretsov, M. A. Kreines, DAN, 144, No. 6 (1962).