MATHEMATICS

Yu. M. BEREZANSKII

ONE GENERALIZATION OF THE MULTIDIMENSIONAL BOCHNER THEOREM

(Presented by Academician S. L. Sobolev, 16 IX 1960)

In this note the \(n\)-dimensional Bochner theorem on positive definite (p.d.) functions is generalized to the case when the role of \(e^{i\lambda_j x_j}\) is played by eigenfunctions of differential (and more general) operators. For \(n=1\) the generalization was obtained by M. G. Krein \((^1)\) and by the author \((^2)\) (in the case of generalized kernels—by Maurin \((^3)\)); for \(n>1\) substantial difficulties arise, connected with the necessity of extending Hermitian commuting operators to self-adjoint commuting ones. The considerations of this article correspond to the case when the closures of the operators are already self-adjoint; they are based on the theory of expansions in generalized eigenvectors (for the literature see \((^4)\)). The note is connected with the work of A. G. Kostyuchenko and B. S. Mityagin \((^5)\), in which similar results were obtained for kernels generating a commutative ring. We do not assume a connection with rings, thanks to which the class of differential operators is substantially enlarged (for example, operators of order higher than the second appear)\(*\).

\(1^\circ\). Let \(H_0\) be a complete Hilbert space with scalar product \((f,g)_0\); let \(H_+\) be a linear set from \(H_0\), dense in it and itself a complete Hilbert space with respect to another scalar product \((u,v)_+\), such that \(\|u\|_0 \leq \|u\|_+\) \((u\in H_+)\). The bilinear form \(B(f,u)=(f,u)_0\) is continuous in \(f\in H_0\) and \(u\in H_+\); therefore it can be written in the form \((f,u)_0=(If,u)_+\), where \(I\) is an operator continuously acting from \(H_0\) into \(H_+\). Introduce in \(H_0\) a new scalar product \((f,g)_-=(If,g)_0\) and, carrying out the completion, obtain the Hilbert space \(H_-\). Thus,

\[ H_- \supset H_0 \supset H_+, \tag{1} \]

where each space of this chain is dense in the one standing to its left, and
\(\|u\|_- \leq \|u\|_0 \leq \|u\|_+\).

It is not difficult to show that a bilinear form \((\alpha,u)_0\) \((\alpha\in H_-,\; u\in H_+)\) is defined, becoming the scalar product in \(H_0\) if \(\alpha\in H_0\). Every linear continuous functional \(l(u)\) on \(H_+\) can be written in the form \(l(u)=(\alpha,u)_0\), where \(\alpha\) is some element of \(H_-\). Thus, \(H_-\) may be interpreted as the space of linear functionals on \(H_+\) (it is clear that it is isometric to \(H_+\)). Elements of the spaces \(H_+\), \(H_0\), and \(H_-\)—spaces with positive, zero, and negative norms—we shall denote respectively by \(u,v,\ldots;\ f,g,\ldots\) and \(\alpha,\beta,\ldots\). The space \(H_-\) may be regarded as the space of “generalized vectors” over the basic space \(H_+\) of “smooth vectors.”

Let the space \(H_{++}\) be in relation to \(H_+\) in the same position as \(H_+\) is in relation to \(H_0\). Then, from \(H_{++}\) and \(H_0\), one can construct the negative space \(H_{--}\). It is easy to see that
\[ H_{--}\supset H_-\supset H_0\supset H_+\supset H_{++} \]
with the same density relations and inequalities as in (1). We call the space \(H_+\) nuclear with respect to \(H_0\) if \(I\), considered as an operator in \(H_0\), has finite trace.

\(*\) The main constructions of the article were reported by the author at a seminar at the Institute of Mathematics of the Academy of Sciences of the USSR in the summer of 1959 (see also \((^4)\)). What is new is Theorem 3, instead of which a cruder result had previously been used, leading to estimates more restrictive than (6).

\(2^\circ\). Tensor-multiplying the spaces of the chain (1) by themselves, we obtain the chain \(H_{-}\times H_{-}\supset H_0\times H_0\supset H_{+}\times H_{+}\). It can be shown that the space \(H_{-}\times H_{-}\) will be negative with respect to the zero space \(H_0\times H_0\) and positive \(H_{+}\times H_{+}\). The elements of the spaces \(H_{+}\times H_{+}\), \(H_0\times H_0\), and \(H_{-}\times H_{-}\), by analogy with the case of spaces of functions, will be called respectively smooth, ordinary, and generalized kernels.

In all that follows we shall assume that in the space \(H_0\) an involution \(f\to \bar f\) has been introduced, which is also an involution for \(H_{+}\). Now the notions of Hermitian and positive definite kernels are introduced in the following way: to each generalized kernel \(K\) there corresponds the bilinear form \(B_K(u,v)=(K,\bar vu)_{H_0\times H_0}\) (the index \(H_0\times H_0\) will henceforth be omitted), where \(u,v\in H_{+}\), and \(\bar vu\) is the tensor product of the vector \(v\) by \(u\). The kernel \(K\) is called Hermitian (positive definite) if the form \(B_K\) is Hermitian (positive definite). Our main problem is the representation of a positive definite kernel in the form of a linear combination of elementary positive definite kernels.

Let in \(H_0\) a system of operators \(A^1,\ldots,A^p\) be defined with domain \(\mathfrak D\), dense in \(H_0\). In what follows we shall consider only separable positive spaces and such spaces that contain \(\mathfrak D\) as their dense part and contain all \(A^j(\mathfrak D)\) \((j=1,\ldots,p)\). Generalized positive definite kernels \(\varphi_\lambda\in H_{-}\times H_{-}\), \(\|\varphi_\lambda\|_{H_{-}\times H_{-}}\leqslant 1\) \((\lambda=(\lambda_1,\ldots,\lambda_p)\) is a parameter) will be called elementary with respect to the system of operators \(A^1,\ldots,A^p\) if \((\varphi_\lambda,(A^j-\lambda_jE)u\cdot\bar v)=0\), \((\varphi_\lambda,\bar v\cdot(A^j-\lambda_jE)u)=0\) \((j=1,\ldots,p)\) for all \(u\in\mathfrak D\) and \(v\in H_{+}\) (from the Hermitian character of the positive definite kernel it follows that one of these equalities entails the other). On the basis of the results of the article \((^4)\) the following is obtained.

Theorem 1. Let \(K\in H_{-}\times H_{-}\) be a generalized positive definite kernel; denote by \(H_K\) the completion of \(H_{+}\) with respect to the scalar product \(\langle u,v\rangle=(K,\bar vu)\) \((u,v\in H_{+})\). Suppose first that \(p=1\). If the operator \(A^1\) is Hermitian in \(H_K\): \(\langle A^1u,v\rangle=\langle u,A^1v\rangle\) \((u,v\in\mathfrak D)\), then the representation is always valid

\[ K=\int_{-\infty}^{\infty}\varphi_\lambda\,d\rho(\lambda), \tag{2} \]

where \(\varphi_\lambda\in H_{--}\times H_{--}\) is some family of elementary positive definite kernels with respect to \(A^1\), and \(d\rho(\lambda)\) is a nonnegative finite measure on the axis; the integral is understood in the weak sense with respect to \(H_{--}\times H_{--}\). Here \(H_{++}\supset H_{+}\) is any positive space, chosen in such a way that it is nuclear with respect to \(H_{+}\). Conversely, if for the kernel \(K\in H_{-}\times H_{-}\) a representation (2) is valid with some family of elementary kernels \(\varphi_\lambda\in H_{--}\times H_{--}\), then \(A^1\) is Hermitian in \(H_K\). In the representation (2) the expression \(\varphi_\lambda\,d\rho(\lambda)\) is determined uniquely if and only if the closure of the operator \(A^1\) is maximal in \(H_K\).

Let now \(p>1\). If the closures of the operators \(A^1,\ldots,A^p\) are self-adjoint in \(H_K\) and their resolutions of the identity \(E^1(\Delta_1),\ldots,E^p(\Delta_p)\) commute for arbitrary intervals \(\Delta_1,\ldots,\Delta_p\), then the representation (2) is valid with the measure \(d\rho(\lambda)\) in \(p\)-dimensional space and with a family of kernels elementary with respect to \(A^1,\ldots,A^p\). The choice of \(H_{++}\) is the same as above; in the representation (2) the expression \(\varphi_\lambda\,d\rho(\lambda)\) is determined uniquely. Conversely, if for the kernel \(K\in H_{-}\times H_{-}\) a representation (2) is valid with some family of elementary kernels \(\varphi_\lambda\in H_{--}\times H_{--}\), with \(\varphi_\lambda\,d\rho(\lambda)\) determined uniquely, then the closures of the operators \(A^1,\ldots,A^p\) are self-adjoint in \(H_K\) and their resolutions of the identity commute.

\(3^\circ\). Let us apply this theorem to differential operators \(A^j\). The case \(p=1\) has been sufficiently studied \((^1,^2)\); everywhere in what follows we assume \(p>1\).

Let \(G\) be a finite or infinite domain of \(n\)-dimensional space; \(H_0=L_2(G)\), the involution being ordinary passage to the complex conjugate; \(\mathfrak D\) contains the set \(C_0^\infty(G)\) of functions finitely and infinitely differentiable relative to \(G\), and \(A^j u=\mathcal L^j[u]\) \((u\in C_0^\infty(G))\), where \(\mathcal L^j\) are linear differential expressions with infinitely differentiable coefficients \((j=1,\ldots,p)\). Then the definition of an elementary positive-definite kernel shows that \(\varphi_\lambda\), in each of the variables, is a generalized solution of a homogeneous linear differential equation, and therefore in the case, for example, of ellipticity of the expressions \(\mathcal L^j\) (or when they are in ordinary derivatives) is, inside \(G\), an ordinary kernel \(\varphi_\lambda(x,y)\) \((x,y\in G)\), satisfying the equations \(\mathcal L_x^j\varphi_\lambda=\lambda_j\varphi_\lambda,\ \mathcal L_y^j\varphi_\lambda=\lambda_j\varphi_\lambda\) \((j=1,\ldots,p)\). If \(\mathcal L^j\) are expressions in partial derivatives, then these equalities still do not make it possible to express \(\varphi_\lambda\) in terms of a standard system of functions depending only on \(\mathcal L^j\) and not depending on \(K\) (cf. (?)). However, in one important case this can be done:

Theorem 2. Let \(G=G_1\times\cdots\times G_n\), where \(G_j\) is a finite or infinite interval of variation of the variable \(x_j\); \(\mathcal L^j\) is an ordinary differential expression in the variable \(x_j\) of order \(r_j\); \(\chi_1^j(x_j,\mu),\ldots,\chi_{r_j}^j(x_j,\mu)\) is a fixed fundamental system of solutions of \(\mathcal L^j[u]=\mu u\). Put
\[ X_{\mathbf j}(x,\lambda)=\chi_{j_1}^1(x_1,\lambda_1)\cdots \chi_{j_n}^n(x_n,\lambda_n) \quad (x=(x_1,\ldots,x_n),\ \lambda=(\lambda_1,\ldots,\lambda_n)). \]
Here \(\mathbf j=(j_1,\ldots,j_n)\) is a combined index, varying over the set \(N\) of points with coordinates \(j_l=1,\ldots,r_l\) \((l=1,\ldots,n)\).

Let \(K\in H_{-}\times H_{-}\) be a generalized positive-definite kernel; if the closures of the corresponding \(\mathcal L^j\)-operators \(A^1,\ldots,A^n\) are self-adjoint in \(H_K\) and their resolutions of the identity commute, then the representation
\[ K=\int_{-\infty}^{\infty}\sum_{\mathbf j,\mathbf k\in N} X_{\mathbf j}(x,\lambda)\overline{X_{\mathbf k}(y,\lambda)}\,d\rho_{\mathbf j\mathbf k}(\lambda), \tag{3} \]
holds, where the matrix \(\|d\rho_{\mathbf j\mathbf k}(\lambda)\|_{\mathbf j,\mathbf k\in N}\) is positive-definite in the sense that
\[ \sum_{\mathbf j,\mathbf k\in N} \rho_{\mathbf j\mathbf k}(\Delta)\xi_{\mathbf j}\overline{\xi_{\mathbf k}}\ge 0 \quad (\rho_{\mathbf j\mathbf k}(\Delta)=\rho_{\mathbf j\mathbf k}(\lambda'')-\rho_{\mathbf j\mathbf k}(\lambda'),\ \Delta=[\lambda',\lambda'')) \tag{4} \]
for any \(\Delta\) and any set of numbers \(\xi_{\mathbf j}\). The integral in (3) converges in the sense of weak convergence in \(H_{-}\times H_{-}\); \(H_{++}\) is chosen in the same way as in Theorem 1.

\(4^\circ\). The principal difficulty that arises in applying Theorems 1 and 2 is the verification of the self-adjointness of the operators \(A^j\) and the commutativity of their resolutions of the identity. We shall now formulate (in a general form) the relevant results. Suppose that for each \(j=1,\ldots,n\) there is given a chain \(H_-^j\supset H_0^j\supset H_+^j\) of type (1); assume that the spaces \(H_+\) and \(H_0\) considered earlier have respectively the form \(H_+^1\times\cdots\times H_+^n\) and \(H_0^1\times\cdots\times H_0^n\); then \(H_-=H_-^1\times\cdots\times H_-^n\). Let \(B^j\) be an operator in \(H_+^j\) with domain of definition \(\mathfrak D(B^j)\) dense in \(H_+^j\); introduce the operator \(A=E\times\cdots\times E\times B^j\times E\times\cdots\times E\) (\(B^j\) stands in the \(j\)-th place) in \(H_+\) with domain of definition
\[ \mathfrak D(A^j)=H_+^1\times\cdots\times H_+^{j-1}\times \mathfrak D(B^j)\times H_+^{j+1}\times\cdots\times H_+^n, \]
dense in \(H_+\), and hence also in \(H_0\). Everywhere below we assume that \(A^j\) is Hermitian in \(H_K\).

Theorem 3. Suppose that for each \(j\), for both equations \(du_t/dt\pm iB^{j*}u_t=0\) \((0\le t<\infty)\), considered in the Hilbert space \(H_+^j\), uniqueness of weak solutions holds. Then the closures of the operators \(A^j\) are self-adjoint in \(H_K\) and their resolutions of the identity commute.

\(5^\circ\). The scheme indicated in item \(4^\circ\) can be realized for ordinary differential operators \(B^j\) on the whole axis with constant coefficients, acting for different \(j\) in different variables; the proof of uniqueness of solutions in this case is carried out

on the basis of the method developed in [6]. Then, applying Theorem 2, we obtain integral representations of positive-definite kernels. We state the main result, restricting ourselves, for simplicity of formulation, to the case of ordinary positive-definite kernels.

Theorem 4. Let in the \(n\)-dimensional space \(E_n\) there be given a continuous positive-definite kernel \(K(x,y)\) \((x,y\in E_n)\), satisfying, in the sense of generalized functions, the relations

\[ \mathscr L^j_{x_j}[K(x,y)]=\overline{\mathscr L^j_{y_j}[K(x,y)]}\qquad (j=1,\ldots,n), \tag{5} \]

where \(\mathscr L^1,\ldots,\mathscr L^n\) are differential expressions with constant coefficients in the variables, respectively, \(x_1,\ldots,x_n\), and of orders \(r_1,\ldots,r_n\). If for some \(\varepsilon>0\) the estimate* holds

\[ |K(x_1,\ldots,x_n,y_1,\ldots,y_n)|\leq C\exp\{|x_1|^{r'_1-\varepsilon}+ \]

\[ +|y_1|^{r'_1-\varepsilon}+\cdots+|x_n|^{r'_n-\varepsilon}+|y_n|^{r'_n-\varepsilon}\} \qquad (C>0;\ x,y\in E_n), \tag{6} \]

where \(r'_j\), for \(r_j\geq 2\), is the number conjugate to \(r_j\) \((1/r_j+1/r'_j=1)\), and for \(r_j=1\) is an arbitrary positive number, then the representation

\[ K(x,y)=\int_{-\infty}^{\infty}\sum_{j,k\in N} X_j(x,\lambda)\overline{X_k(y,\lambda)}\,d\rho_{jk}(\lambda) \qquad (x,y\in E_n) \tag{7} \]

holds, with an absolutely convergent integral. Here the functions \(X_j\) and the matrix \(\|\rho_{jk}\|\) are the same as in Theorem 2.

In this paper we shall not dwell in detail on questions of the uniqueness of the representation (7); we note only that a certain uniqueness of it follows from Theorem 1. It is also clear that the converse assertion to Theorem 4 is valid in the known sense.

From Theorem 4 one can obtain a number of \(n\)-dimensional theorems of Bochner type. Thus, if \(k(x)\) \((x\in E_n)\) is a continuous function for which the kernel \(K(x,y)=k(x-y)\) is positive-definite, then (5) is satisfied with \(\mathscr L^j=i\,\partial/\partial x_j\). Writing the representation (7) for \(K\) and putting \(y=0\), we obtain Bochner’s theorem (here condition (6) is satisfied automatically). If the kernel \(K(x,y)=k(x+y)\) is positive-definite and \(k(x)\) grows at infinity no faster than \(\exp\{|x_1|^{r'_1}+\cdots+|x_n|^{r'_n}\}\), where \(r'_1,\ldots,r'_n\) are some positive numbers, then the representation (7) gives the \(n\)-dimensional theorem of S. N. Bernstein (in this case \(\mathscr L^j=\partial/\partial x_j\)). If the positive-definite kernel \(K(x,y)=k(x+y)+k(x-y)\) and \(k(x)\) grows at infinity no faster than \(\exp\{|x_1|^{2-\varepsilon}+\cdots+|x_n|^{2-\varepsilon}\}\) \((\varepsilon>0)\), then one may take \(\mathscr L^j=\partial^2/\partial x_j^2\); the representation (7) then gives an expansion of \(k(x)\) in terms of \(\cos\sqrt{\lambda_j}x_j\) and \(\sin\sqrt{\lambda_k}x_k\). If, in addition, the evenness of \(k(x)\) is known, then in this representation the terms \(\sin\sqrt{\lambda_k}x_k\) vanish, and we obtain a theorem of M. G. Krein type. One could also consider the case when from \(k(x)\) the kernel \(K(x,y)\) is formed by a combination of the indicated operations, for example
\(K(x,y)=k(x_1-y_1,\ x_2+y_2)+k(x_1-y_1,\ x_2-y_2)\). If \(k(x)\) depends analytically on \(x_j\), then the method given for forming the kernel \(K(x,y)\) can be generalized to the case of expressions \(\mathscr L^j\) of order higher than the second.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
14.IX.1960

CITED LITERATURE

1. M. G. Krein, DAN, 53, No. 1 (1946).
2. Yu. M. Berezanskii, Matem. sborn., 47 (89), No. 2 (1959).
3. K. Maurin, Bull. Acad. Polon. Sci., Ser. Math., Astr. et Phys., 6, No. 3 (1958).
4. Yu. M. Berezanskii, Ukr. matem. zhurn., 11, No. 1 (1959).
5. A. G. Kostyuchenko, B. S. Mityagin, DAN, 131, No. 1 (1960).
6. I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 3, Moscow, 1958.

* The growth of the kernel \(K\) must be somewhat less than the growth of \(\exp\{|x_1|^{r'_1}+|y_1|^{r'_1}+\cdots\}\); instead of subtracting \(\varepsilon\), one could have given finer estimates.