UDC 517.514

MATHEMATICS

B. A. Vostretsov, A. V. Ignat’eva

ON THE EXISTENCE OF A BEST APPROXIMATION OF FUNCTIONS BY SUMS OF A FINITE NUMBER OF PLANE WAVES OF GIVEN DIRECTIONS IN THE METRIC \(L_p\)

(Presented by Academician I. G. Petrovskii on 29 XII 1966)

Let \(O\) be an open set in the space \(R_n=\{\mathbf{x}=(x_1,\ldots,x_n)\}\); let \(\varphi(\mathbf{x})\) be a finite basic function in \(O\); let \(K_O\) be the space of all \(\varphi(\mathbf{x})\); and let \(\{f=(f,\varphi(\mathbf{x}))\}\) be the space of generalized functions on \(K_O\).

We denote by \(L_p(D)\), \(D\subset R_n\) a bounded domain, the space of functions \(f(\mathbf{x})\) for which \(\|f(\mathbf{x})\|_p<\infty\), where

\[ \|f(\mathbf{x})\|_p=\left(\int_D |f(\mathbf{x})|^p\,dx\right)^{1/p}, \]

\[ 1<p<\infty,\qquad |f(\mathbf{x})|_\infty= \inf_{\substack{M\subset D,\,\operatorname{mes} M=0}} \left\{\sup_{\mathbf{x}\in D-M}|f(\mathbf{x})|\right\}. \]

Taking as given \(k\) distinct fixed points \(\mathbf{a}_i\), \(i=1,\ldots,k\), \(\mathbf{a}_i\in \Pi_{n-1}\), \(|\mathbf{a}_i|=1\), where \(\Pi_{n-1}=\{\mathbf{a}=(a_1,\ldots,a_n)\}\) is the real projective space whose points are determined by homogeneous coordinates \(a_1,\ldots,a_n\), consider on each interval \(\Lambda_i=(\alpha_i,\beta_i)=\{z_i:\ z_i=\mathbf{a}_i\mathbf{x}=a_{i1}x_1+\cdots+a_{in}x_n,\ \mathbf{x}\in D\}\) the class \(H_i\) of all functions \(h_i(z_i)\) such that \(h_i(\mathbf{a}_i\mathbf{x})\in L_p^{\mathrm{loc}}(D)\). The symbol \(L_p^{\mathrm{loc}}(D)\) denotes the space of all functions each of which belongs to \(L_p(D')\) on any set \(D'\) compact in \(D\).

Set

\[ \rho_k=\inf\left\|f(\mathbf{x})-\sum_{i=1}^{k}h_i(\mathbf{a}_i\mathbf{x})\right\|_p, \tag{1} \]

where the infimum is taken over all \(h_i\in H_i\), \(\sum_{i=1}^{k}h_i(\mathbf{a}_i\mathbf{x})\in L_p(D)\).

In this paper sufficient conditions are established for the existence of a system of functions \(h_i^0(z_i)\), \(h_i^0(\mathbf{a}_i\mathbf{x})\in L_p^{\mathrm{loc}}(D)\), \(i=1,\ldots,k\), for which the infimum (1) is attained; conditions are given under which \(h_i^0(\mathbf{a}_i\mathbf{x})\in L_p(D)\). At the same time it is shown that the quantity \(\rho_k\) is equal to the supremum of the moduli of the values of the generalized function

\[ f=\int_D f(\mathbf{x})\varphi(\mathbf{x})\,dx \]

on a certain subset \(K_D\).

Let \(\mathbf{a}\in \Pi_{n-1}\), \(|\mathbf{a}|=1\), \(\Delta=\{\mathbf{x}:\alpha<\mathbf{a}\mathbf{x}<\beta\}\), where \(\alpha\) and \(\beta\) \((-\infty\le \alpha<\beta\le +\infty)\) are fixed constants. A generalized function \((f,\varphi(\mathbf{x}))\), \(\varphi(\mathbf{x})\in K_\Delta\), will be called a generalized plane wave of direction \(\mathbf{a}\) if, for every vector \(\mathbf{b}\), \(\mathbf{a}\mathbf{b}=0\), and for every function \(\varphi(\mathbf{x})\in K_\Delta\), the equality

\[ (f,\varphi(\mathbf{x}+\mathbf{b}))=(f,\varphi(\mathbf{x})) \]

holds.

Suppose further that \(h=(h,\psi(z))\) is a generalized function defined on the space \(K_{(\alpha,\beta)}\) of basic functions of the variable \(z\).

We shall call the superposition of \(h\) and the form \(\mathbf{a}\mathbf{x}\) a plane wave of direction \(\mathbf{a}\), defined by the equality

\[ f=\langle f,\varphi(\mathbf{x})\rangle =\left\langle h,\psi(z)=\int_{\mathbf{a}\mathbf{x}=z}\varphi(\mathbf{x})\,ds\right\rangle, \]

where \(\varphi(\mathbf{x})\in K_\Delta\), and \(ds\) is the area element of the hyperplane \(\mathbf{a}\mathbf{x}=z\).

Lemma 1. Every plane wave \(f\) of direction \(\mathbf{a}\) is the superposition of some generalized function \(h=\langle h,\psi(z)\rangle\), \(\psi(z)\in K_{(\alpha,\beta)}\), and the form \(\mathbf{a}\mathbf{x}\).

In proving this lemma one uses

Lemma 2. Let \(\mathbf{a},\mathbf{b}_1,\ldots,\mathbf{b}_{n-1}\), where \(\mathbf{a},\mathbf{b}_i\in R_n\), be an orthonormal frame. If, for \(\varphi(\mathbf{x})\in K_\Delta\),

\[ \int_{\mathbf{a}\mathbf{x}=z}\varphi(\mathbf{x})\,ds=0 \]

for every \(z\), then

\[ \varphi(\mathbf{x})=\partial F_1(\mathbf{x})/\partial b_1+\cdots+ \partial F_{n-1}(\mathbf{x})/\partial b_{n-1}, \qquad F_i(\mathbf{x})\in K_\Delta . \]

By Lemma 1, every generalized plane wave \(f\) of direction \(\mathbf{a}\), by analogy with ordinary functions, can be written in the form \(f=h(\mathbf{a}\mathbf{x})\), and

\[ \partial f/\partial x_i=\partial h(\mathbf{a}\mathbf{x})/\partial x_i =h'_{\mathbf{a}\mathbf{x}}(\mathbf{a}\mathbf{x})a_i,\qquad i=1,\ldots,n . \]

The proof of the existence of an extremal system of functions
\(h_i^0(\mathbf{a}_i\mathbf{x})\), \(i=1,\ldots,k\), is based on one lemma of functional analysis.

Let \(E=\{e\}\) be a linear space of Banach type; \(E^*\) the conjugate space of all linear functionals \(l(e)\) on \(E\); \(\Phi\subset E\) an arbitrary linear manifold; \(U\subset E^*\) the set of linear functionals \(U(e)\) that vanish on \(\Phi\).

Lemma 3 (S. Ya. Khavinson \((^1)\)). For an arbitrary linear functional \(l(e)\),

\[ \|l(e)\|_\Phi=\inf_{u\in U}\|l-u\|_{E^*}, \tag{2} \]

and there exists an element \(u^0\in U\) for which the lower bound in (2) is attained. Here \(\|l\|_\Phi\) is the norm of the functional \(l\) on the manifold \(\Phi\).

The subsequent arguments rest on Theorem 1, which follows directly from Lemma 3.

Given a system of equations

\[ P_i u=0,\qquad i=1,\ldots,r, \tag{3} \]

where \(P_i\) is a homogeneous polynomial in \(\partial/\partial x_1,\ldots,\partial/\partial x_n\) with real constant coefficients, and \(u\) is an unknown generalized function on \(K_D\).

Theorem 1. Let \(f(\mathbf{x})\in L_p(D)\). Denote by \(U=\{u\}\) the set of all functions \(u(\mathbf{x})\in L_p(D)\), each of which generates a regular generalized solution of system (3) on \(K_D\). Then

\[ \inf_{u\in U}\|f(\mathbf{x})-u(\mathbf{x})\|_p = \sup_{\|\widetilde{\varphi}\|_q=1} \left|\int_D f(\mathbf{x})\widetilde{\varphi}(\mathbf{x})\,d\mathbf{x}\right|, \tag{4} \]

where \(\widetilde{\varphi}(\mathbf{x})\) is an arbitrary function representable in the form

\[ \widetilde{\varphi} = P_1\varphi_1(\mathbf{x})+\cdots+P_r\varphi_r(\mathbf{x}), \qquad \varphi_i(\mathbf{x})\in K_D,\qquad 1/p+1/q=1. \]

Moreover, there exists \(u^0(\mathbf{x})\in U\) for which the lower bound in (4) is attained.

For the proof it suffices to set \(E=L_q(D)\), \(1/p+1/q=1\), \(\Phi=\{\widetilde{\varphi}(\mathbf{x})\}\). In this case \(E^*=L_p(D)\), and the set \(U\) of the lemma coinc—

coincides with the set discussed in the theorem. Now taking \(l(e)=\int_D f(x)e(x)\,dx,\ e(x)\in L_p(D)\), we obtain the required assertion. Let us note that for \(n=1,\ r=1\), system (3) reduces to the single equation \(d^m u/dz^m=0\), the set \(U\) becomes the collection of polynomials in \(z\) of degree not exceeding \(m-1\), and Theorem 1 gives the corresponding results for the approximation of functions of one variable \(z\) by polynomials, both in the metric \(L_p(\Lambda)\) and in the Chebyshev metric on \(\Lambda,\ \Lambda=\{\alpha\le z\le \beta\}\).

Now let system (3) be such that the set of forms \(\{P_1(y),\ldots,\ldots,P_r(y)\}\) in the ring of polynomials in the variables \(y_1,\ldots,y_n\) forms a basis of a homogeneous ideal belonging to the set \(\{a_1,\ldots,a_k\}\).

Theorem 2. If every section of the domain \(D\) by a hyperplane orthogonal to the direction \(a_i\) is connected, \(i=1,\ldots,k\), then, in order that the generalized function \(u=(u,\varphi(x)),\ \varphi(x)\in K_D\), be a sum of generalized plane waves of directions \(a_1,\ldots,a_k\), it is necessary and sufficient that the function \(u\) satisfy system (3).

Moreover, if \(u\) is a solution of system (3) representable in the form

\[ u=\sum_{|\gamma|\le m}\int_D f_{\gamma_1,\ldots,\gamma_n}(x)\, \frac{\partial^{|\gamma|}\varphi(x)}{\partial x_1^{\gamma_1}\cdots \partial x_n^{\gamma_n}}\,dx, \]

where \(f_{\gamma_1,\ldots,\gamma_n}(x)\in L_p^{\mathrm{loc}}(D)\), then there exist generalized functions \(h_i\),

\[ h_i=\sum_{j=1}^{m}\int_{\alpha_i}^{\beta_i} h_{ij}(z)\psi^{(j)}(z)\,dz, \]

where \(h_{ij}(z_i)\in L_p^{\mathrm{loc}}(\Lambda_i),\ \psi(z_i)\in K_{\Lambda_i}\), such that

\[ u=\sum_{i=1}^{k} h_i(a_i x), \]

and conversely.

The proof is carried out by induction on \(k\). The following lemmas are used.

Lemma 4. Let \(\Delta_1'=\{x:\ \alpha_1'<x_n<\beta_1'\}\) be an arbitrary strip which, together with its boundary, lies inside \(\Delta_1=\{x:\ \alpha_1<x_n<\beta_1\}\). Under the conditions of Theorem 2 there exist a curve \(x_i=\tau_i(x_n),\ i=1,\ldots,n-1\), where \(\tau_i\) is a polynomial, and a number \(\varepsilon>0\) such that the set of points \(x\) for which

\[ \tau_i(x_n)\le x_i\le \tau_i(x_n)+\varepsilon,\quad i=1,\ldots,n-1,\quad \alpha_1'\le x_n\le \beta_1', \]

belongs to the domain \(D\).

Lemma 5. Every function \(\varphi(x),\ \varphi(x)\in K_{\Delta_1}\), can be written in the form

\[ \varphi(x)=\varkappa_1(x_1-\tau_1(x_n))\cdots \varkappa_{n-1}(x_{n-1}-\tau_{n-1}(x_n)) \int_{-\infty}^{\infty}\varphi(x)\,dx_1\cdots dx_{n-1}+F(x), \]

where

\[ F(x)=\sum_{i=1}^{n-1}\frac{\partial F_i(x)}{\partial x_i},\qquad F_i(x)\in K_{\Delta_1'},\qquad \varkappa_i(z)\in K_{(0,\varepsilon)}, \]

\[ \int_0^\varepsilon \varkappa_i(z)\,dz=1,\qquad i=1,\ldots,n-1. \]

From Theorems 1 and 2 we obtain:

Theorem 3. Let \(f(x)\in L_p(D)\). If every section of the domain \(D\) by a hyperplane orthogonal to the vector \(a_i\) is connected, \(i=1,\ldots,k\), then

\[ \inf \left\| f(x)-\sum_{i=1}^{k} h_i(a_i x)\right\|_p = \sup_{\|\tilde{\varphi}\|_q=1} \left|\int_D f(x)\tilde{\varphi}(x)\,dx\right|, \tag{5} \]

where
\[ \widetilde{\varphi}=\sum_{i=1}^{r} P_i\varphi_i(\mathbf{x}),\qquad \varphi_i(\mathbf{x})\in K_D,\qquad q=\frac{p}{p-1}, \]
and the lower bound is taken over all \(h_i(z_i)\in H_i\),
\[ \sum_{i=1}^{k} h_i(\mathbf{a}_i\mathbf{x})\in L_p(D). \]
In this case there exist functions \(h_i^0(z_i)\in H_i\) that realize the lower bound in (5).

Each of the sets \(\Omega_1=\overline D\cap\{\mathbf{x}:\mathbf{a}\mathbf{x}=\alpha\}\) and \(\Omega_2=\overline D\cap\{\mathbf{x}:\mathbf{a}\mathbf{x}=\beta\}\), where \(\alpha=\inf_{\mathbf{x}\in D}(\mathbf{a}\mathbf{x})\), \(\beta=\sup_{\mathbf{x}\in D}(\mathbf{a}\mathbf{x})\), \(\mathbf{a}\in \Pi_{n-1}\), will be called a supporting set of the domain \(D\) for the direction \(\mathbf{a}\). Any of the supporting sets \(\Omega_{ij}\), \(j=1,2\), of the domain \(D\), satisfying the conditions of Theorem 3, for each direction \(\mathbf{a}_i\), \(i=1,\ldots,k\), is a continuum. For fixed directions \(\mathbf{a}_i\), \(i=1,\ldots,k\), a supporting set \(\Omega_{ij}\) of the domain \(D\) will be called proper in the following cases: 1) \(\Omega_{ij}\cap \Omega_{st}=0\), \(s=1,\ldots,k\); \(t=1,2\); \(s\ne i\); 2) \(\Omega_{ij}\cap\Omega_{s_0t_0}\ne0\), \(s_0\in\{1,\ldots,k\}\), \(t_0\in\{1,2\}\), \(s_0\ne i\), but the set \(\Omega_{ij}\) contains the base of a half-sphere whose interior belongs to \(D\).

Theorem 4. If the requirements of Theorem 3 are satisfied and, among the directions \(\mathbf{a}_i\), \(i=1,\ldots,k\), it is possible to choose \(k-1\) directions so that each supporting set \(\Omega_{ij}\) for every chosen direction is proper, then equality (5) holds under the condition that the lower bound is taken over all \(h_i(z_i)\), \(h_i(\mathbf{a}_i\mathbf{x})\in L_p(D)\). Moreover, there exist functions \(h_i^0(z_i)\), \(h_i^0(\mathbf{a}_i\mathbf{x})\in L_p(D)\), that realize the lower bound.

We give an example of a domain \(D\) and a system \(\mathbf{a}_1,\ldots,\mathbf{a}_k\) not satisfying the conditions of Theorem 4, for which there exists a function \(f(\mathbf{x})\), \(f(\mathbf{x})\in L_p(D)\) for some \(p\), \(1<p<\infty\), such that for it the lower bound (1) is not attained among functions \(h_i(z_i)\), \(h_i(\mathbf{a}_i\mathbf{x})\in L_p(D)\).

Let
\[ S=\sum_{\nu=1}^{\infty}\frac{1}{\nu^{1+\delta}}, \]
where \(\delta\) is a fixed number, \(0<\delta<1/2\);
\[ S_m=\sum_{\nu=1}^{m}\frac{1}{\nu^{1+\delta}}. \]
In the plane \(X_1OX_2\) take the domain \(D\) bounded by the axis \(OX_1\), the line \(x_2=x_1\), and the polygonal line with vertices \((S_m,S_{m-1})\), \(m=1,\ldots\), \(S_0=0\). Put \(k=2\), \(\mathbf{a}_1=(1,0)\), \(\mathbf{a}_2=(0,1)\). The chosen directions do not satisfy the conditions of Theorem 4. For \(1<p<1+\delta\), for the function \(f(x_1,x_2)=h_1(x_1)+h_2(x_2)\), where \(h_1(x_1)=m^{1+2\delta}\), \(S_{m-1}\le x<S_m\); \(h_2(x_2)=-m^{1+2\delta}(1-1/m^{2\delta})\), \(S_{m-1}\le x<S_m\), \(m=1,\ldots\), the lower bound (1) is not attained.

In conclusion, the authors express their warm gratitude to M. A. Kreines for his attention to the work.

Received
27 XII 1966

References

1. S. Ya. Khavinson, Matem. sborn., 36 (78), 3, 445 (1955).