MATHEMATICS

I. A. EZROKHI

ON FUNCTIONALS IN THE SPACES \(C_{s_1\ldots s_n}\) AND \(L^p_{s_1\ldots s_n}\), ANNIHILATING ON GENERALIZED POLYNOMIALS OF MANY VARIABLES

(Presented by Academician V. I. Smirnov, 14 VI 1957)

Let \(V(f)\) be the remainder term of an approximation formula, linear \((^{1})\) on some Banach space \(E\) and exact on some finite-dimensional subspace \(\Omega\). Then \(V(f)\) is linear on \(E\) and annihilates on \(\Omega\).

A number of works \((^{3-10})\) are devoted to a convenient representation of \(V(f)\), both on \(E\) and on its subspaces, in the case when \(\Omega\) is the set of polynomials of degree not exceeding a certain degree.

In the present note, which generalizes the results of E. Ya. Remez \((^{3})\) and of the author \((^{9})\), results are given on the representation of \(V(f)\) in the case when \(E=C_{s_1\ldots s_n}\bigl(L^p_{s_1\ldots s_n}\bigr)\) (for the notation and definitions see \((^{9})\)), and \(\Omega\) is a certain collection of generalized polynomials.

Definition 1. Suppose \(n\) systems of linearly independent functions \(\{u_{i,k_i}(x_i)\}\subset C_{2s_i-1}\), \(k_i=0,1,\ldots,s_i-1\), \(i=1,\ldots,n\), are chosen. Then by a generalized polynomial with respect to \(x_i\) of rank not exceeding \(s_i-1\) we shall mean the function

\[ \omega_{\mu_i}(x_1,\ldots,x_n) = \sum_{k_i=0}^{\mu_i-1} c^i_{k_i}(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)\, u_{i,k_i}(x_i) \quad(\mu_i\le s_i), \]

where all \(c^i_{k_i}\) are continuous.

Definition 2. By \(W_{\mu_i}=W(u_{i,0},\ldots,u_{i,\mu_i-1})\) we shall mean the Wronskian determinant formed from the functions \(u_{i,0},\ldots,u_{i,\mu_i-1}\), and we shall write its nonvanishing on \([a_i,b_i]\) as follows: \(W_{\mu_i}\ne 0\).

Definition 3. We set

\[ D_{\mu_i}(f)=D_{\mu_i,x_i}(f,x_i) = \frac{ W\bigl(u_{i,0}(x_i),\ldots,u_{i,\mu_i-1}(x_i), f(x_1,\ldots,x_n)\bigr) }{ W\bigl(u_{i,0}(x_i),\ldots,u_{i,\mu_i-1}(x_i)\bigr) }. \]

Definition 4. Henceforth \(V_i(f)\) is a functional annihilating on the collection \(\{\omega_{\mu_i}\}\) \((\mu\le s_i)\), and \(V(f)\) is a functional annihilating on the collection of generalized polynomials \(\{\omega_{\mu_1\ldots\mu_n}\}\) of rank not exceeding \(s_i-1\) \((\mu_i\le s_i)\) with respect to \(x_i\), \(i=1,\ldots,n\).

Theorem 1. Suppose \(V\in(C_{s_1\ldots s_n})^*\) (i.e., \(V(f)\) is linear on \(C_{s_1\ldots s_n}\)), respectively \((L^p_{s_1\ldots s_n})^*\). Then, if for every \(i\) \((i=1,\ldots,n)\) \(W_{s_i}\ne0\), then

\[ V(f)=\sum_{i=1}^{n} V_i(f), \]

where \(V_i\in(C_{s_i})^*\), respectively \((L^p_{s_i})^*\).

Theorem 2. Suppose \(V_i\in(C_{s_i})^*\) and \(W_{s_i}\ne0\). Then

\[ V_i(f)=\int_{K_n}\cdots\int D_{s_i}(f)\,d^n g_{s_i}(x_1,\ldots,x_n); \tag{1} \]

\[ g_{s_i}(\bar x_1,\ldots,\bar x_n)=\bar V_i(\gamma_{s_i,\bar x_1\ldots \bar x_n}) =\lim_{m\to\infty}V_i\left[\int_{b_i}^{x_i} \theta_{\bar x_i}^m(z)H_{s_i}(x_i,z)\,dz\cdot \prod_{j\ne i}\theta_{\bar x_j}^m(x_j)\right]^*, \]

where \(H_{s_i}(x_i,z)\) is the Cauchy function \((^2)\) of the differential equation \(D_{s_i}(f)=0\),

\[ \theta_{\bar x_i}^m(x_i)= \begin{cases} 1, & \text{if } x_i\leqslant \bar x_i,\ a_i<\bar x_i,\\ 0, & \text{if } x_i\geqslant \bar x_i+\dfrac1m \text{ or } \bar x_i=a_i,\\ 1-m(x_i-\bar x_i), & \text{if } \bar x_i<x_i<\bar x_i+\dfrac1m\quad (i=1,\ldots,n), \end{cases} \]

and the function \(g_{s_i}(x_1,\ldots,x_n)\) has the same properties as \(g_i(x_1,\ldots,x_n)\) in Theorem 1 \((^9)\).

Theorem 3. Let \(V_i\in (L_{s_i}^p)^*\) and \(W_{s_i}\ne 0\). Then

\[ V_i(f)=\int\cdots\int_{K_{n-1}}\int_{a_i}^{b_i} D_{s_i}(f)\,dx_{j\ne i}^{\,n-1}\,\beta_{s_i}(x_1,\ldots,x_n)\,dx_i, \tag{2} \]

where \({}^{**}\)

\[ \beta_{s_i}(\bar x_1,\ldots,\bar x_n) =\frac{\partial}{\partial x_i}\bar V_i(\gamma_{s_i,\bar x_1\ldots \bar x_n}) \]

and it has the same properties as \(\beta_i(x_1,\ldots,x_n)\) in Theorem 2 \((^9)\).

Theorem 4. Let \(F\in (C_{\mu_i})^*\). Then

\[ F(f)=\sum_{j_i=0}^{\mu_i-1}\int\cdots\int_{K_{n-1}} \frac{\partial^{j_i} f(x_1,\ldots,x_{i-1},b_i,x_{i+1},\ldots,x_n)} {\partial x_i^{j_i}} \times \]

\[ \times\, dx_{j\ne i}^{\,n-1} g_{j_i}(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) +\int\cdots\int_{K_n}\frac{\partial^{\mu_i}f}{\partial x_i^{\mu_i}}\,d^n g_{\mu_i}(x_1,\ldots,x_n), \]

where

\[ g_{j_i}(\ldots,\bar x_{i-1},\bar x_{i+1},\ldots) = \bar F(\psi_{j_i,\bar x_1,\ldots,\bar x_{i-1},b_i,\bar x_{i+1},\ldots,\bar x_n}) \quad (j_i=0,\ldots,\mu_i-1), \]

\[ g_{\mu_i}(\bar x_1,\ldots,\bar x_n)=\bar F(\psi_{\mu_i,\bar x_1,\ldots,\bar x_n}) \]

and, for example \((^9)\),

\[ \bar F(\psi_{\mu_i,\bar x_1\ldots \bar x_n}) = \lim_{m\to\infty}F\left( \int_{b_i}^{x_i}\theta_{\bar x_i}^m(z_i) \frac{(x_i-z_i)^{\mu_i-1}}{(\mu_i-1)!}\,dz_i \prod_{j\ne i}\theta_{\bar x_j}^m(x_j) \right), \]

with all functions \(g\) of bounded variation in all variables and continuous from the right inside \(K_n\) \((^{7,8})\).

Let us now consider the differential complexes \((^{3,7,8})\)

\[ (f,\bar f_{x_i}^{(\mu_i)})_{\mu_i} = \bigl(f(x_1,\ldots,x_n),\, f_{x_i}^{(j_i)}(x_1,\ldots,x_{i-1},b_i,x_{i+1},\ldots,x_n), \]

\[ 0\leqslant j_i\leqslant \mu_i-1,\, \bar f_{x_i}^{(\mu_i)}(x_1,\ldots,x_n)\bigr), \]

* We do not dwell on the fact that the values of the function \(g_{s_i}(x_1,\ldots,x_n)\) may be changed on a countable set of points in each variable inside \(K_n\), without thereby changing the functional. Henceforth this equality serves as the definition of \(g_{s_i}\).

** Obviously, the values of the function \(\beta_{s_i}\) may be changed on a set of measure zero with respect to \(x_i\), without thereby changing the functional (see, moreover, the preceding footnote). Henceforth this equality serves as the definition of \(\beta_{s_i}\).

terms of which are connected by the relation:

\[ f(x_1,\ldots,x_n)= \sum_{j_i=0}^{\mu_i-1} f_{x_i}^{(j_i)}(\ldots x_{i-1}, b_i, x_{i+1},\ldots) \frac{(x_i-b_i)^j}{j!} + \]

\[ +\int_{b_i}^{x_i}\bar f_{x_i}^{(\mu_i)}(\ldots x_{i-1}, z_i, x_{i+1},\ldots) \frac{(x_i-z_i)^{\mu_i-1}}{(\mu_i-1)!}\,dz_i, \]

where the integral is understood in the sense of Lebesgue.

Let \(C_{\mu_i}''\) be the space of complexes for which \(\bar f_{x_i}^{(\mu_i)}\) are Borel-measurable \((^7)\), with norm equal to the greatest of the exact upper bounds of the absolute values of the members of the complex. Then we extend \(F\in(C_{\mu_i})^*\) to \(C_{\mu_i}''\) as follows:

\[ F[(f,\bar f_{x_i}^{(\mu_i)})_{\mu_i}] = \sum_{j_i=0}^{\mu_i-1}\int_{K_{n-1}}\cdots\int f_{x_i}^{(j_i)}\,d_{x_j\ne i}^{\,n-1}g_{j_i} + \int_{K_n}\cdots\int \bar f_{x_i}^{(\mu_i)}\,d^n g_{\mu_i}, \]

where the integrals from Theorem 4 are now understood in the Stieltjes–Lebesgue sense.

Theorem 5. Let \(V_i\in(C_{s_i-k})^*\) \((1\le k\le s_i)\) and \(W_{\mu_i}\ne0\). Then for \(f\in L_{\mu_i}^p\) \((s_i-k<\mu_i\le s_i)\), \(V_i(f)\) has the form (2), and for \(f\in C_{\mu_i}\) also the form (1), everywhere with replacement of the index \(s_i\) by \(\mu_i\). Moreover, for each \(m\) \((1\le m<\mu_i-s_i+k)\) everywhere, and for \(m=\mu_i-s_i+k\), apart from all possible exclusions with respect to \(x_i\),

\[ \frac{\partial^{m-1}\beta_{\mu_i}(\bar x_1,\ldots,\bar x_n)} {\partial\bar x_i^{\,m-1}} = -\,V_i\bigl[(\lambda_{\mu_i-m,\bar x_1\ldots\bar x_n}^{(\mu_i)}, \lambda_{0,\bar x_1\ldots\bar x_n}^{(\mu_i)})_{\mu_i-m}\bigr], \]

where the right-hand side is equal to zero for \(\bar x_i=a_i\), \(\bar x_i=b_i\) and is continuous on the right inside \(K_n\), while

\[ g_{\mu_i}=\int_{a_i}^{x_i}\beta_{\mu_i}\,dx_i \]

and

\[ \lambda_{\mu_i-m,\bar x_1\ldots\bar x_n}^{(\mu_i)}(x_1,\ldots,x_n) = \prod_{j=1}^{n}\theta_{\bar x_j}(x_j) \frac{\partial^{m-1}H_{\mu_i}(x_i,\bar x_i)} {\partial\bar x_i^{\,m-1}}, \]

\[ \lambda_{0,\bar x_1\ldots\bar x_n}^{(\mu_i)}(x_1,\ldots,x_n) = \prod_{j=1}^{n}\theta_{\bar x_j}(x_j) \frac{\partial^{\mu_i-1}H_{\mu_i}(x_i,\bar x_i)} {\partial x_i^{\,\mu_i-m}\partial\bar x_i^{\,m-1}}. \]

Theorem 6. If \(V_i\in(C_{s_i-\mu_i})^*\), \(s_i\ge\mu_i>k\ge1\), \(W_{s_i}\ne0\), \(W_{s_i-k}\ne0\), then

\[ \beta_{s_i}(\bar x_1,\ldots,\bar x_n) = -\int_{a_i}^{x_i} H_{s_i,s_i-k}(z_i,\bar x_i) \beta_{s_i-k}(\bar x_1,\ldots,z_i,\ldots,\bar x_n)\,dz_i, \]

where \(H_{s_i,s_i-k}(x_i,\bar x_i)\) is the Cauchy function of the differential operator \(Q_{s_i,s_i-k}\),

\[ D_{s_i}(f)=Q_{s_i,s_i-k}(D_{s_i-k}(f)). \]

The last two theorems also make it possible to judge the number of sign changes of the functions \(\beta_{\mu_i}'\) \((^4,^8,^9)\).

It remains to note that, under the condition \(u_{i,0}=1\) \((i=1,\ldots,n)\), there hold assertions analogous both in form and in proof

all the results established in (⁹) (pp. 399–413), with the corresponding replacement of words and symbols: “polynomial,” “degree,” \(\varphi_{n-1,\bar x}(x)\), \(\partial^\mu f/\partial x^\mu\) by “generalized polynomial,” “rank,” \(\lambda^{(\mu)}_{\mu-1,\bar x}(x)\), \(D_\mu(f)\) (if \(W_\mu \ne 0\)).

Example. Suppose a cubature formula is given

\[ \frac{1}{\pi^2}\int_{-\pi/2}^{\pi/2}\int_{-\pi/2}^{\pi/2} f(x_1,x_2)\,dx_1\,dx_2 -\sum_{k=1}^{9} A_k f(x_1^k,x_2^k)=V(f), \]

\[ A_1=A_2=A_3=A_4=\left(\frac12-\frac1\pi\right)^2, \]

\[ A_5=A_6=A_7=A_8=\frac{2\pi-4}{\pi^2},\qquad A_9=\frac{12-4\pi}{\pi^2}, \]

\[ -\,x_1^1=x_1^2=x_1^3=-\,x_1^4=-\,x_2^1=-\,x_2^2=x_2^3=x_2^4=\frac{\pi}{2}, \]

\[ -\,x_1^5=x_1^7=-\,x_2^6=x_2^8=\frac{\pi}{3}, \]

\[ x_1^6=x_1^8=x_1^9=x_2^5=x_2^7=x_2^9=0. \]

It is exact for generalized polynomials of rank not higher than 3 in each variable, where for \(i=1,2\)

\[ u_{i,0}(x_i)\equiv 1,\qquad u_{i,1}(x_i)=x_i,\qquad u_{i,2}(x_i)=\cos x_i,\qquad u_{i,3}(x_i)=\sin x_i. \]

Then for \(f\in C_{4,4}\)

\[ V(f)=\frac{1}{144}(105\pi^2-520\pi+644) \left\{\left(\frac{\partial^4}{\partial x^4}+\frac{\partial^2}{\partial x^2}\right)f(\xi_1,\eta_1) +\left(\frac{\partial^4}{\partial y^4}+\frac{\partial^2}{\partial y^2}\right)f(\xi_2,\eta_2)\right\} \]
\[ -\frac{1}{144}(117\pi^2-560\pi+652) \left\{\left(\frac{\partial^4}{\partial x^4}+\frac{\partial^2}{\partial x^2}\right)f(\xi_3,\eta_3) +\left(\frac{\partial^4}{\partial y^4}+\frac{\partial^2}{\partial y^2}\right)f(\xi_4,\eta_4)\right\}. \]

Ukrainian Agricultural Academy

Received
12 VI 1957

REFERENCES

¹ S. S. Banach, Course of Functional Analysis. Radianska shkola, 1946.
² E. Goursat, Course of Mathematical Analysis, 2, part 2, 1933.
³ E. Ya. Remez, Tr. Inst. Mat. AN USSR, 3, Kiev (1939).
⁴ E. Ya. Remez, Tr. Inst. Mat. AN USSR, 4, Kiev (1940).
⁵ A. Sard, Acta Math., 84, 319 (1951).
⁶ A. Sard, Proc. Internat. Congr. Math. Amsterdam, 2, 1954.
⁷ T. G. Ezrokhi, Nauch. zap. Kievsk. ped. inst., 16, No. 5 (1954).
⁸ T. G. Ezrokhi, I. A. Ezrokhi, Izv. Kievsk. politekhn. inst., 19 (1956).
⁹ I. A. Ezrokhi, Matem. sborn., 38 (80), 4 (1956).
¹⁰ S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 16, No. 2, 181 (1952).