MATHEMATICS

O. D. GABISONIYA

ESTIMATION OF THE COMPLETE BEST APPROXIMATION BY PARTIAL BEST APPROXIMATIONS OF FUNCTIONS OF SEVERAL VARIABLES

(Presented by Academician S. N. Bernstein, 8 VI 1963)

Let \(L_p\) \((1 \le p \le \infty)\) be the space of all functions \(f(x_1,\ldots,x_k)\) of period \(2\pi\) in each of the variables \(x_i\) \((i=1,2,\ldots,k)\), whose \(p\)-th power of the modulus is integrable on the \(k\)-dimensional cube of periods. Denote by
\(E^{(p)}_{n_1,\ldots,n_k}(f)\) the complete best approximation \((^1)\) of the function \(f\) by trigonometric polynomials of order \(\le n_i\), respectively in the variables \(x_i\) \((i=1,2,\ldots,k)\), and by \(E^{(p)}_{n_1,\ldots,n_r,\infty}(f)\) the partial best approximation by trigonometric polynomials of order \(n_i\), respectively in the variables \(x_i\) \((i=1,2,\ldots,r;\ r<k)\), with coefficients depending on the variables \(x_i\) \((i=r+1,\ldots,k)\) and belonging to the class \(L_p\) \((1\le p\le \infty)\).

We shall say that a function \(\varphi(x_1,\ldots,x_r)\) of the class \(L_p\) \((1\le p\le \infty)\) belongs to the set \(R_{1,\ldots,r}(f)\), if
\(E^{(p)}_{n_i,\infty}(\varphi)\le E^{(p)}_{n_i,\infty}(f)\) for \(i=1,2,\ldots,r;\ n_i=0,1,2,\ldots\). Put

\[ \bar E^{(p)}_{n_1,\ldots,n_r,\infty}(f) = \inf_{T_{n_1,\ldots,n_r}(x_1,\ldots,x_k)} \left\|f(x_1,\ldots,x_k)-T_{n_1,\ldots,n_r}(x_1,\ldots,x_k)\right\|_{L_p}, \tag{1} \]

where \(T_{n_1,\ldots,n_r}(x_1,\ldots,x_k)\) is a trigonometric polynomial of degree \(n_i\) in the variables \(x_i\) \((i=1,2,\ldots,r)\), with coefficients depending on the variables \(x_i\) \((i=r+1,\ldots,k)\) and belonging to the class \(R_{r+1,\ldots,k}(f)\).

If in this definition \(r=k\), then instead of \(\bar E^{(p)}_{n_1,\ldots,n_k,\infty}(f)\) we shall write
\(\bar E^{(p)}_{n_1,\ldots,n_k}(f)\).

Theorem 1. If \(f\in L_p\) \((1\le p\le \infty)\), then

\[ E^{(p)}_{n_1,\ldots,n_r,\infty}(f) = \bar E^{(p)}_{n_1,\ldots,n_r,\infty}(f). \]

Proof. For \(r=k\), \(R_{r+1,\ldots,k}(f)\) will consist only of constants, and therefore

\[ E^{(p)}_{n_1,\ldots,n_k}(f) = \bar E^{(p)}_{n_1,\ldots,n_k}(f). \tag{1'} \]

Consider the case \(r<k\).

By the definition of \(E^{(p)}_{n_1,\ldots,n_r,\infty}(f)\), for any \(\varepsilon>0\) there exists a trigonometric polynomial
\(T^{(\varepsilon)}_{n_1,\ldots,n_r}(x_1,\ldots,x_k)\) of degree \(n_i\) in the variables \(x_i\) \((i=1,\ldots,r)\), with coefficients depending on \(x_i\) \((i=r+1,\ldots,k)\) and belonging to the class \(L_p\) \((1\le p\le \infty)\), such that

\[ \left\|f(x_1,\ldots,x_k)-T^{(\varepsilon)}_{n_1,\ldots,n_r}(x_1,\ldots,x_k)\right\|_{L_p} \le E^{(p)}_{n_1,\ldots,n_r,\infty}(f)+\varepsilon . \]

Moreover, there is such a natural number \(N(\varepsilon)\) and such a polynomial
\(Q_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\) of degree \(n_i\) in \(x_i\) \((i=1,2,\ldots,k)\), that

\[ \left\|T^{(\varepsilon)}_{n_1,\ldots,n_r}(x_1,\ldots,x_k) - Q_{n_1,\ldots,n_k}(x_1,\ldots,x_k)\right\|_{L_p} <\varepsilon \quad \text{for } n_i>N(\varepsilon) \]
\[ (i=r+1,\ldots,k). \]

Consequently,

\[ \overline{E}^{(p)}_{n_1,\ldots,n_r,\infty}(f) \leq \overline{E}^{(p)}_{n_1,\ldots,n_k}(f) = E^{(p)}_{n_1,\ldots,n_k}(f) \leq \|f-Q_{n_1,\ldots,n_k}\|_{L_p} \leq \]

\[ \leq \|f-T^{(\varepsilon)}_{n_1,\ldots,n_r}\|_{L_p} +\|T^{(\varepsilon)}_{n_1,\ldots,n_r}-Q_{n_1,\ldots,n_k}\|_{L_p} \leq E^{(p)}_{n_1,\ldots,n_r,\infty}(f)+2\varepsilon. \]

Hence, by the arbitrary smallness of \(\varepsilon\),

\[ \overline{E}^{(p)}_{n_1,\ldots,n_r,\infty}(f) \leq E^{(p)}_{n_1,\ldots,n_r,\infty}(f). \tag{2} \]

On the other hand, evidently:

\[ E^{(p)}_{n_1,\ldots,n_r,\infty}(f) \leq \overline{E}^{(p)}_{n_1,\ldots,n_r,\infty}(f). \tag{3} \]

Therefore, by (2) and (3), we obtain (1). The theorem is proved.

Theorem 2. If \(n_i \geq m_i\) \((i=1,2,\ldots,r)\), then

\[ E^{(p)}_{n_1,\ldots,n_r,n_{r+1},\infty}(T_{m_1,\ldots,m_r}) = E^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r}), \tag{4} \]

where \(T_{m_1,\ldots,m_r}(x_1,\ldots,x_k)\) is a polynomial of degree \(m_i\), respectively, in the variables \(x_i\) \((i=1,2,\ldots,r)\), with coefficients depending on the variables \(x_i\) \((i=r+1,\ldots,k)\) and belonging to the class \(L_p\) \((1\leq p\leq \infty)\).

Proof. In view of equality (1), it is enough to prove

\[ \overline{E}^{(p)}_{n_1,\ldots,n_r,n_{r+1},\infty}(T_{m_1,\ldots,m_r}) = \overline{E}^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r}). \tag{5} \]

Obviously, by the definition of \(\overline{E}^{(p)}_{n_{r+1},\infty}(f)\), for the function \(T_{m_1,\ldots,m_r}(x_1,\ldots,x_k)\) there exists a polynomial
\(T^{(\varepsilon)}_{n_{r+1}}(T_{m_1,\ldots,m_r};x_1,\ldots,x_k)\) of degree \(n_{r+1}\) in \(x_{r+1}\), with coefficients belonging to
\(R_{1,\ldots,r,r+2,\ldots,k}(T_{m_1,\ldots,m_r})\), such that

\[ \|T_{m_1,\ldots,m_r}-T^{(\varepsilon)}_{n_{r+1}}(T_{m_1,\ldots,m_r})\|_{L_p} \leq \overline{E}^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r})+\varepsilon. \tag{6} \]

It is not difficult to see that the set
\(R_{1,\ldots,r,r+2,\ldots,k}(T_{m_1,\ldots,m_r})\) will be a subset of all polynomials of degree \(m_i\) in the variables \(x_i\) \((i=1,2,\ldots,r)\) with coefficients depending on the remaining variables \(x_i\) \((i=r+1,\ldots,k)\) and belonging to the class \(L_p\) \((1\leq p\leq \infty)\). The coefficients of the polynomial
\(T^{(\varepsilon)}_{n_{r+1}}(T_{m_1,\ldots,m_r};x_1,\ldots,x_k)\) belong to
\(R_{1,\ldots,r,r+2,\ldots,k}(T_{m_1,\ldots,m_r})\), and therefore they will be polynomials of degree \(m_i\) in \(x_i\) \((i=1,2,\ldots,r)\).

Consequently, taking (6) into account, we may write

\[ \overline{E}^{(p)}_{n_1,\ldots,n_r,n_{r+1},\infty}(T_{m_1,\ldots,m_r}) \leq \|T_{m_1,\ldots,m_r}-T^{(\varepsilon)}_{n_{r+1}}(T_{m_1,\ldots,m_r})\|_{L_p} \leq \]

\[ \leq \overline{E}^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r})+\varepsilon. \]

Hence, by the arbitrary smallness of \(\varepsilon\),

\[ \overline{E}^{(p)}_{n_1,\ldots,n_r,n_{r+1},\infty}(T_{m_1,\ldots,m_r}) \leq \overline{E}^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r}). \tag{7} \]

On the other hand,

\[ \overline{E}^{(p)}_{n_{r+1},\infty}(T_{m_1,\ldots,m_r}) \leq \overline{E}^{(p)}_{n_1,\ldots,n_r,n_{r+1},\infty}(T_{m_1,\ldots,m_r}). \tag{8} \]

From (7) and (8) follows (5), and from (5), in turn, follows (4). The theorem is proved.

Theorem 3. If \(f\in L_p\) \((1\leq p\leq \infty)\), then

\[ E^{(p)}_{n_1,\ldots,n_k}(f) \leq C \sum_{i=1}^{k} E^{(p)}_{n_i,\infty}(f), \tag{9} \]

where \(C\) is a constant \(\leq \dfrac{1}{k}(2^k-1)\).

An inequality of the form (9) for \(1<p<\infty\) was proved by M. F. Timan \({}^{1}\) with an unknown constant depending on \(p\), and for \(p=2,\ k=2\) by S. N. Bernstein \({}^{2}\) with \(C=1\).

Proof. Let \(T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(\varepsilon)}(f;x_1,\ldots,x_k)\) be a trigonometric polynomial of degree \(n_{\nu_i}\), respectively in the variables \(x_{\nu_i}\) \((\nu_s=1,2,\ldots,k;\ s=1,2,\ldots,k;\ i=1,2,\ldots,k-1)\), with coefficients depending on \(x_{\nu_k}\) and belonging to \(L_p\) \((1\le p\le \infty)\), such that

\[ E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f) \le \left\|f-T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(\varepsilon)}(f)\right\|_{L_p} \le E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f)+\varepsilon . \]

Then, taking equality (4) into account, we can write

\[ \begin{aligned} E_{n_1,\ldots,n_k}^{(p)}(f) &\le E_{n_1,\ldots,n_k}^{(p)} \left[f-T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}(f)\right] +E_{n_1,\ldots,n_k}^{(p)} \left[T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(\varepsilon)}(f)\right] \le \\ &\le E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f)+\varepsilon +E_{n_{\nu_k},\infty}^{(p)} \left[T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(\varepsilon)}(f)\right] \le \\ &\le E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f) +E_{n_{\nu_k},\infty}^{(p)} \left[T_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(\varepsilon)}(f)-f\right] +E_{n_{\nu_k},\infty}^{(p)}(f)+\varepsilon \le \\ &\le 2E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f) +E_{n_{\nu_k},\infty}^{(p)}(f)+2\varepsilon . \end{aligned} \]

Hence, by the arbitrariness of the small quantity \(\varepsilon\), we obtain

\[ E_{n_1,\ldots,n_k}^{(p)}(f) \le 2E_{n_{\nu_1},\ldots,n_{\nu_{k-1}}}^{(p)}(f) + E_{n_{\nu_k},\infty}^{(p)}(f). \]

In exactly the same way one can show that

\[ E_{n_{\nu_1},\ldots,n_{\nu_{k-1}},\infty}^{(p)}(f) \le 2E_{n_{\nu_1},\ldots,n_{\nu_{k-2}},\infty}^{(p)}(f) + E_{n_{\nu_{k-1}},\infty}^{(p)}(f). \]

Continuing this process, it is easy to conclude that

\[ E_{n_1,\ldots,n_k}^{(p)}(f) \le \sum_{i=1}^{k} 2^{k-1}E_{n_{\nu_i},\infty}^{(p)}(f). \tag{10} \]

\(\nu_i\) can take \(k\) different values for each \(i\); therefore, from (10) we obtain another \(k-1\) inequalities of the form (10). Adding all these inequalities and dividing by \(k\), we shall have

\[ E_{n_1,\ldots,n_k}^{(p)}(f) \le \frac{1}{k}\left(2^{k-1}+2^{k-2}+\cdots+1\right) \sum_{i=1}^{k} E_{n_i,\infty}^{(p)}(f) = \frac{2^k-1}{k} \sum_{i=1}^{k} E_{n_i,\infty}^{(p)}(f). \]

The theorem is proved.

The question of whether inequality (9) holds for discontinuous functions when \(p=\infty\) was posed by S. N. Bernstein \({}^{2}\). Theorem 3 gives an affirmative answer to this question.

Sukhumi State Pedagogical Institute
named after A. M. Gorky

Received
22 V 1963

References

\({}^{1}\) M. F. Timan, DAN, 112, No. 1, 24 (1957).
\({}^{2}\) S. N. Bernstein, Collected Works, 2, Moscow, 1954, p. 240.