MATHEMATICS

M. F. TIMAN

ON THE RELATION BETWEEN COMPLETE AND PARTIAL BEST MEAN APPROXIMATIONS OF FUNCTIONS OF MANY VARIABLES

(Presented by Academician A. N. Kolmogorov, 7 VIII 1956)

Consider the space (L_p) ((1 \leq p < \infty)) of all measurable 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, with norm

[
|f|_{L_p}
=
\left{
\int_0^{2\pi}\cdots\int_0^{2\pi}
|f(x_1,\ldots,x_k)|^p\,dx_1\cdots dx_k
\right}^{1/p}.
]

Let

[
E_{n_1,\ldots,n_k}(f){L_p}
=
\inf_T
|f(x_1,\ldots,x_k)-T}(x_1,\ldots,x_k)|_{L_p
]

be the complete best approximation of the function (f) by trigonometric
polynomials of order (\leq n_i) in the variables (x_i) ((i=1,2,\ldots,k)).

By Fubini’s theorem, for any (r<k) the function
(f(x_1,x_2,\ldots,x_k)), as a function of the variables (x_1,\ldots,x_r),
for almost all systems ((x_{r+1},\ldots,x_k)), also belongs to the class
(L_p) together with its best approximation
(E_{n_1,\ldots,n_r}(f;x_{r+1},\ldots,x_k)) in the chosen (r) variables.
The quantity

[
E_{n_1,\ldots,n_r,\infty}(f)
=
|E_{n_1,\ldots,n_r}(f;x_{r+1},\ldots,x_k)|_{L_p}
]

may be regarded as the partial best approximation of order (n_i) with
respect to the variables (x_i) ((i=1,2,\ldots,r)). This quantity coincides
with the lower bound

[
\inf
|f(x_1,\ldots,x_k)
-
T_{n_1,\ldots,n_r}
[x_1,\ldots,x_r;\,(x_{r+1},\ldots,x_k)]|_{L_p}
]

over all possible trigonometric polynomials of order (n_i) in the variables
(x_i) ((i=1,2,\ldots,r)), whose coefficients are periodic functions
(\varphi_{i_1,\ldots,i_r}(x_{r+1},\ldots,x_k)) of period (2\pi) in each of
the variables (x_i) ((i=r+1,\ldots,k)), belonging to (L_p). Owing to this,
the inequality

[
E_{n_1,\ldots,n_k}(f) \geq E_{n_1,\ldots,n_r,\infty}(f)
]

always holds.

The following theorem complements this estimate and indicates a closer
connection between complete and partial approximations.

Theorem. For any finite (p>1) there exists a constant (C_p), independent
of the function (f), such that

[
E_{n_1,\ldots,n_k}(f){L_p}
\leq
C_p \min
\left{
E(f)},\ldots,n_{\nu_i},\infty{L_p}
+
E}},\ldots,n_{\nu_k},\infty}(f)_{L_p
\right}
\tag{1}
]

[
(\nu_m=1,2,\ldots,k;\quad m=1,2,\ldots,i).
]

In the cases (p=1), (p=\infty), the inequality

[
E_{n_1,\ldots,n_k}(f)\leq C\min{(E_{n_{\nu_1},\ldots,n_{\nu_i},\infty}(f)+E_{n_{\nu_{i+1}},\ldots,n_{\nu_k},\infty}(f))\ln n_{\nu_1}\cdots \ln n_{\nu_i}}
\tag{2}
]

[
\left(\nu_m=1,2,\ldots,k;\quad m=1,2,\ldots,i;\quad i\leq \left[\frac{k}{2}\right]\right),
]

where (C) is an absolute constant.

For continuous functions of two variables in the case of the uniform metric ((p=\infty)), inequality (2) was obtained by S. N. Bernstein ((^1)), who also indicated the special case of estimate (1), for (p=2) with constant (C_2=1), that follows from Parseval’s equality.

We shall give the proof of inequalities (1) and (2) only for the case of functions of two variables.

Proof of inequality (1). Let (T_{n_1}[x_1;(x_2)]), (T_{n_2}[(x_1);x_2]) be trigonometric polynomials realizing the partial best approximations to the function (f(x_1,x_2)), the first of order (n_1) in (x_1), the second of order (n_2) in (x_2), i.e.

[
E_{n_1,\infty}(f)=|f(x_1,x_2)-T_{n_1}[x_1;(x_2)]|,\qquad
E_{n_2,\infty}(f)=|f(x_1,x_2)-T_{n_2}[(x_1);x_2]|.
]

Denote:

[
S_{n_1}(f;x_1,x_2)=\frac{1}{\pi}\int_0^{2\pi} f(x_1+t_1,x_2)D_{n_1}(t_1)\,dt_1,
]

[
S_{n_1,n_2}(f;x_1,x_2)=\frac{1}{\pi^2}\int_0^{2\pi}\int_0^{2\pi}
f(x_1+t_1,x_2+t_2)D_{n_1}(t_1)D_{n_2}(t_2)\,dt_1dt_2,
]

where

[
D_n(t)=\frac{\sin(2n+1)\frac{t}{2}}{2\sin \frac{t}{2}}.
]

It is obvious that

[
S_{n_1,n_2}(T_{n_1};x_1,x_2)=S_{n_2}(T_{n_1};x_1,x_2)
=\frac{1}{\pi}\int_0^{2\pi}T_{n_1}[x_1,(x_2+t_2)]D_{n_2}(t_2)\,dt_2.
]

It follows from this that

[
E_{n_1,n_2}(f)\leq |f(x_1,x_2)-S_{n_2}(T_{n_1};x_1,x_2)|
\leq |f(x_1,x_2)-S_{n_2}(f;x_1,x_2)|+
]
[
+|S_{n_2}(f;x_1,x_2)-S_{n_2}(T_{n_1};x_1,x_2)|=R_1+R_2.
\tag{3}
]

To estimate each term on the right-hand side of (3), we shall use Riesz’s inequality ((^2))

[
|S_m(f)|{L_p}\leq A_p|f|\qquad (p>1).
]

Then, obviously,

[
R_2\leq A_p|f(x_1,x_2)-T_{n_1}[x_1;(x_2)]|=A_pE_{n_1,\infty}(f);
\tag{4}
]

[
R_1\leq |f(x_1,x_2)-T_{n_2}[(x_1);x_2]|+
|T_{n_2}[(x_1);x_2]-S_{n_2}(f;x_1,x_2)|=
]
[
=E_{n_2,\infty}(f)+|S_{n_2}(f-T_{n_2};x_1,x_2)|
\leq E_{n_2,\infty}(f)+A_pE_{n_2,\infty}(f).
\tag{5}
]

From (4) and (5), (1) follows.

Proof of inequality (2). Considering inequality (3) in the metric (L), we estimate (R_1) and (R_2).

Changing the order of integration and, by virtue of the periodicity of the function in each variable, we obtain:

[
R_2 \leq \int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{\pi}\int_{0}^{2\pi}
\left| f(x_1,x_2+t_2)-T_{n_1}[x_1,(x_2+t_2)] \right|
\cdot |D_{n_2}(t_2)|\,dt_2\,dx_1\,dx_2 \leq
]

[
\leq \left| f(x_1,x_2)-T_{n_1}[x_1;(x_2)] \right|
\cdot \frac{1}{\pi}\int_{0}^{2\pi}|D_{n_2}(t_2)|\,dt_2 =
]

[
= E_{n_1,\infty}(f)\cdot \frac{1}{\pi}\int_{0}^{2\pi}|D_{n_2}(t_2)|\,dt_2,
\tag{6}
]

[
R_1 \leq \left| f(x_1,x_2)-T_{n_2}[(x_1);x_2] \right|
+\left| T_{n_2}[(x_1);x_2]-S_{n_2}(f;x_1,x_2) \right| \leq
]

[
\leq E_{n_2,\infty}(f)+
\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{\pi}\int_{0}^{2\pi}
\left| f(x_1,x_2+t_2)-T_{n_2}[(x_1);x_2+t_2] \right|
\cdot |D_{n_2}(t_2)|\,dt_2\,dx_1\,dx_2 \leq
]

[
\leq E_{n_2,\infty}(f)+\frac{1}{\pi}\int_{0}^{2\pi}|D_{n_2}(t_2)|\,dt_2\cdot E_{n_2,\infty}(f).
\tag{7}
]

From (6) and (7) we obtain

[
E_{n_1,n_2}(f)\leq C{E_{n_1,\infty}(f)+E_{n_2,\infty}(f)}\cdot \ln n_2.
\tag{8}
]

Analogously one can also obtain the inequality

[
E_{n_1,n_2}(f)\leq C{E_{n_1,\infty}(f)+E_{n_2,\infty}(f)}\cdot \ln n_1.
\tag{9}
]

(8) and (9) give (2).

The same method makes it possible to prove inequalities (8) and (9) for the case of the uniform metric.

Dnepropetrovsk
Agricultural Institute

Received
3 V 1956

REFERENCES

1. S. N. Bernstein, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 24 (1951).
2. A. Zygmund, Trigonometric Series, Moscow–Leningrad, 1939.