# Best complete approximation

A best approximation of a function $f (x _ {1} \dots x _ {k} )$ in $k$ variables $(k \geq 2)$ by algebraic or trigonometric polynomials. Let $X$ be the space $C$ or $L _ {p}$ of functions $f (x _ {1} \dots x _ {k} )$, $2 \pi$- periodic in each variable, that are either continuous or $p$- summable ( $p \geq 1$) on the $k$- dimensional period cube with edges of length $2 \pi$.

The best complete approximation of a function $f (x _ {1} \dots x _ {k} ) \in X$ by trigonometric polynomials is the quantity

$$E _ {n _ {1} \dots n _ {k} } (f) _ {X} = \ \inf _ {T _ {n _ {1} \dots n _ {k} } } \ \| f - T _ {n _ {1} \dots n _ {k} } \| _ {X} ,$$

where the infimum is taken over all trigonometric polynomials of degree $n _ {i}$ in the variable $x _ {i}$( $1 \leq i \leq k$). Together with the best complete approximation, one also considers best partial approximations.

A best partial approximation of a function $f (x _ {1} \dots x _ {k} ) \in X$ is a best approximation by functions $T _ {n _ {\nu _ {1} } \dots n _ {\nu _ {r} } } (x _ {1} \dots x _ {k} ) \in K$ that are trigonometric polynomials of degree $n _ {\nu _ {1} } \dots n _ {\nu _ {r} }$( $1 \leq r < k$), respectively, in the fixed variables $x _ {\nu _ {1} } \dots x _ {\nu _ {r} }$ with coefficients depending on the remaining $k - r$ variables, i.e.

$$E _ {n _ {\nu _ {1} } \dots n _ {\nu _ {r} } , \infty } (f) _ {X} = \ \inf _ {T _ {n _ {\nu _ {1} } \dots n _ {\nu _ {r} } } } \ \| f - T _ {n _ {\nu _ {1} } \dots n _ {\nu _ {r} } } \| _ {X} .$$

It is obvious that

$$E _ {n _ {1} \dots n _ {r} \dots n _ {k} } (f) _ {X} \geq \ E _ {n _ {1} \dots n _ {r} , \infty } (f) _ {X} .$$

S.N. Bernstein [S.N. Bernshtein] [1] proved the following inequality for continuous functions in two variables:

$$\tag{1 } E _ {n _ {1} , n _ {2} } (f) _ {C} \leq \$$

$$\leq A \mathop{\rm ln} (2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) (E _ {n _ {1} , \infty } (f) _ {C} + E _ {n _ {2} , \infty } (f) _ {C} ),$$

where $A$ is an absolute constant. It has been shown [3] that the term $\mathop{\rm ln} (2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} )$ in inequality (1) (and in the analogous relation for the space $L _ {1}$) cannot be replaced by a factor with a slower rate of increase as $\mathop{\rm min} \{ n _ {1} , n _ {2} \} \rightarrow \infty$.

In the space $L _ {p}$( $p > 1$) one has the inequality

$$\tag{2 } E _ {n _ {1} \dots n _ {k} } (f) _ {L _ {p} } \leq \ A _ {p, k } \sum _ {i = 1 } ^ { k } E _ {n _ {i} , \infty } (f) _ {L _ {p} } ,$$

where the constant $A _ {p, k }$ depends only on $p$ and $k$.

Similar definitions yield best complete approximations and best partial approximations of functions defined on a closed bounded domain $\Omega \subset \mathbf R ^ {k}$ by algebraic polynomials, and in this case inequalities similar to (1) and (2) have been established.

#### References

 [1] S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) (In Russian) [2] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian) [3] V.N. Temlyakov, "On best approximations of functions in two variables" Dokl. Akad. Nauk SSSR , 223 : 5 (1975) pp. 1079–1082 (In Russian)