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)

References

[a1]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)