Local approximation of functions

A measure of approximation (in particular, best approximation) of a function $ f $ on a set $ E \subset \mathbf R ^ {m} $, regarded as a function of this set. The main interest is in the behaviour of a local approximation of a function as $ \mathop{\rm mes} E \rightarrow 0 $. In certain cases it is possible to characterize the degree of smoothness of the function to be approximated in terms of a local approximation of the function. Let $ E _ {n} ( f ; ( \alpha , \beta ) ) $ be the best approximation of a function $ f \in C [ a , b ] $ by algebraic polynomials of degree $ n $ on an interval $ ( \alpha , \beta ) $, $ a \leq \alpha < \beta \leq b $. The following assertion holds: A necessary and sufficient condition for a function $ f $ to have a continuous derivative of order $ n + 1 $ at all points of $ [ a , b ] $ is that

$$ \frac{E _ {n} ( f ; ( \alpha , \beta ) ) }{( \beta - \alpha ) ^ {n+} 1 } \rightarrow \lambda ( x) ,\ \ a \leq x \leq b , $$

uniformly for $ \beta \rightarrow x $, $ \alpha \rightarrow x $, $ \alpha < x < \beta $, where the continuous function $ \lambda $ is defined by

$$ ( n + 1 ) ! 2 ^ {2n+} 1 \lambda ( x) = | f ^ { ( n + 1 ) } ( x) | . $$

References

[1]	D.A. Raikov, "On the local approximation of differentiable functions" Dokl. Akad. Nauk SSSR , 24 : 7 (1939) pp. 653–656 (In Russian)
[2]	S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) (In Russian)
[3]	Yu.A. Brudnyi, "Spaces defined by means of local approximations" Trans. Moscow Math. Soc. , 24 (1974) pp. 73–139 Trudy Moskov. Mat. Obshch. , 24 (1971) pp. 69–132

Comments

According to [3], which is a valuable survey paper with a rather extensive bibliography, the first result characterizing a space of smooth functions in terms of local approximations was obtained by D.A. Raikov [1].

References

[a1]	J. Peetre, "On the theory of spaces" J. Funct. Anal. , 4 (1969) pp. 71–87