Local approximation of functions

A measure of approximation (in particular, best approximation) of a function on a set , regarded as a function of this set. The main interest is in the behaviour of a local approximation of a function as . 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 be the best approximation of a function by algebraic polynomials of degree on an interval , . The following assertion holds: A necessary and sufficient condition for a function to have a continuous derivative of order at all points of is that

uniformly for , , , where the continuous function is defined by

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