# Favard problem

From Encyclopedia of Mathematics

The problem of calculating the least upper bound

(*) |

where the are trigonometric polynomials of order not exceeding , is the class of periodic functions whose -th derivative in the sense of Weyl (see Fractional integration and differentiation) satisfies the inequality , and . The Favard problem was posed by J. Favard [1]. Subsequently, broader classes of functions have been considered and a complete solution of the Favard problem for and arbitrary has been obtained as a corollary of more general results (see [2], [3]).

#### References

[1] | J. Favard, "Sur les meilleurs procédés d'approximation de certaines classes de fonctions par des polynômes trigonométriques" Bull. Sci. Math. , 61 (1937) pp. 209–224 |

[2] | S.B. Stechkin, "On best approximation of certain classes of periodic functions by trigonometric functions" Izv. Akad. Nauk SSSR Ser. Mat. , 20 : 5 (1956) pp. 643–648 (In Russian) |

[3] | V.K. Dzyadyk, "Best approximation on classes of periodic functions defined by kernels which are integrals of absolutely monotone functions" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 6 (1959) pp. 933–950 (In Russian) |

[4] | N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) |

#### Comments

#### References

[a1] | R.P. Feinerman, D.J. Newman, "Polynomial approximation" , Williams & Wilkins pp. Chapt. IV.4 |

**How to Cite This Entry:**

Favard problem. Yu.N. Subbotin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Favard_problem&oldid=18812

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098