# Steklov function

*for an integrable function on a bounded segment *

The function

(*) |

Functions of the form (*), as well as the iteratively defined functions

were first introduced in 1907 by V.A. Steklov (see [1]) in solving the problem of expanding a given function into a series of eigenfunctions. The Steklov function has derivative

almost everywhere. If is uniformly continuous on the whole real axis, then

where is the modulus of continuity of . Similar inequalities hold in the metric of , provided .

#### References

[1] | V.A. Steklov, "On the asymptotic representation of certain functions defined by a linear differential equation of the second order, and their application to the problem of expanding an arbitrary function into a series of these functions" , Khar'kov (1957) (In Russian) |

[2] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |

#### Comments

Steklov's fundamental paper was first published in French (1907) in the "Communications of the Mathematical Society of Kharkov" ; [1] is the Russian translation, together with additional comments by N.S. Landkof.

#### References

[a1] | E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) |

[a2] | M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978) |

**How to Cite This Entry:**

Steklov function. A.V. Efimov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Steklov_function&oldid=17657