for an integrable function on a bounded segment
Functions of the form (*), as well as the iteratively defined functions
were first introduced in 1907 by V.A. Steklov (see ) 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 .
|||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)|
|||N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)|
Steklov's fundamental paper was first published in French (1907) in the "Communications of the Mathematical Society of Kharkov" ;  is the Russian translation, together with additional comments by N.S. Landkof.
|[a1]||E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)|
|[a2]||M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978)|
Steklov function. A.V. Efimov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Steklov_function&oldid=17657