Steklov function

for an integrable function $f$ on a bounded segment $[ a, b]$

The function

$$\tag{* } f _ {h} ( t) = \frac{1}{h} \int\limits _ { t- h/2} ^ { t+ h/2} f( u) du = \ \frac{1}{h} \int\limits _ { - h/2} ^ { h/2 } f( t+ v) dv.$$

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

$$f _ {h,r} ( t) = \ \frac{1}{h} \int\limits _ { t- h/2} ^ { t+h/2} f _ {h,r-1} ( u) du ,\ \ r = 2, 3 \dots$$

$$f _ {h,1} ( t) = f _ {h} ( t),$$

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 $f _ {h}$ has derivative

$$f _ {h} ^ { \prime } ( t) = \ \frac{1}{h} \left \{ f \left ( t+ \frac{h}{2} \right ) - f \left ( t- \frac{h}{2} \right ) \right \}$$

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

$$\sup _ {t \in (- \infty , \infty ) } | f( t) - f _ {h} ( t) | \leq \omega \left ( \frac{h}{2} , f \right ) ,$$

$$\sup _ {t \in (- \infty , \infty ) } | f _ {h} ^ { \prime } ( t) | \leq \frac{1}{h} \omega ( h, f ),$$

where $\omega ( \delta , f )$ is the modulus of continuity of $f$. Similar inequalities hold in the metric of $L _ {p} (- \infty , \infty )$, provided $f \in L _ {p} (- \infty , \infty )$.

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)