Steklov functions

Steklov functions are functions introduced by the Russian mathematician V. A. Steklov (published in 1907) to solve problems associated with representing functions in the form of series in eigenfunctions of the Sturm-Liouville problem .

Let be $f$ ${\ displaystyle f}$ $f$ Is a function integrable on a segment $[a,b]$ ${\ displaystyle [a, b]}$ $[a, b]$ . Then the function

f_{h}(x)=f_{h,1}(x)={\frac {1}{h}}\int \limits _{x-h/2}^{x+h/2}f(t)\,dt={\frac {1}{h}}\int \limits _{-h/2}^{h/2}f(x+t)\,dt

{\ displaystyle f_ {h} (x) = f_ {h, 1} (x) = {\ frac {1} {h}} \ int \ limits _ {xh / 2} ^ {x + h / 2} f (t) \, dt = {\ frac {1} {h}} \ int \ limits _ {- h / 2} ^ {h / 2} f (x + t) \, dt}

{\ displaystyle f_ {h} (x) = f_ {h, 1} (x) = {\ frac {1} {h}} \ int \ limits _ {xh / 2} ^ {x + h / 2} f (t) \, dt = {\ frac {1} {h}} \ int \ limits _ {- h / 2} ^ {h / 2} f (x + t) \, dt}

called the first order Steklov function for $f$ ${\ displaystyle f}$ $f$ in increments $h>0$ ${\ displaystyle h> 0}$ ${\ displaystyle h> 0}$ .

Induction-defined functions

f_{h,r}(x)={\frac {1}{h}}\int \limits _{x-h/2}^{x+h/2}f_{h,r-1}(t)\,dt,\quad \ r=2,3,\ldots ,

{\ displaystyle f_ {h, r} (x) = {\ frac {1} {h}} \ int \ limits _ {xh / 2} ^ {x + h / 2} f_ {h, r-1} ( t) \, dt, \ quad \ r = 2,3, \ ldots,}

{\ displaystyle f_ {h, r} (x) = {\ frac {1} {h}} \ int \ limits _ {xh / 2} ^ {x + h / 2} f_ {h, r-1} ( t) \, dt, \ quad \ r = 2,3, \ ldots,}

are called Steklov order functions $r$ ${\ displaystyle r}$ $r$ for $f$ ${\ displaystyle f}$ $f$ in increments $h>0$ ${\ displaystyle h> 0}$ ${\ displaystyle h> 0}$ .

Properties

Function $f_{h}(x)$ ${\ displaystyle f_ {h} (x)}$ has a derivative

{\frac {d}{dx}}f_{h}(x)={\frac {1}{h}}{\Bigl (}f(x+h/2)-f(x-h/2){\Bigr )}

{\ displaystyle {\ frac {d} {dx}} f_ {h} (x) = {\ frac {1} {h}} {\ Bigl (} f (x + h / 2) -f (xh / 2 ) {\ Bigr)}}

at almost all points of the segment $[a,b]$ ${\ displaystyle [a, b]}$ .

If a $f$ ${\ displaystyle f}$ is absolutely continuous on the whole material axis, then the following estimates hold:

\sup \limits _{x\in (-\infty ,+\infty )}|f(x)-f_{h}(x)|\leq \omega _{f}(h/2),

{\ displaystyle \ sup \ limits _ {x \ in (- \ infty, + \ infty)} | f (x) -f_ {h} (x) | \ leq \ omega _ {f} (h / 2), }

\sup \limits _{x\in (-\infty ,+\infty )}{\Bigl |}{\frac {d}{dx}}f_{h}(x){\Bigr |}\leq {\frac {1}{h}}\omega _{f}(h),

{\ displaystyle \ sup \ limits _ {x \ in (- \ infty, + \ infty)} {\ Bigl |} {\ frac {d} {dx}} f_ {h} (x) {\ Bigr |} \ leq {\ frac {1} {h}} \ omega _ {f} (h),}

Where $\omega _{f}(\cdot )$ ${\ displaystyle \ omega _ {f} (\ cdot)}$ - modulus of continuity of the function $f$ ${\ displaystyle f}$ .

If a $f\in L^{p}(-\infty ,+\infty ),$ ${\ displaystyle f \ in L ^ {p} (- \ infty, + \ infty),}$ then similar inequalities hold in the norm of this space.

Literature

Akhiezer, N.I. Lectures on the theory of approximation, - M .: Nauka, 1965.
Zhuk V.V., Kuzyutin V.F. Approximation of functions and numerical integration, St. Petersburg: Publishing House of St. Petersburg State University, 1995.

Links

Springer Encyclopaedia of Mathematics.

Steklov functions

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Literature

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