Regulated function

Regulated function

In mathematics, a regulated function (or ruled function) is a "well-behaved" function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations.

Definition

Let "X" be a Banach space with norm || - ||"X". A function "f" : [0, "T"] → "X" is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true harv|Dieudonné|1969|loc=§7.6:

* for every "t" in the interval [0, "T"] , both the left and right limits "f"("t"−) and "f"("t"+) exist in "X" (apart from, obviously, "f"(0−) and "f"("T"+));

* there exists a sequence of step functions "φ""n" : [0, "T"] → "X" converging uniformly to "f" (i.e. with respect to the supremum norm || - ||∞).

It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:

* for every "δ" > 0, there is some step function "φ""δ" : [0, "T"] → "X" such that

::| f - varphi_{delta} |_{infty} = sup_{t in [0, T] } | f(t) - varphi_{delta} (t) |_{X} < delta;

* "f" lies in the closure of the space Step( [0, "T"] ; "X") of all step functions from [0, "T"] into "X" (taking closure with respect to the supremum norm in the space B( [0, "T"] ; "X") of all bounded functions from [0, "T"] into "X").

Properties of regulated functions

Let Reg( [0, "T"] ; "X") denote the set of all regulated functions "f" : [0, "T"] &rarr; "X".

* Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg( [0, "T"] ; "X") is a vector space over the same field K as the space "X"; typically, K will be the real or complex numbers. If "X" is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if "X" is a K-algebra, then so is Reg( [0, "T"] ; "X").

* The supremum norm is a norm on Reg( [0, "T"] ; "X"), and Reg( [0, "T"] ; "X") is a topological vector space with respect to the topology induced by the supremum norm.

* As noted above, Reg( [0, "T"] ; "X") is the closure in B( [0, "T"] ; "X") of Step( [0, "T"] ; "X") with respect to the supremum norm.

* If "X" is a Banach space, then Reg( [0, "T"] ; "X") is also a Banach space with respect to the supremum norm.

* Reg( [0, "T"] ; R) forms an infinite-dimensional real Banach algebra: finite linear combinations and products of regulated functions are again regulated functions.

* Since a continuous function defined on a compact space (such as [0, "T"] ) is automatically uniformly continuous, every continuous function "f" : [0, "T"] &rarr; "X" is also regulated. In fact, with respect to the supremum norm, the space "C"0( [0, "T"] ; "X") of continuous functions is a closed linear subspace of Reg( [0, "T"] ; "X").

* If "X" is a Banach space, then the space BV( [0, "T"] ; "X") of functions of bounded variation forms a dense linear subspace of Reg( [0, "T"] ; "X"):

::mathrm{Reg}( [0, T] ; X) = overline{mathrm{BV} ( [0, T] ; X)} mbox{ w.r.t. } | cdot |_{infty}.

* If "X" is a Banach space, then a function "f" : [0, "T"] &rarr; "X" is regulated if and only if it is of bounded "&phi;"-variation for some "&phi;":

::mathrm{Reg}( [0, T] ; X) = igcup_{varphi} mathrm{BV}_{varphi} ( [0, T] ; X).

* If "X" is a separable Hilbert space, then Reg( [0, "T"] ; "X") satisfies a compactness theorem known as the Fraňková-Helly selection theorem.

* The integral, as defined on step functions in the obvious way, extends naturally to Reg( [0, "T"] ; "X") by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. This extension is well-defined and satisfies all of the usual properties of an integral. In particular, the regulated integral
** is a bounded linear function from Reg( [0, "T"] ; "X") to "X"; hence, in the case "X" = R, the integral is an element of the space that is dual to Reg( [0, "T"] ; R);
** agrees with the Riemann integral whenever both are defined.

References

*.
*.
*.
*.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Regulated integral — In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by… …   Wikipedia

  • Regulated rewriting — is a specific area of formal languages studying grammatical systems which are able to take some kind of control over the production applied in a derivation step. For this reason, the grammatical systems studied in Regulated Rewriting theory are… …   Wikipedia

  • Cocaine and amphetamine regulated transcript — CART prepropeptide Identifiers Symbol CARTPT Entrez 9607 HUGO …   Wikipedia

  • Automotive lighting — Blinker redirects here. For other uses, see Blinker (disambiguation). Not to be confused with Magneti Marelli company AL Automotive Lighting. For lights in seafaring and aviation, see navigation light. The lighting system of a motor vehicle… …   Wikipedia

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

  • Fraňková-Helly selection theorem — In mathematics, the Fraňková Helly selection theorem is a generalisation of Helly s selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana… …   Wikipedia

  • FSMA overview — Introduction Implementation of the Financial Services and Markets Act 2000 (FSMA) completed the vesting of supervisory responsibilities in the regulator, the Financial Services Authority (FSA), and rationalised and substantially replaced the… …   Law dictionary

  • endocrine system, human — ▪ anatomy Introduction  group of ductless glands (gland) that regulate body processes by secreting chemical substances called hormones (hormone). Hormones act on nearby tissues or are carried in the bloodstream to act on specific target organs… …   Universalium

  • Life Sciences — ▪ 2009 Introduction Zoology       In 2008 several zoological studies provided new insights into how species life history traits (such as the timing of reproduction or the length of life of adult individuals) are derived in part as responses to… …   Universalium

  • nervous system, human — ▪ anatomy Introduction       system that conducts stimuli from sensory receptors to the brain and spinal cord and that conducts impulses back to other parts of the body. As with other higher vertebrates, the human nervous system has two main… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”