Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$

By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures on the Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition into non-negative or non-decreasing objects. Using the representation approach to computable analysis, a computable version of the Riesz representation theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we extend this result. We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm. We introduce natural representations for defining computability. We prove that the canonical linear bijections between these spaces and their inverses are computable. We also prove that Jordan decomposition is computable on each of these spaces.


Introduction
Let C[0; 1] be the set of continuous functions h : [0; 1] → R. By the Riesz representation theorem for every linear continuous function F : C[0; 1] → R there is a function g : [0; 1] → R of bounded variation such that F (h) = h dg for every continuous function h ∈ C[0; 1].For every function g : [0; 1] → R of bounded variation there is a signed Borel measure µ on the unit interval of finite variation norm such that h dg = h dµ for every continuous function h ∈ C[0; 1].Finally for every signed Borel measure µ on the unit interval of finite variation norm the function h → h dµ for h ∈ C[0; 1] is linear and continuous.
In this article we study computability of all of these existence theorems.Computability of the Riesz representation theorem and its converse have been proved in [15] with a revised proof in [11].Computability of (µ, h) → h dµ for continuous h and non-negative bounded Borel measure µ has been proved in [19].In this article we extend these results.
We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm.We introduce natural representations for defining computability.We prove that the canonical linear bijections F → g, g → µ and µ → F between these spaces and their inverses are computable.We also prove that (minimal) Jordan decomposition is computable on each of these spaces.
In Section 2 we summarize some definitions and basic facts from classical analysis on linear continuous functionals F : C[0; 1] → R, functions of bounded variation and the Riemann-Stieltjes integral, and on signed measures on the Borel sets of the unit interval.We consider only functions g : [0; 1] → R of bounded variation which are normalized in the sense that g(0) = 0 and for all 0 < y < 1, lim xրy g(x) = g(y).
In Section 3 we outline very shortly some general concepts from the representation approach to computable analysis [20,5].For defining computability we introduce and discuss representations of the functionals, of the functions of bounded variation and of the signed measures and also representations of the subspaces of non-negative or non-decreasing objects, respectively.While in [15,11] partial functions of bounded variation are considered in this article we use total normalized functions with a representation which is very closely related to the one used for the partial functions.
In Section 4 first we prove for the special case of non-negative functionals F , nondecreasing functions g and non-negative measures µ that the mappings F → g, g → µ and µ → F such that F (h) = h dg, h dg = h dµ and h dµ = F (h) are computable w.r.t the "non-negative" representations.Then we prove our main results: On the spaces of linear continuous functionals with operator norm, the space of normalized functions of bounded variation with variation norm and the space of signed measures with finite variation norm the operators F → g, g → µ and µ → F are computable.Furthermore, the Jordan decompositions F → (F + , F − ), g → (g + , g − ) and µ → (µ + , µ − ) are computable.The results can be expressed in such a way that a number of representations of the space of linear continuous functionals are equivalent.
The results can be generalized easily from the unit interval to arbitrary intervals [a; b] with computable endpoints.More generally, the results can be proved computably uniform in a, b, where a and b are given by their standard representation via fast converging Cauchy sequences of rational numbers.
In [13,22] Jordan decomposition of computable real functions and of polynomial time computable functions on the unit interval has been studied.However, they do not investigate computability of the Jordan decomposition operator but ask whether computability or polynomial computability is preserved under Jordan decomposition.Ko [13] has shown that there is a polynomial time computable function f of bounded variation which is not the difference of two non-decreasing polynomial time computable functions.This has been strengthened by Zheng and Rettinger who have proved that there is a polynomial time computable function of bounded variation with polynomial modulus of absolute continuity which is not the difference of two non-decreasing computable functions.

Basics from the classical theory
We summarize some definitions and results about functions of bounded variation and from (non-computable) measure theory which are scattered across many sources [8,9,10,12,14,17,7,1,2,16,15,11] or can be derived easily from there.For convenience we consider only the closed unit interval [0; 1] for functions, measures etc.
Let C[0; 1] be the space of continuous functions h : We shortly introduce functions g : [0, 1] → R of bounded variation and the Riemann-Stieltjes integral h dg for continuous functions h : 2) The function g : [0; 1] → R is of bounded variation if its variation Var(g) := V 1 0 (g) is finite.For a function of bounded variation the total variation function /g/ : [0; 1] → R is defined by /g/(0) := 0 and /g/(x) := V x 0 (g).In the following let h : [0; 1] → R be a continuous function and let g : [0; 1] → R be a function of bounded variation.For any partition Z = (x 0 , x 1 , . . ., x n ) of [0; 1] define Since h is continuous and its domain is compact, it has a (uniform) modulus of continuity, i.e., a function m : . We may assume that the function m is non-decreasing.Notice that by Lemma 2.1 the integral h dg is determined already by the values of the function g on 0 and 1 and on an arbitrary dense set X, since there are partitions of arbitrary precision that contain points only from the set X.If g is of bounded variation, then lim yրx g(y) and lim yցx g(y) exist for all 0 ≤ x ≤ 1. Functions of bounded variation can be normalized without changing the Riemann-Stieltjes integral over continuous functions.
Let BM be the set of signed measures µ with finite variation norm µ m on the Borel subsets of the unit interval The following theorem summarizes the relation between the three spaces introduced above.
The functions T FV , T VM and T FV preserve the norms.Moreover, if F is non-negative then T FV (F ) is non-decreasing, if g is non-decreasing then T VM (g) is non-negative, and if µ is non-negative then T MF (µ) in non-negative.
The three spaces are not separable.Theorem 2.3(1) includes the Riesz representation theorem [10].For real numbers x define x + := (|x| + x)/2 and x − := (|x| − x)/2.Then x + and x − are non-negative numbers such that x = x + − x − .Moreover, x + and x − are minimal, that is, x + ≤ y + and x − ≤ y − if y + , y − are non-negative such that x = y + − y − .By the Jordan decomposition theorem, this kind of decomposition can be generalized to functionals F ∈ C ′ [0; 1], to functions g ∈ BV and to signed measures µ ∈ BM.
) For g ∈ BV the Jordan decomposition is a pair (g + , g − ) of non-decreasing functions in BV such that g = g + − g − , and if t + , t − ∈ BV are non-decreasing functions such that g = t + − t − then g + ≤ t + and g − ≤ t − .
If a Jordan decomposition exists then it is unique by the minimality condition.Notice that some authors do not require minimality for Jordan decomposition.
The other statements can be proved accordingly.

The concepts of computability
In this section we define computability on the three spaces from Theorem 2.3.Since the spaces are not separable, Cauchy representations [20,Chapter 8.1] are not available.
For studying computability we use the representation approach (TTE, Type 2 Theory of Effectivity) for computable analysis [20,5].Let Σ be a finite alphabet.Computable functions on Σ * (the set of finite sequences over Σ) and Σ ω (the set of infinite sequences over Σ) are defined by Turing machines which map sequences to sequences (finite or infinite).On Σ * and Σ ω finite or countable tuplings (injections from cartesian products of Σ * and Σ ω to Σ * or Σ ω ) will be denoted by [20,Definition 2.1.7].The tupling functions and the projections of their inverses are computable.
In TTE, sequences from Σ * or Σ ω are used as "names" of abstract objects such as rational numbers, real numbers, real functions or points of a metric space.We consider computability of multi-functions w.r.t.representations [20,5], [21,Sections 3,6,8,9].A representation of a set X is a function δ that is, h(p) is a name of some f (x), if p is a name of x ∈ dom(f ).The function f is called (γ, γ 0 )-computable, if it has a computable (γ, γ 0 )-realization and (γ, γ 0 )-continuous if it has a continuous realization.The definitions can be generalized straightforwardly to multivariate functions Let There is a representation [δ 1 → δ 2 ] of the set of (δ 1 , δ 2 )-continuous functions f : which is determined uniquely up to equivalence by (U) and (S) [20].
(U) corresponds to the "universal Turing machine theorem" and (S) to the "smn-theorem" from computability theory.Roughly speaking, [δ 1 → δ 2 ] is (up to equivalence) the "weakest" representation of the set of (δ 1 , δ 2 )-continuous functions for which the apply function is computable.The generalized Turing machines in [18] are useful tools for defining new computable functions on represented sets from given ones.We use various canonical notations ν : ⊆ Σ * → X: ν N for the natural numbers, ν Q for the rational numbers, ν Pg for the polygon functions on [0; 1] whose graphs have rational vertices, and ν I for the set RI of open intervals (a; b)⊆(0; 1) with rational endpoints.For functions m : N → N we use the canonical representation δ B : ⊆ Σ ω → B = {m | m : N → N} defined by δ B (p) = m if p = 1 m(0) 01 m(1) 01 m(2) 0 . ... For the real numbers we use the Cauchy representation ρ : ⊆ Σ ω → R, ρ(p) = x if p is (encodes) a sequence (a i ) i∈N of rational numbers such that for all i, |x − a i | ≤ 2 −i , and the lower representation ρ < , ρ < (p) = x iff p is (encodes) a sequence (a i ) i∈N of rational numbers such that x = sup i a i .By the Weierstraß approximation theorem the countable set Pg of polygon functions with rational vertices is dense in C[0; 1].Therefore, C[0; 1] with notation ν Pg of the set Pg is a computable metric space [20] for which we use the Cauchy representation δ C defined as follows: Since the representations ρ and δ C are admissible, a functional G : Since for computations we will need the ρ-name of the norm we include it in the name.
This is the representation of the dual of C[0; 1] space as suggested in Section 15 (see also Definition 3.9) of [V.Brattka: "Computability of Banach Space Principles"] in the case that this dual is not separable.It is admissible and admits computability of scalar multiplication, the norm and the rapid Lim-operator, but vector addition is not computable.This yields a good justification for using δ CF .
In [15,11] a computable version of the Riesz representation theorem is proved.In these articles the concept of bounded variation is generalized straightforwardly to the set BVC with representation δ BV C of partial functions g : ⊆ [0; 1] → R with countable dense domain containing {0, 1} which are continuous on dom(g) \ {0, 1}.Remember that a function of bounded variation has at most countably many points of continuity.The integral h dg for continuous h and an arbitrary function g of bounded variation is defined already by any restriction of g to a countable dense subset containing {0, 1} [11].Every (partial) function g ∈ BVC can be extended uniquely to a normalized (total) function ext(g) ∈ BV by ext(g)(x) := lim yրx, y∈dom(g) g(y) for x ∈ dom(g).Then h dg = h d ext(g) for all h ∈ C[0; 1] an Var(g) = Var(ext(g)).In this article instead of δ BVC we use the representation δ V := ext • δ BVC of the normalized functions.The variation is not (δ V , ρ)-computable but only (δ V , ρ < )-computable.Since for computations we will need the ρ-name of the variation we include it in the name.Notice that for computing the Riemann-Stieltjes integral h dg a δ V -name and an upper bound of Var(g) suffice [15,11].Definition 3.2.Define representations δ V and δ BV of BV as follows: (1) δ V (p) = g iff there are p 0 , q 0 , p 1 , q 1 , . . .∈ Σ ω such that p = p 0 , q 0 , p 1 , q 1 , . . ., ρ(p 0 ) = ρ(q 0 ) = 0, ρ(p 1 ) = 1, g • ρ(p i ) = ρ(q i ) for all i ∈ N, A p := {ρ(p i ) | i ≥ 2} is a dense subset of (0; 1) and g is continuous on A p .(2) δ BV p, q = g : ⇐⇒ δ V (p) = g and ρ(q) = Var(g).
A computable version of the Riesz representation theorem and its converse have been proved in [15,11].The results can be formulated as follows.
Theorem 3.4 ( [15,11]).The operator (g, l) → F , mapping every g ∈ BV and every l ∈ N with Var(g) ≤ 2 l to the functional F defined by F (h) = h dg for all h ∈ C[0; 1], is By a slight generalization of the representation δ m of the probability measures on the Borel sets of the unit interval defined and studied in [19] we obtain a representation of the bounded non-negative Borel measures on the unit interval.

Lemma 2 . 1 (
[15]).Let h : [0; 1] → R be a continuous function with modulus of continuity m : N → N and let g be a function of bounded variation.Then there is a unique number I ∈ R such that |I − S(g, h, Z)| ≤ 2 −k Var(g) for all k ∈ N and for every partition Z of [0; 1] with precision m(k + 1).The number I from Lemma 2.1 is called the Riemann-Stieltjes integral and is denoted by h dg.The operator F g : h → h dg is linear and continuous on C[0; 1].