The paper has been presented at the 12th International Conference on Applications of
Computer Algebra, Varna, Bulgaria, June, 2006.The set of Hausdorff continuous functions is the largest set of
interval valued functions to which the ring structure of the set of continuous
real functions can be extended. The paper deals with the automation of
the algebraic operations for Hausdorff continuous functions using an ultra-
arithmetical approach

Anguelov, Roumen

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 1 (2007), 443–454ALGEBRAIC COMPUTATIONS WITH HAUSDORFFCONTINUOUS FUNCTIONS*Roumen AnguelovAbstract. The set of Hausdorff continuous functions is the largest set ofinterval valued functions to which the ring structure of the set of continuousreal functions can be extended. The paper deals with the automation ofthe algebraic operations for Hausdorff continuous functions using an ultra-arithmetical approach.1. Introduction. CAS typically deal with computations, operationsand manipulations of functions from the set of what is considered elementaryfunctions. These functions are all analytic on their domains of definition. To theauthor’s best knowledge discontinuous functions have not yet been considered inthis context. In this paper we consider algebraic computations with Hausdorffcontinuous (H-continuous) interval valued functions. The H-continuous functionsACM Computing Classification System (1998): I.1.2Key words: Hausdorff continuous function, ultra-arithmetical functoid.*The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006.444 Roumen Anguelovare useful in representing discontinuities of real functions through interval values.They were originally defined within the realm of Approximation Theory but havebeen applied since in many other areas. In particular, recent results have shownthat they are also a powerful tool in the Analysis of PDEs since the solution oflarge classes of nonlinear PDEs can be assimilated with H-continuous functions,[6]. It was also shown recently, see [5], that the algebraic operations for continuousfunctions can be extended to H-continuous functions in such a way that theset of H-continuous functions is a commutative ring. This result is particularlysignificant in view of the fact that interval structures typically do not form linearspaces. In fact, the set of H-continuous functions is the largest set of intervalfunctions which is a linear space, [4]. The present paper deals with the automationof the algebraic operations with H-continuous functions within the structure of afunctoid.2. The algebraic operations with H-continuous functions.The real line is denoted by R and the set of all finite real intervals by IR ={[a, a] : a, a ∈ R, a ≤ a}. Given an interval a = [a, a] ∈ IR, w(a) = a − ais the width of a. An interval a is called a proper interval if w(a) > 0 and apoint interval if w(a) = 0. Identifying a ∈ R with the point interval [a, a] ∈ IR,we consider R as a subset of IR. Let Ω ⊆ R be open. We recall, [10], that aninterval function f : Ω → IR is S-continuous if its graph is a closed subset ofΩ× R. An interval function f : Ω→ IR is Hausdorff continuous (H-continuous)if it is an S-continuous function which is minimal with respect to inclusion, thatis, if ϕ : Ω → IR is an S-continuous function, then ϕ ⊆ f implies ϕ = f . Herethe inclusion is considered in a pointwise sense. We denote by H(Ω) the set ofH-continuous functions on Ω. The following theorem states an essential propertyof the continuous functions which is preserved by the H-continuity [1].Theorem 1. Let f, g ∈ H(Ω). If there exists a dense subset D of Ω suchthat f(x) = g(x), x ∈ D, then f(x) = g(x), x ∈ Ω.H-continuous functions are also similar to the usual continuous real func-tions in that they assume point values on a residual subset of Ω. More precisely,it is shown in [1] that for every f ∈ H(Ω) the set Wf = {x ∈ Ω : w(f(x)) > 0} isof first Baire category and f is continuous on Ω \Wf . Since a finite or countableunion of sets of first Baire category is also a set of first Baire category we have:Algebraic Computations with Hausdorff Continuous Functions 445Theorem 2. Let F be a finite or countable set of H-continuous functions.Then the set DF = {x ∈ Ω : w(f(x)) = 0, f ∈ F} = Ω \⋃f∈F Wf is dense in Ωand all functions f ∈ F are continuous on DF .For every S-continuous function g we denote by [g] the set of H-continuousfunctions contained in g, that is,[g] = {f ∈ H(Ω) : f ⊆ g}.Identifying {f} with f we have [f ] = f whenever f is H-continuous. The S-continuous functions g such that the set [g] is a singleton, that is, it containsonly one function, play an important role in the sequel. In analogy with theH-continuous functions which are minimal S-continuous functions, we call thesefunctions quasi-minimal. The following characterization of the quasi-minimalS-continuous functions is an easy consequence of Theorem 1.Theorem 3. If f is an S-continuous function on Ω which assumes pointvalues on a dense subset of Ω, then f is quasi-minimal S-continuous function.The familiar operations of addition, scalar multiplication and multiplica-tion on the set of real intervals are defined for [a, a], [b, b] ∈ I R and α ∈ R asfollows:[a, a] + [b, b]={a + b : a∈ [a, a], b∈ [b, b]}=[a + b, a + b],α · [a, a]={αa : a∈ [a, a]}=[min{αa, αa},max{αa, αa}],[a, a]×[b, b]={ab :a∈ [a, a], b∈ [b, b]}=[min{ab, ab, ab, ab},max{ab, ab, ab, ab}].Pointwise operations for interval functions are defined in the usual way:(1) (f + g)(x) = f(x) + g(x), (α · f)(x) = α · f(x), (f × g)(x) = f(x)× g(x).It is easy to see that the set of the S-continuous functions is closed underthe above pointwise operations while the set of H-continuous functions is not, see[2], [4]. Hence the following result is significant.Theorem 4. For any f, g ∈ H(Ω) and α ∈ R the functions f + g, α · fand f × g are quasi-minimal S-continuous functions.446 Roumen AnguelovP r o o f. Denote by Dfg the subset of Ω where both f and g assume pointvalues. Then f +g assumes point values on Dfg. According to Theorem 2 the setDfg is dense in Ω which in terms of Theorem 3 implies that f+g is quasi-minimal.The quasi-minimality of α · f and f × g is proved in a similar way. We define the algebraic operations on H(Ω) using Theorem 4. We denotethese operations respectively by⊕,  and⊗ so that distinction from the pointwiseoperations can be made.Definition 5. Let f, g ∈ H(Ω) and α ∈ R. Then(2) f ⊕ g = [f + g], α f = [α · f ], f ⊗ g = [f × g].Theorem 6. The set H(Ω) is a commutative algebra with respect to theoperations ⊕,  and ⊗ given in (2).The proof is partially discussed in [5] and, since it involves standardtechniques, will be omitted here.3. The concept of ultra-arithmetical functoid. Functoid isa structure resulting from the ultra-arithmetical approach to the solution ofproblems in function spaces. The aim of the ultra-arithmetic is the developmentof structures, data types and operations corresponding to functions for directdigital implementation. On a digital computer equipped with ultra-arithmetic,problems associated with functions will be solvable, just as now we solve algebraicproblems [8]. Ultra-arithmetic is developed in analogy with the development ofcomputer arithmetic.LetM be a space of functions and let M be a finite dimensional subspacespanned by ΦN = {ϕk}Nk=0. Every function f ∈ M is approximated by τN (f) ∈M . The mapping τN is called rounding (in analogy with the rounding of numbers)and the space M is called a screen of M. Every rounding must satisfy therequirement (invariance of rounding on the screen): τN(f) = f for every f ∈M .Every function f =∑Ni=0 αiϕi ∈ M can be represented by its coefficient vectorν(f) = (α0, α1, . . . , αN ). Therefore the approximation of the functions in M isrealized through the mappingsMτN−→Mν←→ KN+1, where K is the scalar fieldofM (i.e. K = R or K = C). Since ν is a bijection we can identify M and KN+1and consider only the rounding τN .Algebraic Computations with Hausdorff Continuous Functions 447In M we consider the operations addition (+), scalar multiplication (.),multiplication of functions (×) and integration (∫) defined in the conventionalway. By the semimorphism principle τN induces corresponding operations in M :f b g = τN (fb g) , b ∈ {+, .,×};∫f = τN(∫f).The structure (M, + , . , × ,∫) is called an (ultra-arithmetical) functoid [9].4. A functoid in H(Ω). To simplify the matters we will consider thefunction space of all bounded H-continuous functions on Ω = (−1, 1). Further-more, since we will often use a shift of the argument, we assume that all functionsare produced periodically (period 2) over R. Hence we denote the space underconsideration by Hper(−1, 1). All algebraic operations on Hper(−1, 1) are consi-dered in terms of Definition 5. For simplicity we denote them as the operationsfor reals. Namely, addition is “+” and a space is interpreted as multiplication,where the context shows whether this is a scalar multiplication or a productof functions. In particular, note that indicating the argument of a function ina formula does not mean pointwise operation. Denote by s1 the H-continuousfunction given bys1(x) =x, if x ∈ (−1, 1),[−1, 1], if x = ±1;and produced periodically over the real line. Since the integrals of s1 and s1 areequal over any interval the integral of s1 is a regular real function. We constructiteratively the sequence of periodic splines s1, s2, s3, . . . usingsj+1 =∫sj(x)dx + c,∫ 1−1sj+1(x)dx = sj+2(1)− sj+2(−1) = 0.448 Roumen AnguelovThe first few splines after s1 are given on the interval [−1, 1] as followss2(x) =x22!−16,s3(x) =x33!−16x ,s4(x) =x44!−16x22!+7360,s5(x) =x55!−16x33!+7360x .It should be emphasized that the above splines are periodic functions over R.Furthermore, for any j > 2 the spline sj is a j−2 times continuously differentiablefunction on R and the j − 1 derivative is discontinuous only at the odd integers.Theorem 7. Let f ∈ Hper(−1, 1) be given. Assume that there exists afinite set Λ = {λ1, λ2, . . . , λm} ⊂ (−1, 1] such that f assumes real values and isp times differentiable on (−1, 1] \Λ with the pth derivative in L2(−1, 1). Then fhas a unique representation in the form(3) f(x) = a0 +m∑l=1p∑j=1ajlsj(x + 1− λl) +∞∑k=−∞k 6=0bkeikpix,where∞∑k=−∞k 6=0bkeikpix is p times differentiable with its pth derivative in L2(−1, 1).Furthermore, the coefficients are given by:a0 =12∫ 1−1f(x)dxajl =12(dj−1fdxj−1(λl − 0)−dj−1fdxj−1(λl + 0)),j = 1, . . . , pl = 1, . . . ,mbk =12(ikpi)p∫ 1−1dpf(x)dxpe−ikpixdx , k = ±1,±2, . . .The proof is carried out by considering the Fourier series of the functiong(x) = f(x)− a0 −m∑l=1p∑j=1ajlsj(x + 1− λl)Algebraic Computations with Hausdorff Continuous Functions 449and uses standard techniques. Hence it will be omitted. It should be noted thatexpansion (3) is unique for a given value of p but for different values of p one mayhave different representations of the function f in the form (3).Function f is approximated by(4) ρNp(f ;x) = a0 +m∑l=1p∑j=1ajlsj(x + 1− λl) +N∑k=−Nk 6=0bkeikpixwith a rounding error|f(x)− ρNp(f ;x)| =∣∣∣∣∣∣∑|k|>Nbkeikpix∣∣∣∣∣∣ ≤∑|k|>N|bk|≤ ∑|k|>N(kpi)2p|bk|212 ∑|k|>N1(kpi)2p12(5)≤12∫ 1−1(dpf(x)dxp)2dx−(m∑l=1apl)2−N∑k=−Nk 6=0(kpi)2p|bk|212(2(2p−1)pi2pN2p−1) 12= o(1Np−12)Motivated by the above we consider a screen in Hper(−1, 1) comprisingthe subspace M spanned by the basis{s0(x)} ∪ {sj(x + 1− λl) : j = 0, 1, . . . , p, l = 1, . . . ,m}∪∪{eikpix : k = 0,±1, . . . ,±N},where p,m,N ∈ N and {λ1, λ2, . . . , λm} ⊂ (−1, 1] are parameters with arbitrarybut fixed values. Here s0 is the function which is constant 1 on R. Defining arounding from Hper(−1, 1) to M is still an open problem. However, for functionsof the kind described in Theorem 7 the rounding is defined through ρNp. Further-more, to define a functoid we only need to know how to round the functionsresulting from operations in M . For this purpose the rounding ρNp is sufficient.450 Roumen AnguelovNaturally, since M is a subspace it is closed under the operations addition andscalar multiplication. Furthermore, to define multiplication of functions andintegration we only need to define these operations on the elements of the basis.The products of the functions in the basis are given bysq1(x + 1− λl1)sq2(x + 1− λl2)=q1+q2∑j=q1(j − 1q1 − 1)sq1+q2−j(1 + λl1 − λl2)sj(x + 1− λl1)(6)+q1+q2∑j=q2(j − 1q2 − 1)sq1+q2−j(1− λl1 + λl2)sj(x + 1− λl2)eik1pixeik2pix = ei(k1+k2)pix(7)sq(x + 1− λl)einpix=p∑j=q(−1)nei(λl−1)pi(j − 1q − 1)(inpi)j−qsj(x + 1− λl) +∞∑k=−∞k 6=0βkeikpix(8)where in the formula (8) the coefficients βk are given byβk =(−1)k−n−1np−qkp(ipi)qq−1∑r=0(pr)(nk − n)q−rif k 6= 0, nβn =(pq)(inpi)−qFor the respective integrals we have∫sj(x)dx = sj+1(x) , j = 1, . . . , p(9)∫eikpixdx =1ikpieikpix , k = 0,±1, . . . ,±N(10)Obviously in the formulas (6)–(9) we obtain splines sj with j > p and exponentsAlgebraic Computations with Hausdorff Continuous Functions 451eikpix with |k| > N which need to be rounded. Using thatsj(x + 1− λl) =∞∑k=−∞k 6=0(−1)k−1ei(1−λl)pi(ikpi)jeikpiall that needs to be rounded is a Fourier series which is done by truncation. Notethat each time we truncate a Fourier series of a function which is at least p timesdifferentiable with its pth derivative in L2(−1, 1). Hence the uniform norm of theerror is o(1Np− 12). The integration of s0, when it arises in practical problems,should be handled with special care since∫s0(x)dx = s1(x) holds only on (−1, 1).5. Applications. As an illustrative example we consider the waveequation in the formutt(x, t)− uxx(x, t) = φ(x, t)(11)u(x, 0) = g1(x), ut(x, 0) = g2(x)(12)with periodic boundary conditions at x = −1 and x = 1, assuming that g1, g2,φ or some of their space derivatives may be discontinuous but the functions canbe represented as a spline-Fourier series (3) of the space variable. More precisely,for the function φ we assume a representation of the form(13) φ(x, t) = α0(t) +m∑l=1p∑j=11∑δ=−1αljδ(t)sj(x+δt+1−αl) +N∑k=−Nk 6=0βk(t)eikpixThe solution isu(x, t) =12g1(x + t) + g1(x− t) +∫ x+tx−tg2(θ)dθ +∫∫G(x,t)φ(y, θ)dydθ ,where G(x, t) is the triangle in R2 with vertices (x − t, 0), (x + t, 0) and (x, t).Note that the above form of the solution is an explicit representation of the wellknown property that the value of the solution at (x, t) depends only on the valuesof g1 and g2 on the line segment connecting (x − t, 0) and (x + t, 0) and on thevalues of φ in the triangle G(x, t). The essential part of the above computation452 Roumen Anguelovis evaluating the integral over G(x, t). For that purpose we need to assume somesort of representation of the coefficients in (13) as elementary functions of t. Forsimplicity we assume here that they are polynomials of t. Then the integral overG(x, t) is evaluated through the following formulas:∫∫G(x,t)θqq!sj(y)dydθ = sj+q+2(x+t)+(−1)qsj+q+2(x−t)−2q∑l=0l−eventq−l(q−l)!sj+l+2(x)(14)∫∫G(x,t)θqq!sj(y+θ)dydθ =q+1∑l=0(−12)ltq+1−l(q+1−l)!sj+l+1(x+t)−(−12)q+1sj+q+2(x−t)(15)∫∫G(x,t)θqq!sj(y−θ)dydθ =(12)q+1sj+q+2(x+t)−q+1∑l=0(12)ltq+1−l(q+1−l)!sj+l+1(x−t)(16)∫∫G(x,t)θqq!eikpiydydθ=1(ikpi)q+2eikpi(x+t)+(−1)qeikpi(x−t)−2 q∑l=0l−eventq−l(ikpi)q−l(q−l)!eikpix(17)= 2∞∑l=0l−even(ikpi)ltq+l+2(q+l+2)!eikpixHence the solution is obtained in the form(18) u(x, t) = a0(t) +m∑l=1p∑j=11∑δ=−1aljδ(t)sj(x+δt+1−αl) +N∑k=−Nk 6=0bk(t)eikpixwhere all coefficients are polynomials of t.Let us consider the solution of the problem (11–12) whereg1(x) = 0.05 + 0.1s1(x + 0.75) − 0.1s1(x + 1.25)(19)(a square wave, see Fig. 1)g2(x) = s1(x)(20)φ(x, t) = ts1(x)(21)Using the above method the solution is obtained in the form (18) and it is plottedon Fig. 2. One can observe the accurate representation of the discontinuities ofAlgebraic Computations with Hausdorff Continuous Functions 453the solution and its derivatives.−1 −0.5 0 0.5 1−0.100.10.20.30.4xuFig. 1. Function g1 in (19). Fig. 2. The exact solution of (11)–(12)with data given in (19)–(21).When the right hand side of the equation depends on u, one can use thetechniques derived above to establish an iterative procedure converging to theexact solution. For example, for the equationutt(x, t)− uxx(x, t) = ρ(t)u(x, t) + φ(x, t)the following iterative procedure can be used:(22) u(r+1) = (1− λ)u(r) + λg + 12∫∫G(x,t)ρu(r) , r = 0, 1, 2, . . . ,whereg(x, t) =12g1(x + t) + g1(x− t) +∫ x+tx−tg2(θ)dθ +∫∫G(x,t)φ(y, θ)dydθ .Naturally, successive iterations through (22) will produce functions sj with largeindexes. The obtained expressions can be simplified using the rounding discussedin the preceding section.454 Roumen AnguelovREFERE NCES[1] Anguelov R. Dedekind order completion of C(X) by Hausdorff continuousfunctions. Quaestiones Mathematicae 27 (2004) 153–170.[2] Anguelov R., S. Markov. Extended Segment Analysis. FreiburgerIntervall-Berichte, 10, 1981, 1–63.[3] Anguelov R., S. Markov, B. Sendov. On the Normed Linear Space ofHausdorff Continuous Functions. Lect. Notes in Comput. Sci. 3743, 2006,281–288.[4] Anguelov R., S. Markov, B. Sendov. The Set of Hausdorff ContinuousFunctions – the Largest Linear Space of Interval Functions. ReliableComputing 12 (2006), 337–363.[5] Anguelov R., S. Markov, B. Sendov. Algebraic operations for H-continuous functions. In: Constructive Theory of Functions (Ed. B. Bojanov)Varna 2005, Marin Drinov, Sofia, 2006, 35–44.[6] Anguelov R., E. E. Rosinger. Hausdorff Continuous Solutions ofNonlinear PDEs through the Order Completion Method. QuaestionesMathematicae 28 (2005), 271–285.[7] Anguelov R., J. H. van der Walt. Order Convergence Structure onC(X). Quaestiones Mathematicae 28 (2005), 425–457.[8] Epstein C., W. L. Miranker, T. J. Rivlin. Ultra Arithmetic, Part I:Function Data Types; Part 2: Intervals of Polynomials. Mathematics andComputers in Simulation, vol. XXIV, 1982, 1–18.[9] Kaucher E., W. Miranker. Self-Validating Numerics for Function SpaceProblems. Academic Press, New York, 1984.[10] Sendov B. Hausdorff Approximations, Kluwer, 1990.University of PretoriaPretoria 0002, South Africae-mail: roumen.anguelov@up.ac.zaReceived September 25, 2006Final Accepted October 18, 2007

Algebraic Computations with Hausdorff Continuous Functions

http://sci-gems.math.bas.bg/jspui/bitstream/10525/356/1/sjc034-vol1-num4-2007.pdf

Algebraic Computations with Hausdorff Continuous Functions

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS