Mathematics Subject Classification: Primary 30C40In this paper we give practical and numerical representations of inverse
functions by using the integral transform with the sign kernel, and show
corresponding numerical experiments by using computers. We derive a very
simple formula from a general idea for the representation of the inverse
functions, based on the theory of reproducing kernels

Matsuura, Tsutomu

Saitoh, Saburou

Yamada, Masato

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

REPRESENTATIONS OF INVERSE FUNCTIONS BYTHE INTEGRAL TRANSFORM WITH THE SIGN KERNELMasato Yamada 1, Tsutomu Matsuura 2, Saburou Saitoh 3AbstractIn this paper we give practical and numerical representations of inversefunctions by using the integral transform with the sign kernel, and showcorresponding numerical experiments by using computers. We derive a verysimple formula from a general idea for the representation of the inversefunctions, based on the theory of reproducing kernels.Mathematics Subject Classification: Primary 30C40Key Words and Phrases: inverse function, reproducing kernel, Sobolevspace, sign function, integral transform, solution method of general equa-tions1. IntroductionConsidering arbitrary mapping φ from an arbitrary abstract set intoanother arbitrary set, the third author of this paper tried to represent theinversion φ−1 in terms of the direct mapping φ, and following the generalideas from [1], surprisingly obtained some simple concrete inverse formulas.This paper establishes an analytical and direct relation between a mono-tone function and its inversion. By the invariance relation principle for an-alytic functions, we could derive new local and analytical representations ofthe inverse functions.The method and ideas proposed for the inversion formulas concernrather general non-linear mappings, however, their formulations sound quiteinvolved for such generality. So, we shall state the results concretely for162 M. Yamada, T. Matsuura, S. Saitohtypical cases of mappings and shall give some corresponding numerical ex-periments. First, we discuss the principle of our new method for the rep-resentations of inverses of non-linear mappings, based on the ideas from[1]:We shall consider some representation of the inversion φ−1 in integralform, at this moment needing a natural assumption for the mapping φ.Then, we shall transform the integral representation by the mapping φ tothe original space, that is the defined domain of the mapping φ. Then, wewill be able to obtain the representation of the inverse φ−1 in terms of thedirect mapping φ. In [1], we considered the representation of the inverse φ−1in some reproducing kernel Hilbert spaces. However, here we shall considerthe representations of the inverse φ−1 for a very concrete situation and weshall give a very fundamental representation of the inverse for some generalfunctions on 1-dimensional spaces. Actually, in this paper, we shall considerthe problems by using a simple Sobolev and reproducing kernel space, andin the next section we shall remind the basic background. By using therepresentation of the functions in the reproducing kernel Hilbert space, wewill be able to obtain some natural representation formulas of the inversesof some general and reasonable functions.2. Background resultsNote thatK(y1, y2) =12e−|y1−y2| , y1, y2 ∈ [A,B] (2.1)is the reproducing kernel in the Sobolev-Hilbert space HK whose membersare real-valued and absolutely continuous functions on [A,B] and whoseinner product is given by (see [2]):(f1, f2)HK =∫ BA(f ′1(y)f′2(y) + f1(y)f2(y))dy + f1(A)f2(A) + f1(B)f2(B).(2.2)3. Integral representations of inverse functionsWe start with the most typical case.Consider a function y = f(x) of the C1-class, strictly increasing, andwith f ′(x) not vanishing on [a, b] (f(a) = A, f(b) = B). Then, the inversefunction f−1(y) is a single-valued function and belongs to the space HK ,REPRESENTATIONS OF INVERSE FUNCTIONS BY . . . 163and from the reproducing property we obtain, for any y0 ∈ [f(a), f(b)] therepresentation:f−1(y0) =(f−1(·),K(·, y0))HK=∫ f(b)f(a)((f−1)′(y)Ky(y, y0)+f−1(y)K(y, y0))dy+aK(f(a), y0)+bK(f(b), y0).(3.1)Note the identityK(y1, y2) =12e−|y1−y2| =12pi∫ ∞−∞eiξ(y1−y2)ξ2 + 1dξ,and so, from f(a) ≤ y0 ≤ f(b), we obtainf−1(y0)=12pi∫ f(b)f(a)((f−1)′(y)∫ ∞−∞iξeiξ(y−y0)ξ2 + 1dξ + f−1(y)∫ ∞−∞eiξ(y−y0)ξ2 + 1dξ)dy+a2e(f(a)−y0) +b2e−(f(b)−y0). (3.2)The above integral is calculated by the transformation y = f(x) of thevariables y, x as follows:12pi∫ ba(∫ ∞−∞iξeiξ(f(x)−y0)ξ2 + 1dξ + xf ′(x)∫ ∞−∞eiξ(f(x)−y0)ξ2 + 1dξ)dx (3.3)=12pi∫ ∞−∞(∫ ba(iξeiξ(f(x)−y0)ξ2 + 1+ xf ′(x)eiξ(f(x)−y0)ξ2 + 1)dx)dξ=12pi∫ ∞−∞1ξ2 + 1∫ ba(iξeiξ(f(x)−y0) + xf ′(x)eiξ(f(x)−y0))dxdξ=12pi∫ ∞−∞1ξ2 + 1(iξ∫ baeiξ(f(x)−y0)dx− iξ[xeiξ(f(x)−y0)]ba+iξ∫ baeiξ(f(x)−y0)dx)dξ=12pi∫ ∞−∞1ξ2+1(i(ξ+1ξ)∫ baeiξ(f(x)−y0)dx− iξ(beiξ(f(b)−y0)−aeiξ(f(a)−y0)))dξ=i2pi∫ ∞−∞1ξ∫ baeiξ(f(x)−y0)dxdξ+ia2pi∫ ∞−∞eiξ(f(a)−y0)ξ(ξ2 + 1)dξ− ib2pi∫ ∞−∞eiξ(f(b)−y0)ξ(ξ2 + 1)dξ164 M. Yamada, T. Matsuura, S. Saitoh=i2pi∫ ba∫ ∞−∞eiξ(f(x)−y0)ξdξdx+ia2pi∫ ∞−∞eiξ(f(a)−y0)ξ(ξ2 + 1)dξ− ib2pi∫ ∞−∞eiξ(f(b)−y0)ξ(ξ2 + 1)dξ= − 12pipi∫ basign(f(x)−y0)dx− a2pi2pie− 12(f(a)−y0)sign(f(a)−y0)sinh(f(a)−y02)+b2pi2pie−12(f(b)−y0)sign(f(b)−y0) sinh(f(b)− y02)= −12∫ basign(f(x)−y0)dx− a2(ef(a)−y02 −e− f(a)−y02 )e− 12 (f(a)−y0)sign(f(a)−y0)+b2(ef(b)−y02 − e− f(b)−y02 )e− 12 (f(b)−y0)sign(f(b)−y0)=12∫ basign(y0 − f(x))dx− a2(ef(a)−y02 − e− f(a)−y02 )e 12 (f(a)−y0)+b2(ef(b)−y02 − e− f(b)−y02 )e− 12 (f(b)−y0)=12∫ basign(y0 − f(x))dx− a2(e(f(a)−y0) − 1)+ b2(1− e−(f(b)−y0))=a+ b2+12∫ basign(y0 − f(x))dx− a2e(f(a)−y0) − b2e−(f(b)−y0).The above singular integrals are considered in the sense of the Cauchy prin-cipal values.Thus, we obtain the representationf−1(y0) =a+ b2+12∫ basign(y0 − f(x))dx. (3.4)Using reproducing kernel Hilbert spaces from [2] as in (3.1), we calculatesimilarly with the related assumptions, but surprisingly enough, we obtainthe same formula (3.4). For this formula (3.4), we note that we do not needany smoothness assumptions for the function f(x). Indeed, we need onlythe strictly increasing assumption. Even the assumption of integrability isnot needed.4. Numerical experimentsWe give here several numerical examples. The integrals have been pro-cessed easily by Mathematica.REPRESENTATIONS OF INVERSE FUNCTIONS BY . . . 165-1 -0.5 0.5 1 x-1-0.50.51y-1 -0.5 0.5 1 y-1-0.50.51xFigure 1. The graph of inverse function for y = f(x) = x3.0.2 0.4 0.6 0.8 1 x0.20.40.60.81y0.2 0.4 0.6 0.8 1 y0.20.40.60.81x166 M. Yamada, T. Matsuura, S. SaitohFigure 2. The graph of inverse function fory = f(x) ={x(2− sin log x− cos log x) 0 < x ≤ 10 x = 0-1 -0.5 0.5 1 x246810y2 4 6 8 10 y-1-0.50.51xFigure 3. The graph of inverse function fory = f(x) ={x x ≤ 010x x > 0-1 -0.5 0.5 1 x-1-0.50.511.52yREPRESENTATIONS OF INVERSE FUNCTIONS BY . . . 167-1 -0.5 0.5 1 1.5 2 y-1-0.50.51xFigure 4. The graph of inverse function fory = f(x) ={x x ≤ 0x+ 1 x > 0-1 -0.5 0.5 1 1.5 2 x-1-0.50.51y-1 -0.5 0.5 1 y-1-0.50.511.52xFigure 5. The graph of inverse function fory = f(x) =x x ≤ 00 0 < x ≤ 1x− 1 x > 1168 M. Yamada, T. Matsuura, S. SaitohAcknowledgements: The second author, T. Matsuura is supportedin part by the Grant-in-Aid for Scientific Research (C)(No.18540110) fromthe Japan Society for the Promotion Science. The third author, S. Saitohis supported in part by the Grant-in-Aid for Scientific Research (C)(2)(No.19540164) from the Japan Society for the Promotion Science.References[1] S. Saitoh, Representations of inverse functions. Proc. Amer. Math. Soc.125 (1997), 3633-3639.[2] S. Saitoh, Integral Transforms, Reproducing Kernels and Their Ap-plications. Pitman Res. Notes in Math. Series 369, Addison Wesley -Longman Ltd., UK (1997).Received: September 9, 20071 Division of Electronics and Computing, Graduate School of Engineering,Gunma University, Kiryu 376-8515, JAPAN2 Department of Mechanical Engineering, Graduate School of Engineering,Gunma University, Kiryu 376-8515, JAPAN3 Department of Mathematics, Graduate School of Engineering,Gunma University, Kiryu 376-8515, JAPANe-mails:1 d07e403@gs.eng.gunma-u.ac.jp2 matsuura@me.gunma-u.ac.jp3 ssaitoh@math.sci.gunma-u.ac.jp

Representations of Inverse Functions by the Integral Transform with the Sign Kernel

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Representations of Inverse Functions by the Integral Transform with the Sign Kernel

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