REPRESENTATIONS OF INVERSE FUNCTIONS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 2, December 997, Pages S (97)438-5 REPRESENTATIONS OF INVERSE FUNCTIONS SABUROU SAITOH (Communicated by Theodore W. Gamelin) Abstract. Let φ : Ê E be an arbitrary map from an abstract set Ê into an abstract set E. We shall use the theory of reproducing kernels to provide a general method for representing the inverse map φ in terms of φ. We give several concrete examples of this method.. Introduction We shall consider an arbitrary mapping p = φ(ˆp) (.) φ : Ê E from an abstract set Ê into an abstract set E. Then, we shall consider the formal problem of representing the inverse map φ in terms of φ. Of course, the inverse is, in general, multi-valued. By using the theory of reproducing kernels ([]), we shall first attack this general problem, then, we shall establish a general principle to solve this problem in some general reasonable settings. We shall also give several concrete examples. 2. A general approach Let K(p, q) be a positive matrix on E in the sense of Aronszajn-Moore, so that for any finite number of points. {p j } of E for any complex numbers {c j }, (2.) c k c j K(p k,p j ). k j Then, there exists a uniquely determined functional Hilbert space H K consisting of functions on E admitting the reproducing kernel K(p, q) with the properties that (i) K(,q) H K for any q E, (ii) (f( ),K(,q)) HK = f(q), for any q E for any f H K. We shall assume that K(p, q) is expressible in the form (2.2) K(p, q) =(h(q),h(p)) H on E E Received by the editors May 5, 995, in revised form, July 5, Mathematics Subject Classification. Primary 3C4. Key words phrases. Inverse function, reproducing kernel, the Riemann mapping function, Bergman kernel, Szegö kernel, multi-valued function, harmonic mapping, increasing function c 997 American Mathematical Society

2 3634 SABUROU SAITOH in terms of a Hilbert space H-valued function h(p) one.(cf.[4],[5].)wefurther assume that (2.3) {h(p); p E} is complete in H. Then, for the linear transform of H (2.4) f(p) =(f,h(p)) H, f H, the images f(p) form precisely the Hilbert space H K, furthermore we have the isometrical mapping (2.5) f HK = f H. If the assumption (2.3) is not valid, then we have in (2.5), in general, the inequality (2.6) f HK f H. Now, by using the mapping φ we define the function (2.7) K φ (ˆp, ˆq) =K(φ(ˆp),φ(ˆq)) = (h(φ(ˆq)), h(φ(ˆp))) H on Ê Ê the linear mapping of H (2.8) f(φ(ˆp)) = (f, h(φ(ˆp))) H, f H. Of course, K φ (ˆp, ˆq) is a positive matrix on Ê, so there exists a uniquely determined functional Hilbert space H Kφ admitting the reproducing kernel K φ (ˆp, ˆq). Then, we obtain, from (2.5) (2.6) Theorem 2.. For an arbitrary mapping φ in (.), the functions f(φ(ˆp)) in (2.8) form the reproducing kernel Hilbert space H Kφ we have the inequality (2.9) f(φ) HKφ f HK. Isometry holds here if only if (2.) {h(p); p Range φ} is complete in H. Furthermore, we have Theorem 2.2. Let φ : Ê E be arbitrary let ˆf H Kφ.Iff H K satisfies (2.) ˆf = f (φ) ˆf HKφ = f HK, then (2.2) f (p) =(ˆf( ),K(φ( ),p)) HKφ. Proof. Let N(φ) denotethesetoff Hsuch that (f, h(φ(ˆp))) H = on Ê, let [N(φ)] denote its orthogonal complement in H. Then, in the mapping (2.8) we can take f [N(φ)] satisfying (2.3) ˆf(ˆp)=(f,h(φ(ˆp))) H (2.4) ˆf HKφ = f H.

3 REPRESENTATIONS OF INVERSE FUNCTIONS 3635 Then, for (2.5) f (p) =(f,h(p)) H, we have (2.6) ˆf(ˆp)=f (φ(ˆp)) satisfying (2.). Then, for the orthogonal projection P from H onto [N(φ)] we have the desired result f (p) =(f,h(p)) H = ( P [N(φ)] f, h(p) ) H = ( f,p [N(φ)] h(p) ) H ( ) = ˆf( ),K(φ( ),p) H Kφ. Now, in Theorem 2.2, we shall assume that the mapping φ is onto Ê so isometry between H K H Kφ. If we know the mapping p = φ(ˆp) from Ê to E, then the isometrical mapping from H K onto H Kφ is given by (2.7) f(p) H K f (φ(ˆp)) H Kφ. If we know the (in general, multi-valued) inverse ˆp = φ (p) ofp=φ(ˆp), then for ˆf H Kφ, the function ˆf(φ (p)) is a single-valued function on E the isometrical mapping from H Kφ onto H K is given by (2.8) ˆf H Kφ ˆf (φ (p)) H K. In general, Theorem 2.2 establishes the isometrical mapping (2.9) ˆf H Kφ f H K, explicitly in terms of the reproducing kernel K(p, q) one, the mapping φ the reproducing kernel Hilbert space H Kφ. This fact means that Theorem 2.2 gives, in a sense, a method for constructing the inverse φ. Indeed, for any point p E for any fixed function ˆf H Kφ we first construct the function f by (2.2). Then, we have the inclusion relationship { φ (p) ˆp Ê; ˆf(ˆp)=f } (2.2) (p). Hence, we will be able to look for all the inverses φ (p) in the point set in the righth side in (2.2), by using a suitable Hilbert space H K a suitable function ˆf H Kφ. 3. Reasonable settings We shall analyze the principle in Section 2 to represent the inverse φ in (.) interms of φ.

4 3636 SABUROU SAITOH In order to use the formula (2.2) we need (I) a concrete structure for the Hilbert space H Kφ, admitting its reproducing kernel K φ (ˆp, ˆq) defined by (2.7) (II) for a function ˆf belonging to H Kφ, its inverse ˆf. Then we have the inverse function of φ (3.) ˆp = ˆf (f (p)), in (2.2). For (I), if the Hilbert space H K admitting the reproducing kernel K(p, q) one is concretely given, then the transformed Hilbert space H Kφ will be constructed by a unified method stated in Theorem 2. Theorem 2.2 ([6]). If the inner product in H K is given by an integral form, then the inner product in H Kφ will be given by some integral form induced by the mapping φ, asweshall see in examples. For (II), if the identity mapping belongs to H Kφ, then our situation will become, of course, extremely simple. 4. Examples Following our general principle, we shall give typical examples. These examples will show that concrete reproducing kernels have great value from also the viewpoint of representations of inverse functions. 4.. The Riemann mapping function. Let be the unit disc { z < } D an arbitrary bounded (for simplicity) domain on the Z-plane (Z = X + iy ). Let z = ϕ(z) be a Riemann mapping function from D onto, which is analytic univalent on D. LetK (z,u) be the Bergman kernel (4.) K (z,u) = π( uz) 2 on for the Bergman space H K comprised of all analytic functions f(z) on with finite norms { f(z) 2 2 (4.2) dxdy < (z = x + iy). Then, we can see directly that K,ϕ (ϕ(z), ϕ(u)) = π( ϕ(u)ϕ(z)) 2 is the reproducing kernel for the Hilbert space H K,ϕ comprised of all analytic functions ˆf(Z) =f(ϕ(z))(f H K ) with finite norms { ˆf(Z) 2 ϕ (Z) 2 2 (4.3) dxdy <. D Hence, we have the inverse formula by (2.2) by using the identity for ˆf in (2.2) ϕ (z) = Z ϕ (Z) 2 (4.4) π ( zϕ(z)) dxdy. 2 D

5 REPRESENTATIONS OF INVERSE FUNCTIONS 3637 When Z = ϕ (z) is not one-to-one, if we consider D as a Riemann surface ϕ ( ) spreadoverc, counting multiplicity, then the formula (4.4) is still valid. By this method, we can, in general, overcome the multi-valuedness of the inverse functions. When D is a bounded domain whose boundary D is an analytic Jordan curve, we recall the Szegö reproducing kernel (4.5) 2π( uz) for the Hilbert space consisting of all analytic functions f(z) on such that f(z) 2 has a harmonic majorant with finite norm { f(z) 2 2 (4.6) dz <. Here, f(z) means the Fatou nontangential boundary value on. Then, we obtain similarly the simple inverse formula ϕ (z) = Z ϕ (4.7) (Z) 2π D zϕ(z) dz. The very elementary formulas (4.4) (4.7) can be derived directly easily, but it seems that they cannot be found in any articles. Note that the Riemann mapping function ϕ(z) satisfying ϕ(z )= (Z D) is expressible in the form ϕ(z) = π K D (Z,Z ) Z Z K D (ζ,z )dζ in terms of the Bergman kernel K D (ζ,z) on the domain D (see, for example, [3]) Harmonic mappings. Note that on u 2 z 2 2 uz 2 (4.8) uz 4 u 2 z 2 (4.9) uz 2 are the reproducing kernels for the Hilbert spaces consisting of all harmonic functions u(z) on with finite norm { u(z) 2 2 (4.) dxdy < π { u(z) 2 2 (4.) dz <, 2π respectively. For the latter case we need the assumption that u(z) 2 has a harmonic majorant on u(z)(z ) means the Fatou nontangential boundary value as in the Szegö case (see, for example, [2] [5]). As in the Riemann mapping function we can obtain the representations of the inverse functions for harmonic mappings of D onto.

6 3638 SABUROU SAITOH 4.3. Increasing functions. Note first that (4.2) min(x, y) (x, y > ) is the reproducing kernel for the Hilbert space H(,a)( <a ) consisting of all real-valued functions f(x) on[,a) such that f(x) are absolutely continuous on [,a),f() = with finite norms { a f (x) 2 2 (4.3) dx <. Similarly, for an increasing function x = ϕ(ˆx) from [,b)( <b )onto[,a)of C -class satisfying ϕ (ˆx) > on [,b), the function (4.4) min(ϕ(ˆx),ϕ(ŷ)) on [,b) [,b) is the reproducing kernel for the Hilbert space H ϕ consisting of all functions ˆf(ˆx)= f(ϕ(ˆx))(f H(,a)), such that ˆf(ˆx) are absolutely continuous on [,b), ˆf() = with finite norms { b ˆf (ξ) 2 dξ 2 (4.5) ϕ <, (ξ) as we can see directly. Hence, by (2.2) by using the identity as ˆf in (2.2), we have the formula b ϕ (x) = {min(ϕ(ξ),x)} dξ (4.6) ϕ (ξ) b ( 2 ) sin(ϕ(ξ)t) sinxt dξ = π t 2 dt ϕ (ξ) = 2 b cos(ϕ(ξ)t) sinxt dtdξ. π t In particular, we have (4.7) n x = 2 π cos(ξ n t)sinxt dtdξ. t Acknowledgment This research was partially supported by the Japanese Ministry of Education, Science Culture; Grant-in-Aid Scientific Research, General Research (C) References. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (95), MR 4:479c 2., Green s functions reproducing kernels, Proceedings of the Symposium on Spectral Theory Differential Problems, Oklahoma A. M. College, Oklahoma, (95), pp MR 5:878a 3. S. Bergman, The kernel function conformal mapping, Amer. Math. Soc. Providence, R.I., (97). MR 58:2252

7 REPRESENTATIONS OF INVERSE FUNCTIONS S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 89 (983), MR 84h:44 5., Theory of reproducing kernels its applications, Pitman Res. Notes in Math. Series 89, Longman Scientific & Technical, Engl, (988). MR 9f: , Decreasing principles in transforms of reproducing kernel Hilbert spaces, Mathematica Montisnigri (993), MR 94i:4642 Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, Japan address: