Reproducing Kernel Hilbert Space and Coalescence Hidden-variable Fractal Interpolation Functions

Abstract: Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In thepresent paper, the spaceof Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.


Introduction
The notion of Fractal Interpolation Function (FIF) and its construction was introduced by Barnsley [1] using the theory of Iterated Function System (IFS) and Read-Bajraktarevic operator. Since then, FIFs have been an amazing asset for interpolation of experimental data by a non-smooth curve and has extensive applications in engineering [2], biological sciences [3], planetary science [4] and arts [5]. After the introduction of FIF, different other kinds of FIFs namely Hidden-variable FIFs, Hermite FIFs, Spline FIFs and Super FIFs have also been constructed [6][7][8] and properties such as smoothness [9], approximation property [10,11] and regularity [12] have been discussed. The idea of constructing Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) for simulation of curves that exhibit partly self-affine and partly non-self-affine nature was introduced by Chand and Kapoor [13]. The author had studied the effect of insertion of a new point in the interpolation data on the related IFS and the CHFIF [14] and Riemann-Liouville fractional calculus of CHFIF [15].
Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are significant device which have been found valuable in numerous regions e.g. machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. The theory of RKHS was introduced [16][17][18] and have been used in the statistics literature for the past twenty years. Different kinds of kernels along with their respective RKHS (e.g. Gaussian Kernel) have been in use for a long time. However, most of these kernel spaces consisted of smooth functions. Bouboulis and Mavroforakis [19] showed that the space of any family of FIFs or Recurrent FIFs is an RKHS with a specific associated kernel function. This introduced the self-affine fractal functions to the RKHS universe. Multiresolution analysis arising from CHFIFs which exhibit partly self-affine and partly non-self-affine was developed [20] and as a natural follow-up, they have also been applied to construct orthonormal wavelets [21]. In this paper, the space of CHFIFs is shown to constitute an RKHS and its associated kernel is obtained. This increases the chance of utilizing this new kernel function which is partly self-affine and partly non-self-affine to fields where the structure is not always self-affine.
The organization of the paper is as follows: Section 2 summaries the construction of a CHFIF. Section 3 discusses about the vector space of CHFIFs and its dimension. Section 4 begins with a brief introductory note on RKHS. Then, the space of CHFIFs is shown to be an RKHS and subsequently, its associated kernel is also derived.

Construction of a CHFIF
In this section, the basics of the construction of a CHFIF is discussed. Given interpolation data on R 2 , a CHFIF is constructed as the first component of the attractor of a suitably defined IFS in R 3 with the introduction of generalized interpolation data.
Let the given interpolation data be The IFS required to construct a CHFIF is defined as and Here, αn and n are free variables chosen such that |αn| < 1 and | n| < 1. However, βn are said to be constrained variables as they are chosen such that |βn| + | n| < 1. The functions pn and qn are continuous chosen such that the functions Fn satisfy Fn(x 0 , y 0 , z 0 ) = (y n−1 , z n−1 ) and Fn(x N , y N , z N ) = (yn , zn).
The above conditions are called join-up conditions. It is proved [13] that the above IFS is hyperbolic with respect to a metric d * on R 3 , equivalent to the Euclidean metric. For a hyperbolic IFS, it is known that there exists a unique non-empty compact set A ⊆ R 3 such that A =

Space of CHFIFs
In this section, the vector space of CHFIFs is introduced and its dimension is found.
For f , g ∈ S 0 and a ∈ R, define af + g = (af 1 + g 1 , af 2 + g 2 ). Then, S 0 is a vector space, with usual point-wise addition and scalar multiplication. The set S 0 together with the maximum metric Definition 3.2. Let S 1 0 be the set of functions f 1 : I → R that are first components of functions f ∈ S 0 . Then, S 1 0 is also a vector space with point-wise addition and scalar multiplication.
The following proposition, proved in [20], gives the dimension of S 0 and S 1 0 : Then, V is a vector space with usual point-wise addition and scalar multiplication. Let B(I, R 2 ) denote the set of bounded functions from I to R 2 and C(I, R 2 ) denote the set of continuous functions from I to R 2 . Define the maximum metric on B(I, R 2 ) and C(I, )︁ for x ∈ In, n = 1, 2, . . . , N, where pn and qn are linear polynomials that satisfy the join up conditions: Then Φ t is a contraction map on B(I, R 2 ) and hence by the Banach contraction mapping theorem, Φ t has a unique fixed point f t ∈ B(I, R 2 ). By join-up conditions (2), it follows that f t is continuous.
The above equation gives the following on simplification: Therefore, af t + f¯t is a fixed point of Φ at+t for all a ∈ R and t,t ∈ V. By uniqueness of fixed point of Φ at+t , it follows that 0)). This gives that Now, consider the projection map P : S 0 → S 1 0 . Then, Kernel of P ≡ {f ∈ S 0 such that P(f ) = 0} is a proper subset of S 0 and consists of elements of the form (0, 0) and (0, f 2 ). For the element (0, f 2 ) ∈ KerP, it is observed that βn f 2 (L −1 n (x)) + pn(L −1 n (x)) = 0 for x ∈ In. Hence, for all x ∈ I, it is seen that f 2 (x) = −1 βn pn(x). and for x ∈ [x n−1 , xn]. The linear isomorphism between V and the vector space S 0 together with equations (4) and (5) gives that for f = (f 1 , f 2 ) ∈ S 0 , the function f 1 is the unique CHFIF passing through (x i , y i ), while the function f 2 is the unique AFIF passing through (x i , z i ). By the linear isomorphism between V and the vector space S 0 , it follows using (5) that f 2 is completely determined by f 2 (x i ) for i = 0, 1, . . . N. Further, it follows by (4) that f 1 depends on f 2 . If f 2 is a polynomial of degree of at most 1, then f 1 is affine FIF.

Reproducing kernel Hilbert space of CHFIFs
In this section, a brief introductory note on RKHS is given. Then, the space of S 0 is shown to be an RKHS and its associated kernel function is derived. Subsequently, the space of CHFIFs is also proven to be an RKHS and its associated kernel function is also obtained.
Let H denote a linear class of real-valued functions f defined on a set X and define an inner product ⟨·, ·⟩ with corresponding norm || · || on H such that H is complete with respect to that norm. Then H is a Hilbert space. The space H is said to be a Reproducing Kernel Hilbert Space (RKHS) if there exists a function κ : X × X → R satisfying the following two properties: 1. For every x ∈ X, κ(x, ·) ∈ H and 2. For all f ∈ H, f (x) = ⟨f , κ(x, ·)⟩. In particular, κ(x, y) = ⟨κ(x, ·), κ(y, ·)⟩. Then, κ is called the reproducing kernel of H.
The definition of reproducing kernel says that it depends on the inner product and the Hilbert space. There could be several inner products defined in the same Hilbert space. Hence, the reproducing kernel of a Hilbert space varies if the inner product is changed. The following result is useful in describing the Kernel of a Hilbert space: . Using the above proposition, we shall now derive a kernel for the space S 0 .
Using the above proposition, if we define κ(x, y) = 2N+2 ∑︀ i,k=1 ducing kernel of the space S 0 and hence S 0 is an RKHS.
The above theorem does not help in determining the kernel of space of CHFIFs. It neither induces a norm in the space S 0 . Suppose the inner product on S 0 is chosen as ⟨⟨f ,f ⟩⟩ = ⟨f 1 , Then, it induces a norm on S 0 given by |||f ||| = √︀ ||f 1 || 2 + ||f 2 || 2 . In order to study the kernel of space of CHFIF, assume that y i,j = δ i,j+1 , z i,j = 0 and z N+1+i,j = δ i,j+1 for i = 1, . . . , N + 1 and j = 0, 1, . . . , N. Also, assume that y N+2,j = y 2N+2,j = 0 for j = 0, 1, . . . , N. To derive the kernel of space of CHFIFs, y i,0 and y i,N for i = N + 3, . . . , 2N + 1 are chosen such that y i,0 = y i,N = 0 and for  j = 2, . . . , N, y i,j−1 are real numbers chosen such that ⟨f i1 , f j1 ⟩ = 0. Let, for i = N + 3, . . . , 2N + 1, The Then A is an invertible matrix of order 2N with entry A i,k at (i, k) position and the inverse of A is B. Using the above proposition, it is clear that κ(x, y) = is a reproducing kernel of the space S 1 0 and hence S 1 0 is an RKHS.
With the above choices of free variables and constrained variables and norm inducing inner product on the space S 0 , another Kernel for the space is given as follows:

Conclusions
In this paper, it is shown that the space of CHFIFs is an RKHS and its associated kernel is obtained. This broadens the likelihood of using this new kernel function which is partly self-affine and partly non-selfaffine to fields where the structure is not always self-affine. The Space S 0 consisting of vector valued functions f = (f 1 , f 2 ), where the first component f 1 is a CHFIF and second component f 2 is an AFIF is also shown to be an RKHS with respect to two different inner products and corresponding to each inner product, its associated kernel is also derived.