A Fractal Operator on Some Standard Spaces of Functions 3

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Turning to the layout of our paper, in Section 2, we assemble the requisite general material. In Section 3, we construct α-fractal functions in various spaces of functions. Section 4 concerns development of some foundational aspects of the associated fractal operator.

2. Notation and Preliminaries In this section we provide a rudimentary introduction to function spaces dealt in this article. Further, we review the requisite background material on fractal functions. We recommend the reader to consult the references [1, 8, 14, 15] for further details. The set of natural numbers will be denoted by N, whilst the set of real numbers by R. For a fixed N ∈ N, we shall write NN for the set of first N natural numbers. By a self-map we mean a function whose domain and codomain are same. For a given compact interval I = [c, d] ⊂ R, let B(I) := g : I → R : g is bounded on I . The functional kgk∞ := sup |g(x)| : x ∈ I , I k N termed uniform norm, turns B( ) to a Banach space. For ∈ ∪{0}, consider the space Ck(I) := g : I → R : g is k-times differentiable and g(k) ∈C(I) endowed with the norm (r) kg||Ck := max kg k∞ : r ∈ Nk ∪{0} . In addition to the Banach spaces B(I) and Ck(I), we require the following spaces that have numerous applications in analysis. For 0

Banach space. Note that for p = 2, the space L2(I) is a Hilbert space with respect to the inner product

:= g1g2 dx ZI For 0

u Dkϕ dx = (−1)k vϕ dx ZI ZI for all infinitely differentiable functions ϕ with ϕ(c)= ϕ(d) = 0. For 1 ≤ p ≤ ∞ and k ∈ N ∪{0}, let Wk,p(I) denote the usual Sobolev space. That is, Wk,p(I) := g : I → R : Dj g ∈ Lp(I), j =0, 1,...,k}, where Dj g denotes the j-th weak or distributional derivative of g. The space Wk,p(I) endowed with the norm 1 k j p p j=0 kD gkp , for p ∈ [1, ∞), kgkWk,p :=  h k Djg ,i p .  Pj=0 k k∞ for = ∞ is a Banach space. For p = 2, it is customary to denote Wk,p(I) by Hk(I), which is a Hilbert  P space. A function u : I → R is said to be Hölder continuous with exponent σ if |u(x) − u(y)|≤ C|x − y|σ, for all x, y ∈ I and for some C > 0. A FRACTAL OPERATOR ON SOME STANDARD SPACES OF FUNCTIONS 3

For Hölder continuous functions u with exponent σ, let us define σ-th Hölder semi-norm as |u(x) − u(y)| u [ ]σ = sup σ x,y∈I, x6=y |x − y| and consider the Hölder space Ck,σ(I) := g ∈Ck−1(I): g(k) is Hölder continuous with exponent σ . The space Ck,σ(I) is a Banach space when endowed with the norm k (j) (k) kgkCk,σ := kg k∞ + [g ]σ. j=0 X Having exposed the reader with function spaces that we shall encounter in due course, next we provide an overview of fractal interpolation and related ideas. Assume that N ∈ N, N > 2. Let 2 (xi,yi) ∈ R : i ∈ NN denote the prescribed set of interpolation data with strictly increasing abscissae. Set I = [x , x ] and I = [x , x ] for i ∈ N . Fractal interpolation constructs a 1 N i i i+1 N−1 continuous function g : I → R satisfying g(xi) = yi for all i ∈ NN and whose graph G(g) is a fractal in the sense that G(g) is a union of transformed copies of itself (see (2.1)). Suppose that Li : I → Ii, i ∈ NN−1 are affinities satisfying

Li(x1)= xi,Li(xN )= xi+1.

For i ∈ NN−1, let Fi : I × R → R be continuous functions fulﬁlling ∗ ∗ Fi(x, y) − Fi(x, y ) ≤ ci|y − y |, Fi(x1,y1)= yi, Fi(xN ,yN )= yi+1. ∗ R for y,y ∈ and 0

Cy1,yN (I)= {h ∈C(I): h(x1)= y1,h(xN )= yN }.

It is readily observed that Cy1,yN (I) is a closed (metric) subspace of the Banach space C(I), k.k∞ .

Define the Read-Bajraktarević(RB) operator T : C 1 (I) →C 1 (I) via y ,yN y ,yN −1 −1 N (Th)(x)= Fi Li (x),h ◦ Li (x) , x ∈ Ii, i ∈ N−1. N The nonlinear mapping T is a contraction with a contraction factor c := max{ci : i ∈ N−1}. Consequently, by the Banach fixed point theorem, T has a unique fixed point, say g. Further, g interpolates the data {(xi,yi): i ∈ NN } and enjoys the functional equation

g Li(x) = Fi x, g(x) , x ∈ I, i ∈ NN−1. R R R Let wi : I × → Ii × ⊂ I × be deﬁned by

wi(x, y)= Li(x), Fi(x, y) , i ∈ NN−1.

Consider the iterated function system (IFS) W = I × R; wi,i ∈ NN−1 . Then the attractor G of the IFS W is the graph G(g) of the function g, that is,

(2.1) G(g)= ∪i∈NN−1 wi G(g) . Therefore, g is a self-referential function. Navascu´es [8] observed that the theory of FIF can be used to generate a family of continuous functions having fractal characteristics from a prescribed continuous function. To this end, for a given f ∈C(I), consider a partition ∆ := {x1, x2,...,xN } of I satisfying x1 < x2 < ··· < xN , a continuous map b : I → R such that b 6= f, b(x1)= f(x1) and b(xN )= f(xN ), and N − 1 real numbers αi satisfying |αi| < 1. Deﬁne an IFS through the maps

(2.2) Li(x)= aix + di, Fi(x, y)= αiy + f ◦ Li(x) − αib(x), i ∈ NN−1. α α The corresponding FIF denoted by f∆,b = f is referred to as α-fractal function for f (fractal perturbation of f) with respect to a scale vector α = (α1, α2,...,αN−1), base function b, and partition ∆. Here the interpolation points are (xi,f(xi)) : i ∈ NN . The fractal dimension (Minkowski dimension or Hausdorff dimension) of the function f α depends on the parameter N−1 α α α ∈ (−1, 1) . The function f is the fixed point of the operator T∆,b,f : Cf (I) → Cf (I) defined by α −1 N (2.3) (T∆,b,f g)(x)= f(x)+ αi g − b) ◦ Li (x), x ∈ Ii, i ∈ N−1, 4 P. VISWANATHAN AND M.A. NAVASCUES´ where Cf (I) := g ∈ C(I): g(x1) = f(x1), g(xN ) = f(xN ) . Consequently, the α-fractal function corresponding to f satisfies the self-referential equation α α −1 N (2.4) f∆,b(x)= f(x)+ αi f∆,b − b (Li (x)), x ∈ Ii, i ∈ N−1. Assume that the base function b depends linearly on f, say b = Lf, where L : C(I) →C(I) is a bounded linear map. Then the following map referred to as α-fractal operator α α α F∆,b : C(I) →C(I), F∆,b(f)= f∆,b is a bounded linear operator. To obtain fractal functions with more flexibility, iterated function system wherein scaling factors are replaced by scaling functions received attention in the recent literature on fractal functions. That is, one may consider the IFS with maps

(2.5) Li(x)= aix + di, Fi(x, y)= αi(x)y + f ◦ Li(x) − αi(x)b(x), i ∈ NN−1, where αi are continuous functions satisfying max{kαik∞ : i ∈ NN−1} < 1, and b 6= f is a continuous function that agrees with f at the extremes of the interval I (see, for instance, [17]). The corresponding α-fractal function is the ﬁxed point of the RB-operator α −1 −1 N (2.6) (T∆,b,f g)(x)= f(x)+ αi Li (x) (g − b) Li (x) , x ∈ Ii, i ∈ N−1, and hence obeys the functional equation α −1 α −1 N (2.7) f∆,b(x)= f(x)+ αi Li (x)) f∆,b − b) Li (x) , x ∈ Ii, i ∈ N−1.

3. α-Fractal Functions on Various Function Spaces

In this section, by considering suitable conditions on the scaling functions αi and base function α b, we construct α-fractal function f∆,b corresponding to f in various function spaces. The idea α is to choose the parameters so that the map T∆,b,f becomes a contraction in the space under α consideration and f∆,b is the corresponding fixed point, whose existence is ensured by the Banach fixed point theorem. For the sake of notational simplicity, we may suppress the dependence on α α α α α parameters to denote T∆,b,f by T , f∆,b by f and the corresponding fractal operator F∆,b by F . The following theorems provide sufficient conditions on the scaling functions and base functions for the associate fractal function f α to share the properties of the original function f.

Theorem 3.1. Let f ∈B(I). Assume that ∆ := {x1, x2,...,xN } is a partition of I with strictly increasing abscissae, and Ii = [xi, xi+1) for i ∈ NN−2, IN−1 = [xN−1, xN ] be the corresponding subintervals. Further assume that the maps Li : I → Ii are affinities satisfying Li(x1) = xi, − N N Li(xN ) = xi+1 for i ∈ N−1, and the base function b and scaling functions αi, i ∈ N−1 are real-valued bounded functions on I. Then the RB-operator defined in (2.6) is a well-defined self- map on B(I). Furthermore, if kαk∞ = max{kαik∞ : i ∈ NN−1} < 1, then T is a contraction and has a unique fixed point f α ∈B(I).

Proof. Under the stated hypotheses on functions b, αi and aﬃnities Li, it is straightforward to show that T is well-deﬁned and maps into B(I). Let g1 and g2 be in B(I). For x ∈ Ii, −1 −1 (Tg1)(x) − (Tg2)(x) ≤ αi Li (x) (g1 − g2) Li (x) ≤ k αik∞kg1 − g2 k∞.

Therefore kTg1 − Tg2k∞ ≤kαk∞kg1 − g2k∞, and the assertion follows. In the next theorem, we construct α-fractal functions in Lebesgue space Lp(I), for 0

p Theorem 3.2. Let f ∈ L (I), 0 < p ≤ ∞. Suppose that ∆ = {x1, x2,...,xN } is a partition of I satisfying x1 < x2 < ··· < xN , Ii := [xi, xi+1) for i ∈ NN−2 and IN−1 := [xN−1, xN ]. − Let Li : I → Ii be aﬃne maps Li(x) = aix + di satisfying Li(x1) = xi and Li(xN ) = xi+1 ∞ p for i ∈ NN−1. Choose αi ∈ L (I) for all i ∈ NN−1 and b ∈ L (I). Then the RB operator T p given in (2.6) deﬁnes a self-map on L (I). Furthermore, for the scaling functions αi, i ∈ NN−1 satisfying the following condition, T is a contraction map on Lp(I).

1/p p aikαik∞ < 1, for p ∈ [1, ∞), i∈N −1  N hmaxP kα k < i1, for p = ∞,  N i ∞  i∈ N−1  p aikαik∞ < 1, for p ∈ (0, 1). i∈NN−1  P  Consequently, the corresponding ﬁxed point f α ∈ Lp(I) obeys the self-referential equation (2.7).

Proof. It follows at once from the hypotheses on the function f, aﬃnities Li, scaling functions p αi, and base function b that the operator T is a well-deﬁned self-map on L (I). It remains to p prove that under the stated conditions on scaling functions, T is a contraction. Let g1,g2 ∈ L (I) and 1 ≤ p< ∞.

p p kTg1 − Tg2kp = (Tg1 − Tg2)(x) dx Z I p −1 −1 −1 = αi Li (x) g1 Li (x) − g2 Li (x) dx i∈NN−1 IZi X p = ai αi(˜x) g1(˜x) − g2(˜x) d˜x N i∈ N−1 ZI X p p ≤ aikαik∞ g1(˜x) − g2(˜x) d˜x N i∈ N−1 Z X I p p = aikαik∞ kg1 − g2kp. N hi∈XN−1 i −1 In the third step of the preceding analysis we used the change of variablex ˜ = Li (x), whereas 1/p p the rest follows directly by computations. Hence, we see that for aikαik∞ < 1 the i∈NN−1 map T is a contraction. Proof is similar for the remaining cases. h P i

Recall that, in general, the α-fractal function has noninteger Hausdorﬀ and Minkowski dimen- sions that depend on the scale vector α and hence the map f 7→ f α is a “roughing” operation. In the following theorem we prove that by appropriate choices of scaling functions and base function, the order of continuity of f can be preserved in f α. This extends the construction of smooth fractal function enunciated in [12] with the aid of Barnsley-Harrington (BH) theorem [2]. It is to be stressed that BH theorem does not rescue us in the setting of variable scaling.

k Theorem 3.3. Let f ∈ C (I), where k ∈ N. Suppose that ∆ = {x1, x2,...,xN } is a partition of I satisfying x1 < x2 < ··· < xN , Ii := [xi, xi+1] for i ∈ NN−1 and Li : I → Ii are aﬃne maps Li(x)= aix + di satisfying Li(x1)= xi and Li(xN )= xi+1 for i ∈ NN−1. Suppose that k-times continuously diﬀerentiable scaling functions and base function are selected so that

ai k (r) (r) (r) (r) kα k k ≤ ( ) ; b (x )= f (x ), b (x )= f (x ), i ∈ N , r ∈ N ∪{0}. i C 2 1 1 N N N−1 k Then the RB operator deﬁned in (2.6) is a contraction on the complete metric space

k k (r) (r) (r) (r) N Cf (I) := g ∈C (I): g (x1)= f (x1), g (xN )= f (xN ), r ∈ k ∪{0} . 6 P. VISWANATHAN AND M.A. NAVASCUES´

Furthermore, the derivative (f α)(r) of its unique ﬁxed point f α satisﬁes the self-referential equation r r (f α)(r)(x)= f (r)(x)+ a−r α(r−j) L−1(x) (f α − b)(j) L−1(x) , x ∈ I , i ∈ N , i j i i i i N−1 j=0 h X i and consequently, f α agrees with f at the knot points up to k-th derivative.

Proof. By the continuity conditions imposed on the maps αi and b, it follows at once that Tg is continuous on each subinterval Ii. By the Leibnitz rule of diﬀerentiation we obtain r r (Tg)(r)(x)= f (r)(x)+ a−r α(r−j) L−1(x) (g − b)(j) L−1(x) , x ∈ I , i ∈ N . i j i i i i N−1 j=0 h X i −1 −1 Using Li (xi)= x1 and Li−1(xi)= xN we infer that (r) + (r) − (r) N (Tg) (xi ) = (Tg) (xi )= f (xi), i =2, 3,...,N − 1, r ∈ k ∪{0}. Further, it is plain to see that (r) (r) (r) (r) (Tg) (x1)= f (x1), (Tg) (xN )= f (xN ) for r ∈ Nk ∪{0}. k Consequently, Tg ∈Cf (I) and T is a self-map. k We will now demonstrate that T is a contraction map. For g1,g2 ∈Cf (I), we obtain (r) (r) −r r N N (Tg1) (x) − (Tg2) (x) ≤ ai 2 kαikCr kg1 − g2kCr , x ∈ Ii, i ∈ N−1, r ∈ k ∪{0}. The previous inequality implies

2 k N kTg1 − Tg2kCk ≤ max ( ) kαikCk : i ∈ N−1 kg1 − g2kCk . ai n o The assumption on the scaling functions now ensure that T is a contraction map, and, hence by the Banach ﬁxed point theorem, T has a unique ﬁxed point f α. Further, it follows that (f α)(r) α (r) (r) obeys the functional equation given in the statement and (f ) (xi)= f (xi) for r ∈ Nk ∪{0} and i ∈ NN , completing the proof.

To ease the exposition, we assume constant scaling function αi(x) = αi for all x ∈ I in the following theorem. However, we note in passing that one can handle the setting of scaling functions in a similar manner by using the following Leibnitz formula for weak derivatives which states [4]: Assume u ∈Wk,p(I) and ϕ is inﬁnitely diﬀerentiable. Then ϕu ∈Wk,p(I) and for j ≤ k

j j Dj (ϕu)= DrϕDj−ru. r r=0 X To prepare for the next theorem let us recall the chain rule for weak derivative adapted to our situation: Let u ∈ W1,1(I) and ϕ : V ⊂ R → I be diﬀeomorphism such that ϕ and (ϕ−1)′ are bounded. Then u ◦ ϕ ∈W1,1(V ) and D(u ◦ ϕ) = (Du ◦ ϕ).Dϕ.

k,p Theorem 3.4. Let f ∈ W (I). Suppose that ∆ = {x1, x2,...,xN } is a partition of I with increasing abscissae, Ii := [xi, xi+1) for i ∈ NN−2 and IN−1 := [xN−1, xN ]. Let Li : I → Ii be − N aﬃne maps Li(x)= aix + di satisfying Li(x1)= xi and Li(xN )= xi+1 for i ∈ N−1. Assume that b ∈Wk,p(I) and the scaling factors satisfy

p 1/p |αi| kp−1 < 1, for p ∈ [1, ∞), ai i∈NN−1  h i P |αi| < , p .  max { ak } 1 for = ∞  i∈NN−1 i Then the RB operator (2.3) is a contraction map on Wk,p(I) and the unique ﬁxed point f α ∈ Wk,p(I) obeys the self-referential equation (2.4). A FRACTAL OPERATOR ON SOME STANDARD SPACES OF FUNCTIONS 7

Proof. From the stated assumptions on the scaling factors and base function it follows that T is a well-deﬁned self-map on Wk,p(I). Next we shall show that T is a contraction. Following the proof of Theorem 3.2, for 1 ≤ p< ∞, we obtain

1 p p kTg1 − Tg2kp ≤ ai|αi| kg1 − g2kp. N h i∈XN−1 i For 1 ≤ j ≤ k, straightforward computations yield

j j p j j p kD (Tg1) − D (Tg2)kp = D (Tg1)(x) − D (Tg2)(x) dx ZI

j − 1 p = D [αi(g1 − g2) ◦ Li (x)] dx N Ii i∈ N−1 Z X p 1 j −1 p = |αi| jp D (g1 − g2) Li (x) dx N a Ii i∈ N−1 i Z X p 1 j p = |αi| jp−1 D (g1 − g2)(˜x) d˜x N a I i∈ N−1 i Z X p 1 j p = |αi| jp−1 kD (g1 − g2)kp, N ai h i∈XN−1 i which implies that for 1 ≤ j ≤ k,

p 1 j j |αi| p j kD (Tg1) − D (Tg2)kp = jp−1 kD (g1 − g2)kp. N ai h i∈XN−1 i Consequently, for 1 ≤ p< ∞,

k 1 j p p kTg1 − Tg2kWk,p = kD (Tg1 − Tg2)kp h j=0 i X 1 k p p |αi| j p = kD (g1 − g2)k . ajp−1 p " j=0 N −1 i # X h i∈XN i Bearing in mind that 0

1 p 1 k p |αi| p j p kTg1 − Tg2kWk,p ≤ kD (g1 − g2)kp . akp−1 N −1 i " j=0 # h i∈XN i X p 1 |αi| p k,p = kp−1 kg1 − g2kW N −1 ai h i∈XN i Therefore, for 1 ≤ p< ∞, the stated assumption on scaling factors ensures that T is a contraction on Wk,p(I). Next consider the case p = ∞. For x ∈ Ii, i ∈ NN−1

−1 (Tg1)(x) − (Tg2)(x) = αi(g1 − g2) Li (x) ≤ | αi|kg1 − g2k∞, from which one can read

kTg1 − Tg2k∞ ≤ max{|αi| : i ∈ NN−1}kg1 − g2k∞. 8 P. VISWANATHAN AND M.A. NAVASCUES´

Similarly, for 1 ≤ j ≤ k, j j j −1 (D Tg1)(x) − (D Tg2)(x) = αiD (g1 − g2) ◦ Li (x) 1 j −1 = | αi| j D (g1 − g2) ◦ Li (x ) ai

|αi| j ≤ j kD (g1 − g2)k∞, ai which asserts α j j | i| N j kD (Tg1) − D (Tg2)k∞ ≤ max j : i ∈ N−1 kD (g1 − g2)k∞. ai Thus we have k j kTg1 − Tg2kWk,∞ = kD (Tg1 − Tg2)k∞ j=0 X |αi| N N ≤ max max j : i ∈ N−1 : j ∈ k ∪{0} kg1 − g2kWk,p ai n o |αi| N k,p = max k : i ∈ N−1 kg1 − g2kW , ai n o which with prescribed conditions on the scaling factors implies that T is a contraction.

k,σ Theorem 3.5. Let f ∈C (I). Suppose that ∆= {x1, x2,...,xN } is a partition of I satisfying x1 < x2 < ··· < xN , Ii := [xi, xi+1] for i ∈ NN−1 and Li : I → Ii are aﬃne maps Li(x) = k,σ aix+di satisfying Li(x1)= xi and Li(xN )= xi+1 for i ∈ NN−1. Let the base function b ∈C (I) and the scaling factors αi satisfy

|αi| max : i ∈ NN−1 < 1, aσ+k n i o (r) (r) (r) (r) b (x1)= f (x1), b (xN )= f (xN ), r ∈ Nk ∪{0}. k,σ k,σ Then the RB operator T defined in (2.3) is a contraction map on Cf (I) ⊂C (I) defined by k,σ k,σ (r) (r) (r) (r) Cf (I)= g ∈C (I): g (x1)= f (x1), g (xN )= f (xN ), 0 ≤ r ≤ k and the unique fixed point f α ∈Ck,σ(I) obeys the self-referential equation (2.4). Proof. Following the proof of Theorem 3.3 we assert that Tg ∈ Ck(I) whenever g ∈ Ck,σ(I). Further, (r) (r) αi (r) −1 N N (3.1) (Tg) (x)= f (x)+ r (g − b) Li (x) , x ∈ Ii, i ∈ N−1, r ∈ k ∪{0}. ai The σ-th Hölder seminorm of (Tg)(k) is given by |(Tg)(k)(x) − (Tg)(k)(y)| Tg (k) [( ) ]σ = sup σ x,y∈I,x6=y |x − y| αi g b (k) L−1 x g b (k) L−1 y ak [( − ) i ( ) − ( − ) i ( ) ] = max sup i i∈N −1 σ N x,y∈Ii,x6=y |x − y|

(k) −1 (k) −1 (k) −1 (k) −1 |αi| g Li (x) − g Li (y) b Li (x)) − b Li (y) ≤ max ( k ) sup σ + σ i∈NN−1 a |x − y| |x − y| i x,y∈Ii,x6=y h (k) −1 (k) −1 (k) −1 (k) −1 i |αi| |g Li (x) − g Li (y) | |b Li (x)) − b Li (y) | = max ( ) sup −1 −1 + −1 −1 i∈N −1 k σ σ σ σ N ai x,y∈Ii,x6=y a |L (x) − L (y)| a |L (x) − L (y)| h i i i i i i i |α | |g(k)(˜x) − g(k)(˜y)| |b(k)(˜x) − b(k)(˜y)| = max ( i ) sup + N k+σ σ σ i∈ N−1 a x,˜ y˜∈I,x˜6=˜y |x˜ − y˜| |x˜ − y˜| i h i |αi| (k) (k) = max ( ) [g ]σ + [b ]σ . i∈N −1 k+σ N ai A FRACTAL OPERATOR ON SOME STANDARD SPACES OF FUNCTIONS 9

k,σ (k) Since b and g are in C (I), the previous estimate ensures that [(Tg) ]σ < ∞ and hence that k,σ Tg ∈C (I). Again from (3.1), for x ∈ Ii,

(r) |αi| (r) −1 (Tg1 − Tg2) (x) = r (g1 − g2) Li (x) ai |αi| (r) ≤ r k(g1 − g2) k∞ ai and hence one readily obtains

(r) |αi| N (r) k(Tg1 − Tg2) k∞ ≤ max r : i ∈ N−1 k(g1 − g2) k∞. ai n o (k) On lines similar to the estimation of [(Tg) ]σ, we get

(k) |αi| (k) [(Tg1 − Tg2) ]σ ≤ max : i ∈ NN−1 [(g1 − g2) ]σ. ak+σ n i o From these computations we gather that

k (r) (k) kTg1 − Tg2kCk,σ = k(Tg1 − Tg2) k∞ + [(Tg1 − Tg2) ]σ r=0 X k |α | |α | ≤ max i : i ∈ N k(g − g )(r)k + max i : i ∈ N [(g − g )(k)] ar N−1 1 2 ∞ k+σ N−1 1 2 σ r=0 i ai X n o n o k |αi| (r) |αi| (k) ≤ max max r k(g1 − g2) k∞ + max k+σ [(g1 − g2) ]σ r∈Nk∪{0} i∈NN−1 a i∈NN−1 i r=0 ai X n o k |αi| N (r) (k) ≤ max k+σ : i ∈ N−1 k(g1 − g2) k∞ + [(g1 − g2) ]σ ai r=0 h X i |αi| N k,σ = max k+σ : i ∈ N−1 kg1 − g2kC . ai k,σ The assumption on scaling factors now yields that T is a contraction on Cf (I), and hence the proof.

4. Fractal operator on Function Spaces Theorems 3.1-3.5 established in the previous section illustrate, albeit indirectly, the existence of an operator that assigns a function f to its self-referential (fractal) analogue f α. To be precise, for a fixed set of scaling functions (factors) αi and for suitable choice of base function, there exists a map F α : X → X; F α(f)= f α, where X is one of the Banach spaces B(I), Ck(I), Lp(I), Wk,p(I), or Ck,σ(I). In this section, we offer a couple of fundamental results on this fractal operator F α. For definiteness, we shall work with X = Wk,p(I), other cases can be similarly dealt with. Further, we assume that the base function b depends on f linearly and b = Lf, where L : X → X is a bounded linear operator with respect to the norm in the corresponding space. Proposition 4.1. For f ∈Wk,p(I) and scaling factors satisfying conditions prescribed in The- orem 3.4, we have

p 1 |αi| p α −1 kf − bk k,p , for p ∈ [1, ∞), akp W α i∈N −1 i kf − fkWk,p ≤  N P |αi| N α  max k : i ∈ N−1 kf − bkWk,p , for p = ∞.  ai Proof. We have the functional equation α α −1 N f (x)= f(x)+ αi(f − b) Li (x) , x ∈ Ii, i ∈ N−1. 10 P. VISWANATHAN AND M.A. NAVASCUES´

Therefore, k α αi k α −1 N D (f − f) (x)= k (D (f − b)) Li (x) , x ∈ Ii, i ∈ N−1. ai Assume that 1 ≤ p< ∞. By a series of self-evident steps

k α p k α p kD (f − f)kp = |D (f − f)(x)| dx ZI |αi| p k α −1 p = k D (f − b) Li (x) dx N ai Ii i∈ N−1 Z X

|αi| p k α p = k ai D (f − b)(˜x) d˜x N ai I i∈ N−1 Z X p |αi| k α p = kp−1 kD (f − b)kp, N −1 ai i∈XN and consequently, p 1 k α |αi| p k α kD (f − f)kp ≤ kp−1 kD (f − b)kp. N −1 ai h i∈XN i Thus α α k α kf − fkWk,p = kf − fkp + kD (f − f)kp 1 p 1 p p α |αi| p k α ≤ ai|αi| kf − bkp + kp−1 kD (f − b)kp N −1 N −1 ai h i∈XN i h i∈XN i 1 p 1 p p |αi| p α k,p ≤ max ai|αi| , kp−1 kf − bkW N −1 N −1 ai nh i∈XN i h i∈XN i o p 1 |αi| p α k,p = kp−1 kf − bkW . N −1 ai h i∈XN i The proof for p = ∞ is similar.

k,p Theorem 4.1. Let Id be the identity operator on W (I), b = Lf, where L is a bounded linear operator on Wk,p(I), and the scaling factors satisfy conditions prescribed in Theorem 3.4. Then

p 1 |αi| p kp−1 N ai α i∈ N−1 kf − fk k,p ≤ 1 kI − Lk kfk k,p , for 1 ≤ p< ∞, W p d W P |αi| p 1 − kp−1 ai i∈NN−1 P |αi| N max k : i ∈ N−1 α ai kf − fkWk,p ≤ kId − Lk kfkWk,p , for p = ∞. |αi| N 1 − max k : i ∈ N− 1 ai Proof. The previous proposition in conjunction with the triangle inequality yields α α kf − fkWk,p ≤ Kkf − bkWk,p α ≤ K kf − fk∞ + kf − Lfk∞ α ≤ Kkf − fk∞ + kId − Lkkfk∞ , where p 1 |αi| p kp−1 , for p ∈ [1, ∞), N ai K =  i∈ N−1 P |αi| N  max k : i ∈ N−1 , for p = ∞.  ai from which the result follows at once.  A FRACTAL OPERATOR ON SOME STANDARD SPACES OF FUNCTIONS 11

Theorem 4.2. For scaling factors satisfying conditions prescribed in Theorem 3.4, the self- referential operator F α : Wk,p(I) → Wk,p(I) is a bounded linear operator which reduces to the identity operator for α =0.

k,p α α Proof. Let f1 and f2 be in W (I) and β1, β2 reals. The functional equation for f1 = F (f1) α α and f2 = F (f2) are given by α α −1 N f1 (x)= f1(x)+ αi(f1 − Lf1) Li (x) , x ∈ Ii, i ∈ N−1, α α −1 N f2 (x)= f2(x)+ αi(f2 − Lf2)Li (x), x ∈ Ii, i ∈ N−1. Therefore α α α α −1 (β1f1 + β2f2 )(x) = (β1f1 + β2f2)(x)+ αi β1f1 + β2f2 − L(β1f1 + β2f2) Li (x) , α α from which it follows that β1f1 + β2f2 is a ﬁxed point of the operator −1 (Tg)(x) = (β1f1 + β2f2)(x)+ αi g − L(β1f1 + β2f2) Li (x) By the uniqueness of the ﬁxed point we see that α α α α α α F (β1f1 + β2f2) = (β1f1 + β2f2) = β1f1 + β2f2 = β1F (f1)+ β2F (f2), proving the linearity of F α. In view of the previous theorem we have α α α kF (f)kWk,p = kf kWk,p ≤ kf − fkWk,p + kfkWk,p K ≤ kI − Lk kfk k,p + kfk k,p 1 − K d W W K = 1+ kI − Lk kfk k,p , 1 − K d W which ensures that F α is bounded and h i K kF αk≤ 1+ kI − Lk. 1 − K d The last part of the theorem follows by noting that for α = 0, f α = f. Theorem 4.3. Consider a scale vector α ∈ RN−1 whose components satisfy

p 1/p |αi| −1 kp−1 < min 1, kLk , for p ∈ [1, ∞), ai i∈NN−1  h i P |αi| < , L −1 , p .  max { ak } min 1 k k for = ∞  i∈NN−1 i α α The corresponding fractal operator F is bounded below. In particular, F is injective and has a closed range. In fact, F α : Wk,p(I) →F α Wk,p(I) is a topological isomorphism. Proof. From Proposition 4.1 α α α α kfkWk,p −kf kWk,p ≤kf − f kWk,p ≤ Kkf − LfkWk,p ≤ K kf kWk,p + kLkkfkWk,p , where K is as prescribed in Theorem 4.1. Also by the stated assumption on scale vector we have K < kLk−1. Thus 1+ K α kfk k,p ≤ kf k k,p . W 1 − KkLk W That is, the operator F α is bounded below and hence, in particular, injective. To prove that α k,p α α k,p α F W (I) is closed, let fn be a sequence in F W (I) such that fn → g. In particular, f α = F α(f ) is a Cauchy sequence in F α Wk,p(I) . Since n n 1+ K α α kf − f k k,p ≤ kf − f k k,p , n m W 1 − KkLk n m W k,p k,p it follows that {fn} is a Cauchy sequence in W (I). Since W (I) is a complete space, there k,p exists an f ∈ W (I) such that fn → f and consequently, by the continuity of the fractal operator F α we conclude that g = F α(f) = f α. Now from the bounded inverse theorem (see, for instance, [3]) it follows that the inverse of the map F α : Wk,p(I) → F α Wk,p(I) is a bounded linear operator, completing the proof. 12 P. VISWANATHAN AND M.A. NAVASCUES´

Theorem 4.4. For a scale vector α ∈ RN−1 whose components satisfy

p 1/p |αi| −1 kp−1 < 1+ kId − Lk , for p ∈ [1, ∞), ai i∈NN−1  h i −1 P |αi| < I L , p ,  max { ak } 1+ k d − k for = ∞  i∈NN−1 i α k,p the fractal operator F is a topological automorphism on W (I). Furthermore, with K as in Theorem 4.1

1 − KkLk α K kfk k,p ≤ kF (f)k k,p ≤ 1+ kI − Lkkfk k,p. 1+ K W W 1 − K d W Proof. From Theorem 4.1 it can be easily seen that K kI −F αk≤ kI − Lk. d 1 − K d −1 In view of the assumption on the scaling vector α, we get K < 1+ kId − Lk and hence kI −F αk < 1. Consequently, the Neumann series ∞ (I −F α)j is convergent in the operator d j=0 d norm and F α = I − (I −F α) is invertible (see, for instance, [3]). Bearing in mind that d d P kLk = Id − (Id − L)k≤ 1+ kId − Lk, α the bounds on kF (f)kWk,p follow from Theorem 4.2 and 4.3.

The existence of Schauder bases for Sobolev spaces is desired for demonstrating the existence of solutions of nonlinear boundary value problems. Sometimes it is interesting to look for the global structure involved in a given problem and hence self-referentiality may be advantageous. Therefore, it is worth to search for a Schauder basis for Wk,p(I) consisting of fractal functions. It is to this that we now turn. First let us recall the following theorem which asserts the existence of Schauder basis for Wk,p(I), where p ≥ 1 is a real number. We reproduce its proof here briefly for the sake of completeness. Without loss of generality assume that I = [0, 1]. Theorem 4.5. (Fuˇc´ık).(see [5] Theorem 4.7). For 1 ≤ p< ∞, Wk,p(I) has a Schauder basis. Proof. Proof is by induction with respect to k. For k = 0, W0,p(I)= Lp(I), and has a Schauder k k,p k basis. Let {fn } be a Schauder basis for W (I) and {βn} be a sequence of continuous linear k,p ∞ k k k,p functionals such that for each f ∈W (I), f = n=1 βn(f)fn . For f ∈W (I), set k+1 k+1 f1 (x) ≡ 1, β1 (f)= f(0)P x k+1 k k+1 k ′ fn (x)= fn−1(t) dt, βn (f)= βn−1(f ), n ≥ 2. Z0 k+1 k+1,p Then {fn } is a Schauder basis in W (I). Theorem 4.6. For 1 ≤ p < ∞, the space Wk,p(I) admits a Schauder basis consisting of self- referential functions. k,p Proof. Let {fn} be a Schauder basis for W (I) with associated coefficient functionals {βn}. Consider the scaling factors αi, i ∈ NN−1, such that the condition prescribed in Theorem 4.4 is satisfied so that F α is a topological automorphism on Wk,p(I). Let f ∈ Wk,p(I). Then (F α)−1(f) ∈Wk,p(I) and ∞ α −1 α −1 (F ) (f)= βn F ) (f) fn. n=1 X Since F α is a bounded linear map we obtain ∞ α −1 α f = βn F ) (f) fn . n=1 X ∞ α Next we prove that the representation is unique. Let f = n=1 γnfn be another represen- α −1 α −1 ∞ tation of f. Continuity of (F ) ensures that (F ) (f) = n=1 γnfn and hence γn = α −1 N α P k,p βn (F ) (f) for all n ∈ . Thus {fn } is a Schauder basis for W (I) comprising of self- referential functions. P A FRACTAL OPERATOR ON SOME STANDARD SPACES OF FUNCTIONS 13

For p = 2, the Sobolev space Wk,2(I)= Hk(I) is a Hilbert space and hence we can talk about the adjoint (in the usual sense) (F α)∗ of the fractal operator F α. We have the following.

1 2 2 N−1 |αi| −1 Theorem 4.7. α R 2 −1 < , L For a scale vector ∈ satisfying i∈NN−1 k min 1 k k , ai h i Hk(I)= rg(F α) ⊕ ker (FPα)∗ , α ∗ where rg(A) and ker(A) represent range and kernel of an operator A. Also, (F ) is surjective. Proof. From the proof of Theorem 4.3 it follows that for a scale vector α satisfying the stated hypothesis, rg(F α) is closed. By the orthogonal decomposition theorem for a Hilbert space (see, for instance, [3]) Hk(I)= rg(F α) ⊕ rg(F α)⊥ = rg(F α) ⊕ ker (F α)∗ . Again by the orthogonal decomposition, ⊥ ⊥ Hk(I)= rg (F α)∗ ⊕ rg (F α)∗ = rg (F α)∗ ⊕ rg (F α)∗ = rg (F α)∗ ⊕ ker(F α). α α For a scale vector satisfying the given condition, F is injective. That is, ker(F ) = {0}. Consequently, Hk(I)= rg (F α)∗ , i.e., rg (F α)∗ is dense in Hk(I). For a linear and bounded operator of a Hilbert space, its range is closed if and only if the range of its adjoint is closed (see, for instance, [3]). Therefore, rg (F α)∗ is closed and hence rg (F α)∗ = Hk(I).

1 2 2 |αi| −1 Theorem 4.8. α 2 −1 < I L For a scale vector satisfying i∈NN−1 k 1+k d − k , the fractal ai α k k operator F : H (I) → H (I) is Fredholm withh P index zero. i Proof. With the stated assumption on α, Theorem 4.4 asserts that the operator F α is an isomorphism. Therefore, by the following theorem (see [13]) F α is Fredholm. A linear bounded operator A is Fredholm if and only if A = B + F , where B is an isomorphism and F has finite rank. Also, ind F α := dim ker(F α) − codim rg(F α)=0, delivering the proof. Remark 4.1. Throughout the article we have confined our discussion to real-valued functions. However, we remark that many of our results apply immediately to complex-valued functions defined on a real compact interval. For instance, denoting the real and imaginary parts of f by fre and fim respectively, one may consider the operator α k,p C k,p C α α α α FC : W (I, ) →W (I, ), FC(f)= FC (fre + ifim)= F (fre)+ iF (fim). α α It is not hard to show that FC is a bounded linear operator. Other properties of FC can be similarly dealt with. References

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