Abstract
In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying classical multiquadric function. Based on the translates of a fractal multiquadric function defined on a grid, we propose two fractal quasi-interpolants \(L_C^{\alpha }f\) and \(L_D^{\alpha }f\) to approximate smooth and irregular functions. We study the convergence of \(L_C^{\alpha }f\) and \(L_D^{\alpha }f\) to f using uniform error estimates. We investigate the linear polynomial reproducing property, convexity/concavity and monotonicity features of these quasi-interpolation operators. The advantages of fractal quasi-interpolants over the classical quasi-interpolants are demonstrated by various examples.
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1 Introduction
Classical interpolation theory deals with the continuous representation of discrete data through various types of interpolants which are sufficiently smooth with the possible exception of a finite number of points. On the other hand, a great number of scientific experimental data and real-world signals are highly complex and their graphs are effectively never smooth; see, for instance, (Wang et al. 2021; Geng et al. 2021; Sun et al. 2006; Barnsley 1996; Timbo et al. 2009; Onali and Goddard 2011).
In order to describe such irregular functions or signals more accurately, Barnsley (1986) introduced the concept of fractal interpolation function (FIF) based on the theory of iterated function system (IFS) (Hutchinson 1981; Barnsley and Demko 1985). The construction of various types of FIFs and the investigation into their properties comprises one of the foremost topics in the theory of fractal functions. It has gone through several generalizations and extensions over the last few decades to generate smooth fractal splines (Barnsley and Harrington 1989; Chand and Kapoor 2006; Massopust 2010; Navascués and Sebastián 2006), fractal rational and trigonometric interpolations (Chand et al. 2014; Tyada et al. 2021; Viswanathan and Chand 2015), zipper FIFs (Chand et al. 2020), and so forth.
A suitable subclass of FIFs elucidates connections between the concept of univariate FIF and classical approximation theory. This subclass was proposed by Barnsley (1986, Example 2, Page 309) and its elements were later coined \(\alpha \)-fractal functions to emphasize their dependency on the vertical scaling vector \(\alpha \) which affects the shape and dimension of the graph of the associated fractal function. Navascués and her collaborators extensively investigated the various approximation theoretic issues in their work (Navascués 2005; Navascués and Chand 2008; Vijender et al. 2021; Navascués and Massopust 2019; Viswanathan et al. 2016).
Quasi-interpolation is an indispensable tool in the approximation of functions which has been investigated assiduously both in theory and in engineering applications; see for instance (Chen and Wu 2006; Duan and Rong 2013; Zhang et al. 2019; Wu and Zhang 2013) and the references therein. The major advantage of quasi-interpolation is that it provides a solution directly without solving a system of linear equations. Based on the multiquadric (MQ) functions which were conceptualized by Hardy (1971), Beatson and Powell (1992) first proposed three different types of univariate MQ quasi-interpolants associated with scattered centers in a bounded interval. Wu and Schaback (1994) made some improvements to one of this quasi-operators and studied its approximation and shape-preserving aspects.
By utilizing the definition of MQ B-splines, Beatson and Dyn (1996) investigated MQ quasi-interpolation theoretically. The quasi-interpolation operator defined via MQ functions has many applications such as the solution of PDEs (Chen and Wu 2007; Bao and Song 2014), the approximation of derivatives of a function (Ma and Wu 2009), and the solution of shock wave equations (Hon and Wu 2000).
The quasi-interpolant preserves several basic properties of MQ quasi-interpolants such as the smoothness, simplicity, efficiency, and the capability of approximating higher order derivatives. It is found that both fractal interpolation and fractal quasi-interpolation are useful in the approximation of functions. But in the literature, there has not been a common modality between them. The objective of this paper is to create a bridge between these two diverse fields and propose a new object called fractal quasi-interpolation to approximate either very irregular functions or smooth functions having a highly irregular derivative.
In this article, we first present the construction of multiquadric fractal functions from the classical multiquadric functions. Further, we investigate their differentiability and suitability for constraint approximation. Using the translates of a fractal MQ function as base functions, we construct a fractal approximant for a function f similar to the classical analog given in Wu and Schaback (1994). We propose two quasi-interpolation operators in this article. The first quasi-interpolation operator \(L^{\alpha }_Cf\) requires the values of the derivative of f at the endpoints of the domain whereas the second quasi-interpolation operator \(L^{\alpha }_Df\) does not.
Shape preserving aspects and convergence results for these two operators are investigated and it is shown that the order of convergence is \(O(h^2)\) in the case of \(L^{\alpha }_Cf\) given some restrictions on the scaling vector \(\alpha \). A similar order of convergence for \(L^{\alpha }_Df\) can be obtained provided certain conditions on the scaling vector and the shape preserving constant c of the multiquadric functions are imposed. Examples are provided to support the theoretical results of the approximation of functions by the operators \(L_C^{\alpha }f\) and \(L_D^{\alpha }f\).
In various research domains, a multitude of challenges emerge where traditional functions characterized by simple geometric structures prove insufficient. These challenges cut across diverse fields, encompassing finance, various types of differential equations, ecology, turbulence modeling, and more (Liu et al. 2020; Ortmann and Buhmann 2024; Brambila 2017). The primary aim of this article is to address theoretical issues concerning the approximation of irregular functions, as well as regular functions exhibiting irregular derivatives. Furthermore, it seeks to tackle approximation problems associated with irregular or fractal functions, such as delay differential equations (Rihan 2021), ECG dynamics (Golany et al. 2021), and financial modeling (Pan and Zhang 2023). Moreover, a plethora of other applications can be explored in related articles and their respective references (Wu and Duan 2016; Gao and Chi 2014; Gao and Zhang 2018).
The structure of this article is arranged as follows. Section 2 provides some basics and introduces the theory of fractal functions to the degree that it is required in the subsequent parts. Section 3 introduces the construction of the fractal MQ functions and their smoothness and shape-preserving properties are presented. In Sect. 4, a novel fractal quasi-interpolation operator \(L_C^{\alpha }f\) is introduced and its shape preserving and convergence properties are derived. In Sect. 5, we introduce the second fractal quasi-interpolation operator \(L_D^{\alpha }f\) and study results similar to those in the previous section using various examples.
2 Background and preliminaries
For the benefit of the reader, we provide in this section the basics of IFSs and the construction of fractal functions from a suitable IFS. For more details, the reader may consult (Hutchinson 1981; Barnsley and Demko 1985; Barnsley 1986; Massopust 2016; Navascués 2005).
In the sequel, we use the following notation. For \(N\in {\mathbb {N}}:= \{1,2,3,\ldots \}\), we set \({\mathbb {N}}_N := \{1, \ldots , N\}\), \({\mathbb {N}}_N^- := -{\mathbb {N}}_N\), and \({\mathbb {N}}_{N,0} :={\mathbb {N}}_N\cup \{0\}\).
An IFS \(\mathcal {F}:= \{X;w_i: i\in {\mathbb {N}}_N \}\) consists of a collection of continuous functions from a metric space (X, d) to itself. \(\mathcal {F}\) is called a hyperbolic IFS if each \(w_i\) is contractive on X, i.e., its Lipschitz constant
As we deal exclusively with hyperbolic IFSs, we will drop the adjective “hyperbolic" in the following unless it is necessary.
Let \(\mathcal {H}(X) := \{ A\subseteq X : A \text { is non-empty and compact} \}\). The Hausdorff-Pompeiu metric h on \(\mathcal {H}(X)\) is defined by
where \(d(A,B):=\max \{d(x,B):x \in A\}\) and \(d(x,B):=\min \{d(x,y):y\in B\}\). It is known that if (X, d) is a complete metric space, then \((\mathcal {H}(X),h)\) is also a complete metric space, named the space of fractals in Barnsley (1993).
Define the Huchinson map (Hutchinson 1981) \(W : \mathcal {H}(X)\rightarrow \mathcal {H}(X)\) by
If the IFS \({\mathcal {F}}\) is hyperbolic then W is a contraction on \((\mathcal {H}(X),h)\) with contraction factor \(s :=\max \limits _{i\in {\mathbb {N}}_N} |s_i|\). Thus, by the Banach fixed point theorem, there exists a unique \(G \in \mathcal {H}(X)\) such that
where \(W^{\circ m}\) denotes the m-fold composition of W with itself. The fixed point G is known as the attractor of or deterministic fractal generated by the IFS \({\mathcal {F}}\).
Next, we would like to construct a suitable IFS whose attractor is the graph of a continuous function interpolating a given set of data or interpolation points.
For this purpose, consider a set of interpolation points
Let \(\{u_i\}_{i\in \mathbb {N}_{N}}\) be a set of homeomorphisms from \(I:=[x_0,x_N]\) to \(I_i:=[x_{i-1},x_{i}]\) satisfying
For \(i\in {\mathbb {N}}_{N}\), construct functions \(v_i: I \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) such that \(v_i\) is continuous in the first variable and Lipschitz continuous in the second variable with Lipschitz constant \(s_i<1\):
Further, impose the following join-up conditions:
Now, define an IFS whose maps \(w_i: I\times {\mathbb {R}}\rightarrow I_i \times {\mathbb {R}}\) are given by
Such an IFS is sometimes called an interpolation IFS and we denote it by
As the maps \(w_i\) in the IFS \(\mathcal {I}\) are all contractive, \(\mathcal {I}\) has a unique attractor G that satisfies the self-referential set-valued equation
Let \(C(I)=\{f:I \rightarrow \mathbb {R}: f ~ \text {is continuous on}~ I\}\) and \( \mathcal {G}:=\{g\in C(I):g(x_0)=y_0 \wedge g(x_N)=y_N\}.\) Define a metric on \( \mathcal {G}\) by \(\rho (h,g) :=\max \{ | h(x)-g(x) | : x\in I \}\), \(g,h\in \mathcal {G}\). Since \( \mathcal {G}\) is closed in C(I), \((\mathcal {G},\rho )\) is a complete metric space. Define a Read-Bajraktarevic (RB) operator (Massopust 2016) T on \((\mathcal {G},\rho )\) by
where \(\chi _K\) denotes the characteristic function of a set K, \(I_i^*=[x_{i-1},x_i), i\in {\mathbb {N}}_{N-1}\), \(I_N^*=[x_{N-1},x_N]\). The following theorem provides the existence of a fractal function arising from an interpolating IFS.
Theorem 1
(Barnsley 1986, Theorem 1 Massopust 2016) The IFS \(\mathcal {I}\) has a unique attractor G which is the graph of a continuous function \(f^*:I\rightarrow {\mathbb {R}}\) satisfying the interpolation conditions \(f^*(x_j)=y_j\), \(j\in {\mathbb {N}}_{N,0}\). Moreover, if \(\mathcal {G}\) is endowed with the uniform metric \( \rho \), then \(f^*\) is the unique fixed point of the operator T acting on \({\mathcal {G}}\).
Since \(f^*\) is the fixed point of T given in (5), \(f^*\) obeys the self-referential functional equation
on I. This function \(f^*\) is termed as a fractal interpolation function (FIF) associated with the IFS \(\mathcal {I}\) as its graph \(G(f^*)\) satisfies a similar self-referential set-valued Eq. (4).
The concept of FIF may be used to define a specific class of fractal functions (Barnsley 1986; Massopust 2010; Navascués 2005) associated with any function \(f \in C(I)\) as described in the following.
To this end, let \(I:= [x_0,x_N]\subset {\mathbb {R}}\). For a given \(f\in C(I)\), consider a partition \(\Delta :=\{ x_0, x_2,\dots , x_N\}\) of I satisfying \(x_0<x_1<\ldots <x_N\) and a continuous function \(b:I \rightarrow \mathbb {R}\) with \(b\ne f\) that satisfies the endpoint interpolation conditions \(b(x_0)=f(x_0)\) and \(b(x_N)= f(x_N)\).
Choose a scaling vector \(\beta := (\beta _1, \ldots , \beta _{N}) \in (-1,1)^{N}\). The components of \(\beta \) are called scaling factors. For \(i\in \mathbb {N}_{N}\), we set
The constants \(a_i\) and \(b_i\) are computed via the conditions (1). Then, the IFS \(\{[x_0,x_N]\times \mathbb {R};w_i(x,y)=(u_i(x),v_i(x,y)), i\in \mathbb {N}_{N}\}\) possesses an attractor which is the graph of a fractal function \(f^{\beta }_{\Delta ,b} =: f^{\beta }\). The function \(f^{\beta }\) is referred to as a \(\beta \)-fractal function for f with respect to the scaling vector \(\beta \), the base function b, and the partition \(\Delta \).
If \(C_f(I)=\{ g\in C(I): g(x_0)=f(x_0) \wedge g(x_N)=f(x_N)\},\) then \(f^{\beta }\) is the unique fixed point of the RB operator \(T:C_f(I)\rightarrow C_f(I)\) defined by Massopust (2016)
Consequently, \(f^{\beta }\) satisfies the self-referential functional equation
Using (9), it is straight-forward to show that
where \(|\beta |_\infty = \max \{ \beta _i \; : \, i \in \mathbb {N}_{N}\}.\) The above inequality shows that \(f^{\beta }\) converges uniformly to f if either \(|\beta |_{\infty } \rightarrow 0\) or/and \(\Vert f-b\Vert _{\infty } \rightarrow 0\).
3 Fractal multiquadric functions
In this section, we introduce a novel fractal multiquadric (MQ) function and also construct convexity preserving fractal MQ-functions. The construction of a symmetric fractal MQ-function is based on the following steps:
-
(i)
From a given set of abscissas \(\{x_0,x_1,\dots x_n\}\) contained in the domain \([a,b]\subset {\mathbb {R}}\) of the approximating function, we construct another closed set I which is symmetric about 0 and a partition for it.
-
(ii)
This partition must contain all values of \(|x_i-x_j|\), for all \(i,j \in {\mathbb {N}}_{0,n}\).
-
(iii)
We define affine mappings \(u_i\) such that \(u_i (r) = -u_{-i}(r)\), for all \(r\in I\).
-
(iv)
We require the scaling factors \(\alpha _i\) to satisfy \(\alpha _i = \alpha _{-i}\).
-
(v)
We construct a suitable iterated function system to obtain a symmetric fractal function given a symmetric MQ-function.
Construct a closed set \( I:=[-K,K]\), where \(K:= {\max }_{0\le j\le n} {\sup |x-x_j|}_{ x_0 \le x \le x_n}\). If the abscissas are not ordered, we can order them. Thus, without loss of generality, let \(P:=\{x_0<x_1< \dots <x_n\}\) be the set of abscissas in \(\Omega :=[x_0,x_n]\).
Next, we partition [0, K] into N parts \( 0 = r_0< r_1< \dots < r_N\) such that the ensuing partition \(\Pi \) contains all the values
We sometimes refer to the \(r_i\) as nodal points. Then, we extend the partition \(\Pi \) to the left of \(r_0=0\) in a symmetric way to obtain the required partition
Let \(I_i:=[r_{i-1},r_i]\), and \(I_{-i}:=[-r_i,-r_{i-1}]\), \( i \in \mathbb {N}_N.\) For \(i\in {\mathbb {N}}_N\), define affine maps \( u_i:I \rightarrow I_i\) and \( u_{-i}:I \rightarrow I_{-i}\) by setting
If \(u_i(r)=a_ir+b_i\) and \(u_{-i}(r)=a_{-i}r+b_{-i}\), \(i\in \mathbb {N}_N\), then one finds that
It is clear from these equations that \( u_i=-u_{-i}\) and \(|a_i| < 1\), for all \(i\in {\mathbb {N}}_N\cup {\mathbb {N}}_N^-\).
To explain the construction of the \(u_i\) from a set P using steps (i) – (iii) above, we provide the following example.
Example 1
Consider the set of abscissas \(P:=\{\tfrac{1}{2},1,\tfrac{3}{2}\}\) in \(\Omega =[\tfrac{1}{2},\tfrac{3}{2}]\). Then \(I=[-1,1]\) and \(\Delta =\{-1,-\tfrac{1}{2},0,\tfrac{1}{2},1\}\) is the smallest partition satisfying (11). For this \(\Delta \), we have
Computing \(a_i\) and \(b_i\) for \( i\in {\mathbb {N}}_2\cup {\mathbb {N}}_2^-\), yields
Now, we briefly explain the distribution of points under these maps around the center 0. Observe that \(u_i:I \rightarrow I_i\) for \(i\in \mathbb {N}_2 \cup {\mathbb {N}}_2^-\), are homeomorphic and contractive. Note that
is a just-touching IFS and its attractor is \([-1,1]\).
Let \(r\in [0,1]\) be generated by a finite sequence of maps \(u_{n_1},u_{n_2},\dots ,u_{n_k}\), i.e.,
for some \(x\in [-1,1]\) and \(n_1 \in {\mathbb {N}}_2\), \(n_i\in {\mathbb {N}}_2 \cup {\mathbb {N}}_2^-\). Then, an easy computation shows that the symmetric point \(-r\in [-1,0]\) can be obtained as \(-r=u_{-n_1}\circ u_{n_2}\circ \dots \circ u_{n_k}(x)\). Thus, each point in \([-1,1]\) is generated symmetrically around 0 by the IFS \(\tilde{I}\). For example, some points symmetric about \(r_0 = 0\) are obtained by similar indices which differ only in the first digit:
The last two steps (iv)–(v) in the construction of fractal MQ-functions depend on the maps \(v_i:I \times \mathbb {R} \rightarrow \mathbb {R}\) for an MQ-function. Here, we take as an MQ-function the most popular multiquadric function, namely \({\phi }:{\mathbb {R}}\rightarrow {\mathbb {R}}^+\), \(r\mapsto \sqrt{c^2+r^2}\), where c is the so-called shape parameter.
Let \(y_i:={\phi }(r_i)\), \(i\in {\mathbb {N}}_{N,0}\). For \(i\in \mathbb {N}_N \), we define maps \(v_i:I \times \mathbb {R} \rightarrow \mathbb {R}\) by
where \( \alpha _i=\alpha _{-i}\) with \( |\alpha _i|\in [0,1)\).
Finally, for \(i\in {\mathbb {N}}_N\cup {\mathbb {N}}_N^-\), we define maps \(w_i: I \times \mathbb {R} \rightarrow I_i \times \mathbb {R}\) by
Following the procedure in Sect. 2, the IFS
has a unique attractor \(G\in \mathcal {H}(\mathbb {R}^2)\). This G is the graph of a continuous fractal function \({\phi }^*:I \rightarrow \mathbb {R}\) satisfying the interpolating conditions \({\phi }^*(r_i)={\phi }^*(-r_i)=y_i\), for \(i\in {\mathbb {N}}_{N,0}\). In order to get the functional equation for \({\phi }^*\), we proceed as follows.
Let \(\mathcal {G}=\{g\in C[-K,K]: g(K)=g(-K)=y_N\}\) and \(\rho \) be the sup-norm on \(\mathcal {G}\). Then \((\mathcal {G},\rho )\) is a complete metric space. Define an RB-operator \(T:(\mathcal {G},\rho ) \rightarrow (\mathcal {G}, \rho )\) by
where \(I_i^*:= [r_{i-1}, r_i)\), \(i \in {\mathbb {N}}_{N-1}\), \(I^*_N := [r_{N-1},r_{N}]\), \( I^{\dagger }_{-i} := ( -r_i , -r_{i-1}]\), \(i \in {\mathbb {N}}_{N-1}\), and \(I^{\dagger }_{-N} := [-r_N, -r_{N-1}].\)
In order to obtain the fractal MQ-function, we take \(v_i(r,y):=\alpha _iy+q_i(r)\) with \(q_i(r) :=\phi (u_i(r))-\alpha _i b(r)\), \(\alpha _i=\alpha _{-i}\) with \(|\alpha _i|<1\), and any continuous b with \(b(-r_N)=b(r_N)=\phi (r_N)=y_N\). Then, it is easy to prove that T is a contractive map. By the Banach fixed point theorem, T has a unique continuous fixed point \({\phi }^{\alpha }\). Finally, we will show that \( \phi ^{\alpha }\) is symmetric about 0.
To this end, as \(\phi ^{\alpha }\) is the fixed point of (16), it satisfies the self-referential equation
Using the symmetry of \(\phi \) together with the form of the scaling parameters in (17), we have
Definition 1
(Multiquadric Fractal Function) The fractal function \(\phi ^{\alpha }\) defined in (17) is termed a fractal multiquadric (MQ) function. For \(\alpha =0\), \(\phi ^0\) agrees with the classical multiquadratic function \(\phi \).
Note that the maps \(u_{-i}, i \in \mathbb {N}_N\), are defined with negative scaling factors and around \(r_0=0\), \(u_{i}\) and \(u_{-i}\) are negatives of each other. For this reason, not all the results from the theory of fractal functions transfer easily to fractal MQ-functions. Here, we wish to check conditions on the scaling factors \(\alpha _i\) and the base function b such that \({\phi }^{\alpha }\) is p-times (\(p\in {\mathbb {N}}\)) differentiable and preserves the convexity of \({\phi }\).
Theorem 2
Let \({\phi }(r) =\sqrt{r^2+c^2}\) be multiquadric function with shape parameter \(c > 0\). Let \(b\in C^p(I)\), \(p\in {\mathbb {N}}\), be such that
and let \(\alpha _i\) obey \(|\alpha _i|<|a_i|^p\), for all \(i\in {\mathbb {N}}_{N}\cup {\mathbb {N}}_{N}^{-}\). Then, \({\phi }^{\alpha }\in C^{p}(I)\) and satisfies the following self-referential functional equations
Furthermore, for \(k\in {\mathbb {N}}_{p,0}\), we have the estimate
where \(|a|:=\min \{|a_i|:i\in \mathbb {N}_N\}\) and \(|\alpha |_{\infty }:=\max \{|\alpha _i|:i\in \mathbb {N}_N\}\).
Proof
It is known (cf., for instance, Barnsley and Harrington 1989) that for the existence of a p-times differentiable fractal function, it is necessary that the scaling factors satisfy \(|\alpha _i| < a_i^p\), where \(a_i\) is the horizontal contraction of the map \(u_i\), \(i \in \mathbb {N}_{N}.\) Define
Then \(\mathcal {G}^p(I)\) becomes a complete metric space when endowed with the metric
Define an RB operator \(T:\mathcal {G}^p(I) \rightarrow \mathcal {G}^p(I)\) by
Clearly, \(Th \in C^p(I_i)\) for \(p\in {\mathbb {N}}\). For \(k\in {\mathbb {N}}_{p,0}\), (21) implies
To show the continuity of \((Th)^{(k)}\) on I, it suffices to prove continuity at the points of intersection of the subintervals \(I_i\).
First, by using (22), we show the continuity of \((Th)^{(k)}\) at 0 and \(r_i \) for \(i \in {\mathbb {N}}_{N-1} \cup {\mathbb {N}}_{N-1}^-\).
To prove continuity at \(r_i \) is similar to proving it at \(-r_i \) for \(i \in {\mathbb {N}}_{N-1}\). Thus, we verify the continuity at \(-r_i \) for \(i \in {\mathbb {N}}_{N-1}\):
The above two equations yield that \((Th)^{(k)}\in C(I)\), for \(k\in {\mathbb {N}}_{p,0}\).
Now, we will show that the RB-operator T is a contraction on \(\mathcal {G}^p(I)\). By (21), we can write
where \(\hat{\alpha }=\max \left\{ \frac{|\alpha _i|}{|a_i|^k}:i\in {\mathbb {N}}_N\right\} <1\). According to the assumption in the theorem, (24) ensures that
Hence, by the Banach fixed point theorem, there exists a unique fixed point \({\phi }^{\alpha }\in \mathcal {G}^p(I) \) of T and this fixed point satisfies the self-referential functional Eq. (19).
Using (19), a uniform error estimate for the fractal MQ-function is obtained via
The above inequality simplifies to the desired estimate (20). \(\square \)
As \({\phi }''(r)=\frac{c^2}{(r^2+c^2)^{\frac{3}{2}}}\ge 0\), \({\phi }\) is convex. Now, we prove that convexity is preserved when going from \(\phi \) to the MQ-fractal function \({\phi }^\alpha \).
Theorem 3
Let \({\phi }(r) =\sqrt{r^2+c^2}\), where \(c>0\) is fixed. Let the base function \(b\ne {\phi }\) be any function satisfying
If the scaling factors satisfy \(|\alpha _i|< |a_i|^2\) for \(i\in \mathbb {N}_{N} \cup \mathbb {N}_{N}^-\), and
where \(m_i^{(2)}:=\min \left\{ {\phi }''(u_i(r)):r\in I\right\} \) and \(M^{(2)}:=\min \left\{ b''(r):r\in I\right\} \), then the fractal MQ-function \(\phi ^{\alpha }\) is convex, i.e., \( (\phi ^{\alpha })''\ge 0\) on I.
Proof
Since the base function b and the scaling functions \(\alpha \) fulfil all conditions of Theorem 2, \({\phi }^{\alpha }\in C^2(I)\). The fractal MQ-function is constructed iteratively from nodal points by using Theorem 2. Thus, it is sufficient to prove that \(({\phi }^{\alpha })'' \ge 0\) on I at the j-th iteration implies \(({\phi }^{\alpha })'' \ge 0\) on I at the \((j+1)\)-st iteration.
To this end, assume that r is nodal point at the j-iteration. Then, \(({\phi }^{\alpha })''(r) = {\phi }''(r) \ge 0\). Now, we have to show that \({\phi }^{\alpha }(u_i(r)) \ge 0\) for \(i\in {\mathbb {N}}_{N} \cup {\mathbb {N}}_{N}^-\). Employing (26) and (19), we obtain
As
we have
Hence, we have the desired convexity result of \({\phi }^\alpha \): \(({\phi }^{\alpha })''(u_{i}(r)) \ge 0, \text { for } i\in {\mathbb {N}}_N \cup {\mathbb {N}}_N^-\). \(\square \)
Remark 1
In the above theorem, we need a base function b that has to satisfy 2-point Hermite interpolation conditions at \(-r_N\) and \(r_N\) for \({\phi }\). For this reason, we use for b the Hermite polynomial \(H_5\) in the next two sections. The Hermite polynomial \(H_5(r)\) is given by
where the constants \(h_1\), \(h_2\), and \(h_3\) are determined by
4 The fractal quasi-interpolation operator \(L_{C}^{\alpha }\)
This section deals with the shape-preserving approximation of irregular or/and self-similar functions by means of a novel fractal quasi-interpolation operator. These quasi-interpolants are useful for approximating data generated from an irregular fractal function or a smooth function whose first or second derivative is irregular. Several examples are provided to demonstrate the importance of this quasi-interpolation operator when compared to the classical quasi-interpolation operators defined by the translates of an MQ-function.
Let \(f\in C^1(I)\), where \(I\subseteq {\mathbb {R}}\) is an interval as defined in Sect. 3. In the sequel, we employ the following notation: \( f'_0:=f'(x_0),\; f'_n:=f'(x_n), f_j:=f(x_j)\), and \({\phi }_j(x):=\sqrt{c^2+(x-x_j)^2}\), for \(j\in {\mathbb {N}}_{n,0}\).
We would like to approximate f using a fractal quasi-interpolation operator \(L_C^{\alpha }\). First, we choose a scaling vector \(\alpha \) according to Theorem 3 to ensure differentiability and convexity of the ensuing fractal MQ-function \({\phi }^{\alpha }\). We define a fractal quasi-interpolation operator \(L_{C}^{\alpha }\) using the translates of the fractal MQ-function \(\phi ^{\alpha }\) along the interpolation points \(x_i\in P\), i.e., we set
where
with \(\phi _j^{\alpha }(x):=\phi ^{\alpha }(x-x_j)\).
Remark 2
Note that \(\phi _j^{0}(x)=\phi (x-x_j)\) for \(j=0,1,\dots ,N\). Thus, for \(\alpha =0\), we can write \(\psi _j(x) := \psi _j^0(x)\), \(\beta _0(x) := \beta _0^0(x)\), \(\beta _n(x) := \beta _n^0(x)\), \(\gamma _n(x) := \gamma _n^0(x)\), and \(\gamma _n(x) := \gamma _n^0(x)\). Then, the above operator reduces to the classical quasi-interpolation operator \(L_{C}\) (Wu and Schaback 1994; Beatson and Powell 1992) which is given by
We require the following estimate to compute uniform error bounds for fractal quasi-approximants. For \(j\in \mathbb {N}_{n,0}\), we have that
Utilising the above inequality, we derive the following uniform error bounds between the classical and fractal approximants for the computation of \(\Vert L_C^{\alpha }f-f\Vert _{\infty }\) below. (See also Theorem 4.)
where we set \(\delta :=\min \limits _{1\le j\le n} |x_j-x_{j-1}|\).
Next, we present a computational example to show the advantages of fractal quasi-interpolants over the classical quasi-interpolants.
Example 2
Suppose we are given a function \(f^*\in C^1[0,1]\) whose first derivative is highly irregular. The graphs of \(f^*\) and \((f^*)'\) are depicted in Fig. 1b and a, respectively. We approximate \(f^*\) by \(L_C^{\alpha }f^*\). For this purpose, let \({\phi }= \sqrt{r^2 + c^2}\) with \(c=0.01\) and let \((\phi ^{\alpha })^{(k)}\), \(k=1,2\), be MQ-fractal functions corresponding to \(P:=\{0,\frac{1}{10},\frac{2}{10},\dots , \frac{9}{10},1\}\subseteq [0,1]\), \(b=H_5\), and \(\Delta :=\{-1<-\frac{9}{10}<\dots<-\frac{1}{10}<0<\frac{1}{10}< \dots<\frac{9}{10}<1\}\) with two different scaling vectors, both elements of \({\mathbb {R}}^{20}\):
Note that the scaling vectors \(\alpha ^k\) satisfy the conditions specified in (26) for \(k=1,2\).
Using these two scaling vectors, we construct the translates of the fractal multi-quadric functions \(\phi _j^{\alpha ^k}(x)=\phi ^{\alpha ^k}(x-x_j)\), for \(k =1,2\) and all j, and thus obtain the fractal quasi-interpolants \(L_C^{\alpha ^1}f^*\) and \(L_C^{\alpha ^2}f^*\). These, along with the classical quasi-interpolant \(L_Cf^*\) are shown in Fig. 1b.
The error functions \(|f^*(x)-L_C^{\alpha ^k}f^*(x)|\), \(k=1,2\), together with the error function \(|f^*(x)-L_Cf^*(x)|\) are plotted in Fig. 1c. One observes that the absolute error is higher for the classical quasi-interpolant than for the fractal quasi-interpolants. It is evident from Fig. 1d that the derivative of \(L_Cf^*\) does not contain sharp irregularities in the derivative of the original function \(f^*\). However, the derivatives of the fractal quasi-interpolants \(L_C^{\alpha ^1}f^*\) and \(L_C^{\alpha ^2}f^*\) capture more accurately and efficiently the derivative of the original functions \(f^*\) as compared to \(L_Cf^*\). This is demonstrated in Fig. 1e–f. Figure 1b–d also show that the fractal quasi-interpolants better approximate the original function by (i) providing a lower absolute error, and (ii) preserving the irregularities in the derivative of the original function \(f^*\) more accurately than the classical quasi-interpolant. If the derivative of the original function f is a smooth then one can take as the scaling vector \(\alpha = \textbf{0}\) to obtain the classical smooth quasi-interpolant \(L_Cf\).
Remark 3
In the case of fractal quasi-interpolation, we can relax (11), i.e., the partition \(\Pi \) need not contain all values of \(|x_i-x_j|\), for all \(i,j \in {\mathbb {N}}_{0,n}\), as the interpolation matrix does not arise in the construction. However, by experimenting with various numerical examples, it was found that fractal quasi-interpolants for which \(\Pi \) satisfies (11) yield better accuracy. This was used in the above example.
Theorem 4
Let \(f\in C^2[x_0,x_n]\) and let the interpolation data set be \(P:=\{x_0,x_1,\dots ,x_n\}\). Further, let \(h:= \max |x_j-x_{j-1}|_{1\le j\le n}\), \(\omega :=\sup _{x_0\le x \le x_N} |f{''}(x)|\), and \(\Vert f\Vert _1:=\max \{\Vert f\Vert _{\infty },\Vert f'\Vert _{\infty }\}\). Then, the uniform error between \( L_{C}^{\alpha }f\) and f is given by
- (i):
-
$$\begin{aligned} \begin{aligned} \Vert L_C^{\alpha }f-f\Vert _{\infty }&\le \frac{1}{4} c^2\omega \left\{ 1+2\log \left( 1+\frac{x_N-x_0}{c}\right) \right\} \\ {}&\quad + \frac{1}{8}h^2 \omega +\frac{\Vert \phi ^{\alpha }-\phi \Vert _{\infty }}{\delta }\Vert f\Vert _1(2n+1). \end{aligned} \end{aligned}$$(33)
- (ii):
-
If \(c\le Dh\) for some positive constant D, then the above estimate reduces to
$$\begin{aligned} \begin{aligned} \Vert L_C^{\alpha }f-f\Vert _{\infty }&\le \left\{ \frac{1}{8}+\frac{1}{4}D^2+\frac{1}{2}D^2 \log (1+\frac{x_n-x_0}{Dh})\right\} \omega h^2 \\ {}&\quad +\frac{\Vert \phi ^{\alpha }-\phi \Vert _{\infty }}{\delta } \Vert f\Vert _1(2n+1). \end{aligned} \end{aligned}$$(34)
Proof
Utilising (32) together with (30), yields
Without loss of generality, we can assume that \(\delta \le 1\). By assumption, we have
By Beatson and Powell (1992), it is known that
and
Thus, (37) and (38) establish the error estimate (33), and (37) and (39) give the estimate (34). \(\square \)
Remark 4
If we consider a set of uniformly distributed data points \(P:=\{x_0,x_1,\dots ,x_n\}\), i.e., with \(h:= x_j-x_{j-1}\), for \(1\le j\le n\), and additionally assume that \(\Vert \alpha \Vert _{\infty }=O(h^4)\), then we have
where \(A=3\frac{\Vert {\phi }-b_{MQ}\Vert _{\infty }}{1-\Vert \alpha \Vert _{\infty }} \Vert f\Vert _1 \) for small h. From above and together with (34), we conclude that
We now present a numerical example to verify the order of convergence \(O(h^2)\) derived in the last theorem for uniformly distributed data points and compare the fractal quasi-interpolation operator \(L_C^{\alpha }f(x)\) to its classical counterpart.
Example 3
Consider a function \(f^*\in C^2[0,1]\) whose second derivative, which is plotted in Fig. 2a, is highly irregular. The graph of \(f^*\) is depicted in Fig. 2b. We approximate \(f^*\) by using the fractal quasi-interpolant \(L_C^{\alpha ^k}f^*\), for \(k=1,2\). As in the previous example, we construct fractal MQ-functions \(\phi ^{\alpha ^k}\), for \( k=1,2\) from \({\phi }(r)= \sqrt{r^2 + c^2}\) with \(c = 0.01\). Further, we use again the partition \(P:=\{0,\frac{1}{10},\frac{2}{10},\dots , \frac{9}{10},1\}\subseteq [0,1]\), \(b=H_5\), and \(\Delta :=\{-1<-\frac{9}{10}<\dots<-\frac{1}{10}<0<\frac{1}{10}< \dots<\frac{9}{10}<1\}\) but choose as scaling functions the following two vectors in \({\mathbb {R}}^{20}\):
The scaling factors \(\alpha _i^1\) and \(\alpha _i^2\) were chosen according to condition (26).
Using the translates \(\phi _j^{\alpha ^k}(x)=\phi ^{\alpha ^k}(x-x_j)\), the fractal quasi-interpolant \(L_C^{\alpha ^1}f^*\) and \(L_C^{\alpha ^2}f^*\), along with the classical quasi-interpolant \(L_Cf^*\), are plotted in Fig. 2b. The error functions \(|f^*(x)-L_C^{\alpha ^k}f^*(x)|\), for \(k=1,2\), and \(|f^*(x)-L_Cf^*(x)|\) are depicted in Fig. 2c. One observes that the error estimates are of order \(O(h^2)\) as claimed in Theorem 6. Note that the second derivative of \(L_Cf^*\) in Fig. 2d is very much different from that of the original function in Fig. 1a. The second derivative of \(L_C^{\alpha ^1}f^*(x)\) and \(L_C^{\alpha ^2}f^*(x)\), illustrated in Fig. 2e–f, shows a better approximation of the second derivative of \(L_Cf^*(x)\). Employing diverse optimization techniques for scaling vectors, one can get more accurate fractal quasi approximants \(L_C^{\alpha }f^*(x)\). These fractal quasi-approximants are tailored to capture intricate patterns and structures within the data, enhancing the fidelity of the approximation process.
Theorem 5
If the data points \(\{f(x_j)\}_{j=0}^{n}, f{'}(x_0),f{'}(x_n)\) stem from a convex (concave, linear) function \( f\in C^1[x_0,x_n]\) and the scaling factors are chosen according to Theorem 3, then \(L_C^{\alpha }f(x)\) is a convex (concave, linear) function.
Proof
Let f be convex.
As the data points \(\{f(x_j)\}_{j=0}^{n}, f{'}(x_0),f{'}(x_n)\) are taken from a convex function, all the terms in square brackets in (40) are non-negative. Due to the choice of the scaling parameters, we know that \((\phi ^{\alpha }){''}(x)\ge 0\). Thus, the right hand side of (40) is non-negative confirming that \(L_C^{\alpha }f\) is convex. The other cases are proven similarly. \(\square \)
Now, we want to illustrate the results in Theorem 5 by a few examples.
Example 4
Let \(\phi _j^{\alpha }(x)=\phi ^{\alpha }(x-x_j)\), \(c=0.01\), and let \(\phi ^{\alpha }\) be a fractal MQ-function corresponding to \(P:=\{0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\}\subseteq [0,1]\), \(b=H_5\), \(\Delta =\{-1<-\frac{4}{5}<\dots<-\frac{1}{5}<0<\frac{1}{5}< \dots<\frac{4}{5}<1\}\), and
The scaling vector \(\alpha \) satisfies the conditions in (26).
Choose the convex function \(f(x):=(x-1/2)^2\), \(x\in [0,1]\). The graphs of f and \(L_C^{\alpha }f\) are shown in Fig. 3a. One observes that \(L_C^{\alpha }f\) is a convex function.
Let us consider now the concave function \(f(x)=\sin (2\pi x)\), \(x\in [0,1]\). The graphs of f and \(L_C^{\alpha }f\) are depicted in Fig. 3b. Since \(\alpha \) obeys (26), it is verified that \(L_C^{\alpha }f\) is concave.
5 The fractal quasi-interpolation operator \(L_{D}^{\alpha }\)
In this section, we require the scaling vector \(\alpha \) for the fractal MQ-function \({\phi }^{\alpha }\) to satisfy Theorem 3, hence \({\phi }^{\alpha }\) is convex. Since the quasi-interpolation operator \(L_C^\alpha f\) requires the values of the derivative of the function f at the end points, we propose another fractal quasi-operator \(L_D^\alpha f\) which avoids this requirement.
Let \(f\in C[x_0,x_n]\) and let \(P:=\{ x_0,x_1,\dots ,x_n\}\) be a set of interpolation points. Define a new fractal quasi-interpolation operator by
where
As \({\phi }^0_j(x) = {\phi }(x)\), for \( j \in \mathbb {N}_{n,0}\), the proposed fractal quasi-interpolation operator reduces to its classical counterpart \(L_Df\) defined by Wu and Schaback (1994) whenever \(\alpha ={0}\).
Next, we discuss convexity, concavity, and linear polynomial reproduction for \(L_{D}^{\alpha }\).
Theorem 6
Suppose that a set of data points \(\{f(x_j)\}_{j=0}^n\) is obtained from a convex, concave, or linear function f. If the scaling factors and the base function are chosen according to Theorem 3, then the fractal quasi-interpolant \(L_{D}^{\alpha }f(x)\) is also convex, concave, or linear, respectively.
Proof
Computing the second derivative of (41), yields
Simplifying (42) produces
Note that each term in square brackets is positive or negative according to whether the original function f is convex or concave, respectively. By our assumption on \(\alpha \), \((\phi ^{\alpha }){''} \ge 0\). Thus, (43) confirms that \(L_{D}^{\alpha }f\) is convex (concave) if f is convex (concave). If f is linear on its domain, then each term in square brackets in (42) is zero, that is, \(L_{D}^{\alpha }f\) is a linear polynomial. \(\square \)
Example 5
Consider the convex function
and the concave function
f and g are approximated by the fractal quasi-operators \(L_{D}^{\alpha }f\) and \(L_{D}^{\alpha }g\), respectively. Let \(\phi ^{\alpha }\) be a fractal MQ-function corresponding to \(P=\{0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\}\), \(b=H_5\), \(c=0.01\), \(\Delta =\{-1<-\frac{4}{5}<\dots<-\frac{1}{5}<0<\frac{1}{5}< \dots<\frac{4}{5}<1\}\), and
where the scaling factors were chosen according to Theorem 3. Using the translates \(\phi _j^{\alpha }(x)=\phi ^{\alpha }(x-x_j)\), \( j \in \mathbb {N}_{n,0}\), the fractal quasi-approximants \(L_D^{\alpha }f\) and \(L_D^{\alpha }g\) are shown in Fig. 4a and b, respectively. One observes that \(L_D^{\alpha }f\) is a convex and \(L_D^{\alpha }g\) a concave function. As f is not linear on its domain, \(L_D^{\alpha }f\) is also not linear; see Fig. 4a.
In the next theorem, we derive an additional condition on the scaling vector \(\alpha \) so that the fractal quasi-interpolation \(L_D^{\alpha }f\) preserves the monotonicity of a function \(f \in {C}[x_0,x_n]\).
Theorem 7
The fractal quasi-interpolation operator \(L_D^{\alpha }\) defined in (41) preserves the monotonicity of f if the scaling factors and the base function are chosen according to Theorem 3 and, in addition,
where |a| and K are defined in Sect. 3.
Proof
As \((\phi ^{\alpha }){''}\ge 0\), \((\phi ^{\alpha }){'}\) is monotonic increasing. Now \({\phi }'(r) = \frac{r}{\sqrt{r^2+c^2}} \le 1.\) Thus,
The above inequality ensures that for \( x\in [x_0,x_n]\),
From (20), for \(r\in [-K,K]\), we obtain
which implies
Now, \({(\phi ^{\alpha }}){'}(r)\le \phi {'}(r) +\frac{|\alpha |_{\infty }}{|a|-|\alpha |_{\infty }} \Vert \phi {'}-b{'}\Vert _{\infty } < 1\) provided that
Thus, we conclude that \({(\phi ^{\alpha }}){'}(r) < 1\) if
Differentiating \(L_D^{\alpha }f\), yields
Rearranging the above expression, we obtain
Now, (46) ensures the non-negativity of the first square bracket in (47), and (45) the non-negativity of the second and third square bracket in (47). As the set of data points \(\{f_j\}_{j=0}^{n}\) satisfies \(f_j\le f_{j+1}\), all the terms in (47) are non-negative. Hence, \(L_D^{\alpha }f\) is monotone. \(\square \)
Example 6
Consider a monotone function
Let \(\phi ^{\alpha }(x)\) be a fractal MQ-function corresponding to the base function \(b=H_5\), \(P:=\{0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\}\), \(c=0.01\), \(\Delta =\left\{ -1<-\frac{4}{5}< \dots<-\frac{1}{5}<0<\frac{1}{5}< \dots<\frac{4}{5}<1\right\} \), and
Note that \(\alpha \) is chosen according to Theorem 3 and (44). The graphs of \(L_D^{\alpha }f\) and f are plotted in Fig. 5a, and their zoomed-in versions are shown in Fig. 5b. The monotonicity of \(L_D^{\alpha }f\) confirms the statement in Theorem 7.
In the following theorem, the rate of convergence of the fractal quasi-operator \(L_D^{\alpha }f\) to an \(f\in C^2[x_0,x_n]\) is derived.
Theorem 8
Let \(f\in C^2[x_0,x_n]\) and let \(h := \max \limits _{1 \le j \le n} |x_j - x_{j-1}|\). Then, a bound for the uniform error between f and \(L_D^{\alpha }f\) is given by
for some positive constants \(K_0,K_1,K_2,K_3,K_4\). If \(c=O(h)\), then
Proof
We can rewrite \(L_D^{\alpha }\) as
where \(\Delta \) and \(\Delta ^2\) denotes the first and second order divided difference operator, respectively. Let Lf be the piecewise interpolant of f. Then, for \( x \in [x_0, x_n]\),
By Wu and Schaback (1994), it is known that
for some positive constants \(M_0, M_1\). Let
Substituting (52) into (51), produces
Thus,
By standard interpolation theory, we know that \(\Vert Lf-f\Vert _{\infty }=M_3h^2\), for some positive constant \(M_3\). Therefore, (53) reduces to
where \(K_0=M_3\), \(K_1=M_1\), \(K_2=8\), \(K_3=M_0\), and \(K_4=2M_2\). As the scaling factors are chosen according to \( |\alpha _i| < a_i^2\), for \( i \in \mathbb {N}_N\), we have that \(\Vert \alpha \Vert _{\infty }=O(h^2)\). If \(c=O(h)\), then we get the desired estimate in (49). \(\square \)
The above convergence rate for \(L_D^{\alpha }f\) makes \(L_D^{\alpha }f\) more valuable than the fractal quasi-operator \(L_C^{\alpha }f\). We provide an example to demonstrate the superior performance of the fractal quasi-interpolation method over the classical quasi-interpolants. Both interpolants work equally well for smooth functions but when we approximate a self-similar and/or irregular function, then the fractal quasi-interpolation operator \(L_D^{\alpha }\) has definite advantages over its classical counterpart \(L_D\).
Example 7
Consider a fractal function \(f^*(x), x\in [0,1]\), which is constructed from the IFS (7) with germ function \(f(x)=\sin (\pi x)\), partition \(P:=\{0, \frac{1}{5}, \frac{2}{5},\frac{3}{5}, \frac{4}{5}, 1\}\subseteq [0,1]\), a base function b which is the Bernstein polynomial \(B_2f\), and \(\beta _i=0.2\), for all \(i\in \mathbb {N}_{5}\).
The fractal function \(f^*\) is depicted in Fig. 6a and b in blue color. Applying the procedure outlined in Sect. 2, we construct a fractal MQ-function \({\phi }^{\alpha }\) corresponding to \(\Omega :=\{0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\}\),
\(c=0.01\), and \(b(r):=\frac{r-1}{-2}{\phi }(-1)+\frac{r+1}{2}\).
The approximants \(L_Df^*\) and \(L_D^{\alpha }f^*\) are plotted in Fig. 6 for different values of the scaling factors (\(\alpha \in \{0.05,0.06,0.07\}\)) along with the function \(f^*\). From Fig. 6a and b, it is clear that the quasi-interpolant \(L_Df^*\) is not a good approximant for \(f^{*}\), for any value of the shape parameter c. However, \(L_D^{\alpha }f^*\) very nicely captures the irregularities of \(f^*\) and is even close to \(f^*\); see Fig. 6b. One can employ an optimization technique to obtain the optimal value for the scale factor \(\alpha \) to get a best approximation for \(f^*\). Thus, the operator \(L_D^{\alpha }\) can be used to approximate non-smooth functions effectively in addition to any smooth function.
In Theorem 8, we derived an upper bound for the uniform error between f and \(L_D^{\alpha }f\) in order to obtain the convergence result. We note that it is not necessarily true that \(\Vert f - L_Df \Vert _\infty \le \Vert f -L_D^{\alpha }f \Vert _\infty \) even for a smooth function f. The following example shows that \(L_D^{\alpha }f\) is a much better approximant than the classical quasi-approximant \(L_Df.\)
Example 8
Consider the smooth function \(f(x)=\sin ( \pi x)\), \(x\in [0,1]\). The smooth fractal MQ-function \(\phi ^{\alpha }\) is constructed according to Sect. 3, where \(\Omega :=\{0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\}\subseteq [0,1]\),
\(\alpha _j=\alpha _{-j}, j\in {\mathbb {N}}_5\), and \(b=H_5\) on \([-1,1]\).
The fractal MQ-approximants \(L_D^{\alpha }f\) are plotted in Fig. 7a for different values of \(\alpha \) together with \(L_D\) and f. It is difficult to distinguish between these quasi-approximants graphically. Thus, we plotted in Fig. 7b the graphs of the errors \(|f- L_D^{\alpha }f|\) and \(|f - L_Df|\). The error between the quasi-interpolants and f is less than that between the classical quasi-approximant and f. This example shows that fractal quasi-approximants are superior to the classical quasi-approximants.
6 Conclusion
In this work, we proposed the novel concepts of a fractal multiquadric function \({\phi }^\alpha (r)\) and a fractal quasi-interpolation operator. The former generalizes the classical multiquadric functions \({\phi }(r)=\sqrt{r^2+c^2}\), \(c>0\), and the latter the classical quasi-interpolation operators. We derived suitable conditions for the scaling factors to preserve the smoothness and convexity of a germ function \({\phi }\). Two quasi-interpolations operators \(L_C^{\alpha }f\) and \(L_D^{\alpha }f\) that we introduced have been investigated concerning their shape-preserving properties, such as linear polynomial reproduction, convexity, concavity, and monotonicity. Through our analysis, we provided evidence showcasing the advantages of fractal quasi-approximants over their classical counterparts. Additionally, we delved into the rate of convergence of these fractal quasi-interpolants to the original function. Our findings suggest that these newly introduced fractal quasi-interpolants are adept at approximating both smooth and irregular functions effectively.
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References
Bao W, Song Y (2014) Multiquadric quasi-interpolation methods for solving partial differential algebraic equations. Numer Methods Partial Differ Equ 30(1):95–119. https://doi.org/10.1002/num.21797
Barnsley MF (1993) Fractals Everywhere. In: 2nd edn., Academic Press Professional, Boston, MA, Boston, p 534. Revised with the assistance of and with a foreword by Hawley Rising, III
Barnsley MF (1986) Fractal functions and interpolation. Constr Approx 2(4):303–329. https://doi.org/10.1007/BF01893434
Barnsley MF (1996) Fractal image compression. Notices Am Math Soc 43(6):657–662
Barnsley MF, Demko S (1985) Iterated function systems and the global construction of fractals. Proc R Soc Lond A 399:243–275
Barnsley MF, Harrington AN (1989) The calculus of fractal interpolation functions. J Approx Theory 57(1):14–34. https://doi.org/10.1016/0021-9045(89)90080-4
Beatson RK, Dyn N (1996) Multiquadric \(B\)-splines. J Approx Theory 87(1):1–24. https://doi.org/10.1006/jath.1996.0089
Beatson RK, Powell MJD (1992) Univariate multiquadric approximation: quasi-interpolation to scattered data. Constr Approx 8(3):275–288. https://doi.org/10.1007/BF01279020
Brambila F (2017) Fractal analysis—applications in physics. Engineering and Technology, IntechOpen, Rijeka
Chand AKB, Kapoor GP (2006) Generalized cubic spline fractal interpolation functions. SIAM J Numer Anal 44(2):655–676. https://doi.org/10.1137/040611070
Chand AKB, Vijender N, Navascués MA (2014) Shape preservation of scientific data through rational fractal splines. Calcolo 51(2):329–362. https://doi.org/10.1007/s10092-013-0088-2
Chand AKB, Vijender N, Viswanathan P, Tetenov AV (2020) Affine zipper fractal interpolation functions. BIT 60(2):319–344. https://doi.org/10.1007/s10543-019-00774-3
Chen R, Wu Z (2006) Applying multiquadratic quasi-interpolation to solve Burgers’ equation. Appl Math Comput 172(1):472–484. https://doi.org/10.1016/j.amc.2005.02.027
Chen R, Wu Z (2007) Solving partial differential equation by using multiquadric quasi-interpolation. Appl Math Comput 186(2):1502–1510. https://doi.org/10.1016/j.amc.2006.07.160
Duan Y, Rong F (2013) A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation. Eng Anal Bound Elem 37(1):89–94. https://doi.org/10.1016/j.enganabound.2012.08.006
Gao F, Chi C (2014) Numerical solution of nonlinear Burgers’ equation using high accuracy multi-quadric quasi-interpolation. Appl Math Comput 229:414–421. https://doi.org/10.1016/j.amc.2013.12.035
Gao W, Zhang R (2018) Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective. Numer Algorithms 77(1):243–259. https://doi.org/10.1007/s11075-017-0313-1
Geng Y, Sun W, Ying P, Zheng Y, Ding J, Sun K, Li L, Li M (2021) Bioinspired fractal design of waste biomass-derived solar-thermal materials for highly efficient solar evaporation. Adv Funct Mater 31(3):2007648. https://doi.org/10.1002/adfm.202007648
Golany T, Freedman D, Radinsky K (2021) ECG ODE-GAN: Learning ordinary differential equations of ECG dynamics via generative adversarial learning 35(1):134–141
Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. Geo Res 76(8):1905–1915
Hon YC, Wu Z (2000) A quasi-interpolation method for solving stiff ordinary differential equations. Int J Numer Methods Eng 48(8):1187–1197
Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30(5):713–747. https://doi.org/10.1512/iumj.1981.30.30055
Liu ST, Zhang YP, Liu CA (2020) Applications of fractal control in biologies. Springer, Singapore, pp 163–234
Ma L, Wu Z (2009) Approximation to the \(k\)-th derivatives by multiquadric quasi-interpolation method. J Comput Appl Math 231(2):925–932. https://doi.org/10.1016/j.cam.2009.05.017
Massopust P (2010) Interpolation and approximation with splines and fractals. Oxford University Press, Oxford
Massopust P (2016) Fractal functions, fractal surfaces, and wavelets, 2nd edn. Elsevier/Academic Press, London, p 405
Navascués MA (2005) Fractal polynomial interpolation. Z Anal Anwend 24(2):401–418. https://doi.org/10.4171/ZAA/1248
Navascués MA, Chand AKB (2008) Fundamental sets of fractal functions. Acta Appl Math 100(3):247–261. https://doi.org/10.1007/s10440-007-9182-2
Navascués MA, Massopust PR (2019) Fractal convolution: a new operation between functions. Fract Calc Appl Anal 22(3):619–643. https://doi.org/10.1515/fca-2019-0035
Navascués MA, Sebastián MV (2006) Smooth fractal interpolation. J Inequal Appl. https://doi.org/10.1155/JIA/2006/78734
Onali E, Goddard J (2011) Are european equity markets efficient? New evidence from fractal analysis. Int Rev Finan Anal 20(2):59–67
Ortmann M, Buhmann M (2024) High accuracy quasi-interpolation using a new class of generalized multiquadrics. J Math Anal Appl 538(1):128359. https://doi.org/10.1016/j.jmaa.2024.128359
Pan G, Zhang S (2023) A meshless multiquadric quasi-interpolation method for time fractional Black-Scholes model. Int J Financ Eng 10(2):2350008–12. https://doi.org/10.1142/S2424786323500081
Rihan FA (2021) Delay differential equations and applications to biology. Springer, Singapore
Sun W, Xu PGG, Liang S (2006) Fractal analysis of remotely sensed images: a review of methods and applications. Int J Remote Sens 27(22):4963–4990. https://doi.org/10.1080/01431160600676695
Timbo C, Rosa LAR, Gonçalves M, Duarte SB (2009) Computational cancer cells identification by fractal dimension analysis. Comput Phys Commun 180(6):850–853
Tyada KR, Chand AKB, Sajid M (2021) Shape preserving rational cubic trigonometric fractal interpolation functions. Math Comput Simul 190:866–891. https://doi.org/10.1016/j.matcom.2021.06.015
Vijender N, Chand AKB, Navascués MA, Sebastián MV (2021) Quantum Bernstein fractal functions. Comput. Math. Methods 3(3):1118–13. https://doi.org/10.1002/cmm4.1118
Viswanathan P, Chand AKB (2015) A \(C^1\)-rational cubic fractal interpolation function: convergence and associated parameter identification problem. Acta Appl Math 136:19–41. https://doi.org/10.1007/s10440-014-9882-3
Viswanathan P, Navascués MA, Chand AKB (2016) Associate fractal functions in \(L^p\)-spaces and in one-sided uniform approximation. J Math Anal Appl 433(2):862–876. https://doi.org/10.1016/j.jmaa.2015.08.012
Wang X, Liu C, Gao C, Yao K, Masouleh SSM, Berté R, Ren H, Menezes L, Cortés E, Bicket IC, Wang H, Li N, Zhang Z, Li M, Xie W, Yu Y, Fang Y, Zhang S, Xu H, Vomiero A, Liu Y, Botton GA, Maier SA, Liang H (2021) Self-constructed multiple plasmonic hotspots on an individual fractal to amplify broadband hot electron generation. ACS Nano 15(6):10553–10564. https://doi.org/10.1021/acsnano.1c03218
Wu H-Y, Duan Y (2016) Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation. Appl Math Comput 274:83–92. https://doi.org/10.1016/j.amc.2015.10.044
Wu ZM, Schaback R (1994) Shape preserving properties and convergence of univariate multiquadric quasi-interpolation. Acta Math Appl Sin (English Ser) 10(4):441–446. https://doi.org/10.1007/BF02016334
Wu Z, Zhang S (2013) Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations. Eng Anal Bound Elem 37(7–8):1052–1058. https://doi.org/10.1016/j.enganabound.2013.04.011
Zhang S, Yang H, Yang Y (2019) A multiquadric quasi-interpolations method for CEV option pricing model. J Comput Appl Math 347:1–11. https://doi.org/10.1016/j.cam.2018.03.046
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The second author would like to thank Indian Institute of Technology Madras for the ERP with Grant No.: RF23241503MARFER008323.
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Kumar, D., Chand, A.K.B. & Massopust, P.R. Shape preserving fractal multiquadric quasi-interpolation. Comp. Appl. Math. 43, 281 (2024). https://doi.org/10.1007/s40314-024-02802-7
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DOI: https://doi.org/10.1007/s40314-024-02802-7
Keywords
- Fractals
- Fractal interpolation
- Multiquadric function
- Quasi-interpolation
- Monotonicity
- Convexity
- Convergence