<p>In this paper, we discuss univariate and bivariate constrained interpolation problems using the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-rational cubic fractal interpolation function (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-RCFIF) based solely on function values. The <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-RCFIF takes the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\tfrac{P_i(\textbf{u})}{Q_i(\textbf{u})}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <msub> <mi>Q</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mstyle> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$P_i(\textbf{u})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a cubic polynomial determined by the interpolation conditions and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q_i(\textbf{u})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a preassigned quadratic polynomial. We establish suitable conditions on the associated iterated function system (IFS) parameters to ensure that the graph of the corresponding <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-RCFIF (the attractor) lies between two specified piecewise functions. For appropriate choices of tension parameters and vertical scaling factors, the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-RCFIF remains confined within a rectangle. Furthermore, we develop a bivariate cubic rational fractal model (BCRFM) by applying the Coons blending technique. By selecting appropriate shape parameters and scaling factors, the BCRFM is shown to lie above a prescribed plane. With properly chosen IFS parameters, the BCRFM is also contained within a cuboid. Several numerical examples are provided to illustrate the effectiveness of the proposed interpolation scheme. Additionally, we incorporate ARIMA-based forecasting to complement the fractal interpolation framework.</p>

A geometric approach to constrained fractal interpolation and forecasting: piecewise bounds, rectangles, and cuboids

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发表日期：

2026年7月2日

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Mahipal Reddy K,
Rajesh L

摘要

In this paper, we discuss univariate and bivariate constrained interpolation problems using the $\beta $ $β$ -rational cubic fractal interpolation function ( $\beta $ $β$ -RCFIF) based solely on function values. The $\beta $ $β$ -RCFIF takes the form $\tfrac{P_i(\textbf{u})}{Q_i(\textbf{u})}$ $\frac{P_{i} (u)}{Q_{i} (u)}$ , where $P_i(\textbf{u})$ $P_{i} (u)$ is a cubic polynomial determined by the interpolation conditions and $Q_i(\textbf{u})$ $Q_{i} (u)$ is a preassigned quadratic polynomial. We establish suitable conditions on the associated iterated function system (IFS) parameters to ensure that the graph of the corresponding $\beta $ $β$ -RCFIF (the attractor) lies between two specified piecewise functions. For appropriate choices of tension parameters and vertical scaling factors, the $\beta $ $β$ -RCFIF remains confined within a rectangle. Furthermore, we develop a bivariate cubic rational fractal model (BCRFM) by applying the Coons blending technique. By selecting appropriate shape parameters and scaling factors, the BCRFM is shown to lie above a prescribed plane. With properly chosen IFS parameters, the BCRFM is also contained within a cuboid. Several numerical examples are provided to illustrate the effectiveness of the proposed interpolation scheme. Additionally, we incorporate ARIMA-based forecasting to complement the fractal interpolation framework.