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Let D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) , ( − 1 − 1 ) } and M ∈ M 2 (ℤ) be an expansive matrix. In this paper, we have proved that there exists Λ such that the set ℰ Λ : = { e 2 π i 〈 λ , x 〉 : λ ∈ Λ } is an orthonormal basis of L 2 (μ M , D) for a matrix which has all even entries. Also, we have found a spectrum Λ of L 2 (μ M , D) for some specific M and D. [ABSTRACT FROM AUTHOR]

In this paper, a new notion of super coalescence hidden-variable fractal interpolation function (SCHFIF) is introduced. The construction of SCHFIF involves choosing an IFS from a pool of several non-diagonal IFS at each level of iteration. Further, the integral of a SCHFIF is studied and shown to be a SCHFIF passing through a different set of interpolation data. [ABSTRACT FROM AUTHOR]

Riemann-Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order of a CHFIF defined on any interval is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order of a CHFIF defined on any interval is also a CHFIF. [ABSTRACT FROM AUTHOR]

In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [xo, xN], the interpolating Cubic Spline SFIF gσ(x) ≡ gσ(0)(x) and their derivatives gσ(j)(x) converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ε( 0 < ε < 1), j = 0,1, 2, as the norm h of the partition of [xo, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives. [ABSTRACT FROM AUTHOR]

In the present paper, the bounds on fractal dimension of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) in ℝ3 on a equispaced mesh are found. These bounds determine the conditions on the free parameters for fractal dimension of the constructed CHFIS to become close to 3. The results derived here are tested on a tsunami wave surface by computing the lower and upper bounds of the fractal dimension of its CHFIS simulation. [ABSTRACT FROM AUTHOR]

In the present paper, the smoothness of a Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS), as described by its Lipschitz exponent, is investigated. This is achieved by considering the simulation of a generally uneven surface using CHFIS. The influence of free variables and Lipschitz exponent on the smoothness of CHFIS is demonstrated by considering interpolation data generated from a sample surface. [ABSTRACT FROM AUTHOR]