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Section Innovation in Computer Science

Generalized Jacobi Chebyshev Wavelet Approximation

Perkiraan Wavelet Jacobi Chebyshev Umum
Vol. 26 No. 3 (2025): July:

Mohammad Zarif Mehrzad (1), Abdulwali Azamsafi (2), Abdul Mohammad Qudosi (3)

(1) Department of Mathematics, Parwan University, Parwan, Afghanistan
(2) Department of Mathematics, Parwan University, Parwan, Afghanistan
(3) Department of Mathematics, Parwan University, Parwan, Afghanistan

Abstract:

General Background: Wavelet approximations are fundamental in numerical analysis and signal processing, with classical orthogonal polynomials like Jacobi and Chebyshev serving as key tools due to their strong approximation properties. Specific Background: The use of Chebyshev wavelets has been extended through generalized polynomial frameworks, such as Koornwinder’s generalization of Jacobi polynomials, offering more flexibility for function approximation on finite intervals. Knowledge Gap: Despite existing wavelet frameworks, the integration of generalized Jacobi and Chebyshev structures into a unified wavelet approximation scheme remains underexplored. Aims: This study introduces the Generalized Jacobi Chebyshev Wavelet (GJCW) approximation, establishing its theoretical foundations and demonstrating convergence and approximation capabilities. Results: It is shown that for a uniformly bounded function expanded in the GJCW basis, the partial sums yield both convergent and best uniform polynomial approximations. Novelty: The formulation of a new wavelet approximation based on a hybrid of generalized Jacobi and Chebyshev polynomials constitutes a novel contribution, supported by rigorous recurrence relations and multiresolution analysis. Implications: This work enhances the theoretical landscape of wavelet-based function approximation, with potential applications in computational mathematics, signal analysis, and numerical solutions of differential equations.


Highlight :




  • Wavelet Construction: The paper defines and constructs generalized Jacobi Chebyshev wavelets using orthogonal polynomials.




  • Approximation Theory: It proves that if the wavelet series converges, then a uniform best polynomial approximation exists.




  • Multiresolution Framework: The approach is grounded in Mallat’s multiresolution analysis, enabling efficient function approximation.




Keywords : Jacobi Polynomials, Chebyshev Wavelets, Multiresolution Analysis, Polynomial Approximation, Orthonormal Basis

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INTRODUCTION

Jacobi polynomials P_n^(α,β) (x) constitute a category of classical orthogonal polynomials. They are orthogonal about the weight (1-x)^α (1+x)^β on the interval [-1,1]. . We have P_n^((α,β)) (x) Chebyshev polynomials with |x|≤1.

Figure 1.

also for n=4,5,⋯

Figure 2.

The Jacobi polynomials are generated by the three-term recurrence relation, for scalars a_n^((α,β)),b_n^((α,β)) and , c_n^((α,β)),

Figure 3.

where

Figure 4.

also

Figure 5.

The Jacobi polynomials y=P_n^((α,β)) (x) solve the linear second-order differential equation

Figure 6.

A. Jacobi polynomials

Theorem 1.1 There are scalars a_0^((α,β)),a_1^((α,β)),⋯,a_n^((α,β))∈R such that

P_n^((α,β)) (x)=∑_(m=0)^n▒‍ a_(n,m)^((α,β)) x^m.

Proof. We use mathematical induction.

For this

P_0^((α,β)) (x)=1,

P_1^((α,β)) (x)=1/2(α+β+2)x+1/2(α-β).

The hypothesis of mathematical induction:

Suppose for 0≤k<n , we have

Figure 7.

The rule of mathematical induction:

For scalars a_n^((α,β)),b_n^((α,β)) and c_n^((α,β))

Figure 8.

Theorem 1.2

Figure 9.

B. Generalized Jacobi Chebyshev Wavelets

Figure 10.

Figure 11.

It is necessary to study multiresolution analysis and Mallat’s Theorem for wavelet approximation.

Definition 2.1 Multiresolution Analysis: An MRA with scaling function ϕ constitutes a collection of closed subspaces

Figure 12.

Figure 13.

We now present Mallat’s theorem, which ensures that in the context of an orthogonal multiresolution analysis (MRA), an orthonormal basis exists for L^2 (R) exists. These basis functions are essential in wavelet theory, facilitating the development of sophisticated computational algorithms.

Lemma 2.1 (Mallat's Theorem) In the context of an orthogonal multiresolution analysis (MRA) characterised by a scaling function ϕ, a corresponding wavelet exists ψ∈L^2 (R) such that for each j∈Z , the family {ψ_(j,k) }_(k∈Z) is an orthonormal basis for W_j . Hence the family {ψ_(j,k) }_(k∈Z) is an orthonormal basis for L^2 (R) .

Definition 2.2 (i) Let family be an orthonormal basis for L^2 (R) and P_n (f) the orthogonal projection of L^2 ([-1,1]) onto V_n . Then

P_n (f)=∑_(-∞)^∞▒‍<f,ψ_(n,k)>ψ_(n,k),n=1,2,3,⋯

Figure 14.

Figure 15.

Definition 2.3 Suppose f∈L^2 [a,b] and P_n is a set of all polynomials of degree and smaller n. If there exists a function q^*∈P_n such that lim_(n→∞) P_n (f)=0 , where P_n (f)=〖inf〗_(p∈P_n )∥f-p∥_2 . Then is called best uniform polynomial approximation to f on [a,b] .

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Declarations

Ethical Approval: not applicable

Funding: No funding

Availability of data and materials: Data sharing is not applicable to this article.

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