Mohammad Zarif Mehrzad (1), Abdulwali Azamsafi (2), Abdul Mohammad Qudosi (3)
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
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.
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.
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.
Ethical Approval: not applicable
Funding: No funding
Availability of data and materials: Data sharing is not applicable to this article.
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