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
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