<bold id="bold-3daadef412b0c58283315ee9101ed262">INTRODUCTION</bold>

GENERALIZED JACOBI CHEBYSHEV WAVELET APPROXIMATION

APROKSIMASI WAVELET JACOBI CHEBYSHEV UMUM

Hamid

Siti Nur Cholisa

sitinur@gmail.com Muis

Lidya Shery

lidyasherymuis@umsida.ac.id

Indonesia Indonesia 25 10 2024

<bold id="bold-3daadef412b0c58283315ee9101ed262">INTRODUCTION</bold>

Jacobi polynomials constitute a category of classical orthogonal polynomials. They are orthogonal about the weight on the interval . We have Chebyshev polynomials with .

Figure 1

also for

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The Jacobi polynomials are generated by the three-term recurrence relation, for scalars and ,

Figure 3

where

Figure 4

also

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The Jacobi polynomials solve the linear second-order differential equation

Figure 6

<bold id="bold-63df22ff0671105680574d7cf49540e7">A. Jacobi polynomials</bold>

Theorem 1.1 There are scalars such that

Proof. We use mathematical induction.

For this

The hypothesis of mathematical induction:

Suppose for , we have

Figure 7

The rule of mathematical induction:

For scalars and

Figure 8

Theorem 1.2

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<bold id="bold-0a3ab999d2eab8ff10a7dbec929e0c0a">B. Generalized Jacobi Chebyshev Wavelets</bold> Figure 10

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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

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We now present Mallat’s theorem, which ensures that in the context of an orthogonal multiresolution analysis (MRA), an orthonormal basis exists for 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 such that for each , the family is an orthonormal basis for . Hence the family is an orthonormal basis for .

Definition 2.2 (i) Let family be an orthonormal basis for and the orthogonal projection of onto . Then

ψ_(n,k),n=1,2,3,⋯]]>

Figure 14

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Definition 2.3 Suppose and is a set of all polynomials of degree and smaller n. If there exists a function such that , where . Then is called best uniform polynomial approximation to on .

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<bold id="_bold-21">Declarations</bold>

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