Numerical examples concerning the comparison between the approximation of functions in $L^2$ norm and $W^{1,2}$ norm are also presented.
📌 3 Key Takeaways
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Objective & Hypothesis:Let ${q_n^{(alpha,beta)}}_{ngeq0}$ be the sequence of polynomials orthonormal with respect to the Sobolev inner product$langle f,g rangle s := int_{-1}^1 f(x)g(x)w^{(alpha,beta)}(x)dx + int_{-1}^1 f'(x)g'(x)w^{(alpha+1,beta+1)}(x)dx $,where $w^{(alpha,beta)} (x) = (1-x)^{alpha} (1+x)^{beta}, x in [-1,1]$ and $alpha, beta > −1$ .
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Key Findings & Data:This paper explores the convergence in the $W^{1-p} biggl((-1,1),(w^{(alpha,beta)}, w^{(alpha +1,beta +1)})biggr)$ norm of the Fourier expansion in terms of ${q_n^{(alpha,beta)}}_{ngeq0}$with 1 < p < ∞ , using the Pollard decomposition method.
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Conclusion & Impact:Numerical examples concerning the comparison between the approximation of functions in $L^2$ norm and $W^{1,2}$ norm are also presented.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
Let ${q_n^{(alpha,beta)}}_{ngeq0}$ be the sequence of polynomials orthonormal with respect to the Sobolev inner product$langle f,g rangle s := int_{-1}^1 f(x)g(x)w^{(alpha,beta)}(x)dx + int_{-1}^1 f'(x)g'(x)w^{(alpha+1,beta+1)}(x)dx $,where $w^{(alpha,beta)} (x) = (1-x)^{alpha} (1+x)^{beta}, x in [-1,1]$ and $alpha, beta > −1$ . This paper explores the convergence in the $W^{1-p} biggl((-1,1),(w^{(alpha,beta)}, w^{(alpha +1,beta +1)})biggr)$ norm of the Fourier expansion in terms of ${q_n^{(alpha,beta)}}_{ngeq0}$with 1 < p < ∞ , using the Pollard decomposition method. Numerical examples concerning the comparison between the approximation of functions in $L^2$ norm and $W^{1,2}$ norm are also presented.
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Cite This Article
MARCELLAN, F., QUINTANA, Y., URIELES, A. (2022). On the Pollard decomposition method applied to some Jacobi–Sobolev expansions. Turkish Journal of Mathematics, Cilt: 37 - Sayı: 6, 934–948. https://ajindex.com/articles/608337
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