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Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. (English) Zbl 1057.41003
The paper is devoted to Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. The approximation results are of the form $$ \left\Vert Q_N v-v \right\Vert _{B_1} \leq d_{N,\alpha,\beta}\vert v \vert_{B_2}, $$ where $B_1$ and $B_2$ are non-uniformly Jacobi weighted Sobolev spaces, $Q_N$ is an orthogonal projection upon the set of polynomials of degree at most $N$, and $d_{N,\alpha,\beta}$ is an explicit function of $N$, $\alpha$ and $\beta$, independent of $v$. The results are general and all estimates are as sharp as possible. First some basic results on Jacobi approximation are dissussed. Then several orthogonal approximations in non-uniformly Jacobi weighted Sobolev spaces, and also Jacobi-Gauss-type interpolations are studied. These results are useful tools in the numerical solutions of differential and integral equations.

MSC:
41A10Approximation by polynomials
41A25Rate of convergence, degree of approximation
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References:
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