Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces.

*(English)*Zbl 1057.41003The paper is devoted to Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. The approximation results are of the form
\[
\left\| Q_N v-v \right\| _{B_1} \leq d_{N,\alpha,\beta}| v |_{B_2},
\]
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.

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.

Reviewer: Margit Lenard (Safat)

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\textit{B.-y. Guo} and \textit{L.-l. Wang}, J. Approx. Theory 128, No. 1, 1--41 (2004; Zbl 1057.41003)

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