## 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 $\left\| Q_N v-v \right\| _{B_1} \leq d_{N,\alpha,\beta}| v |_{B_2},$ 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.

### MSC:

 41A10 Approximation by polynomials 41A25 Rate of convergence, degree of approximation
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### References:

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