## Interpolation on sparse grids and tensor products of Nikol’skij-Besov spaces.(English)Zbl 0945.41002

In this paper, the authors investigate errors of periodic interpolation of functions of two variables in terms of smoothness properties of those functions. The smoothness is measured by tensor products of Besov and Nikolskij-Besov norms. The interpolation is undertaken on sparse periodic grids; the latter has been well studied before, but the essentially new feature of this paper are the spaces of functions on which interpolation is performed, and errors are analysed. The authors methodically define the spaces involved, and then go on to present errors of interpolation of the uniform and $$L_p$$ norms. Relations to $$n$$-widths and earlier results are discussed, and as examples, they discuss interpolation involving periodized $$B$$-splines.

### MSC:

 41A05 Interpolation in approximation theory 42A15 Trigonometric interpolation 65D05 Numerical interpolation

### Keywords:

error estimates; sparse grids; interpolation
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