Approximation of Bivariate Functions via Smooth Extensions

Abstract EN

For a smooth bivariate function defined on a general domain with arbitrary shape, it isdifficult to do Fourier approximation or wavelet approximation.

In order to solve these problems, in this paper,we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodicfunction in the whole space or to a smooth, compactly supported function in the whole space.

These smoothextensions have simple and clear representations which are determined by this bivariate function and somepolynomials.

After that, we expand the smooth, periodic function into a Fourier series or a periodic waveletseries or we expand the smooth, compactly supported function into a wavelet series.

Since our extensions aresmooth, the obtained Fourier coefficients or wavelet coefficients decay very fast.

Since our extension tools arepolynomials, the moment theorem shows that a lot of wavelet coefficients vanish.

From this, with the help ofwell-known approximation theorems, using our extension methods, the Fourier approximation and the waveletapproximation of the bivariate function on the general domain with small error are obtained.