A variational approach to the Dirichlet-Gabor wavelet-distributed approximating functional (Q1580227)

The authors show that an interpolating Dirichlet-Gabor wavelet distributed approximating functional (DAF) can be obtained by the same variational principle they used to derive non-interpolating DAF. An interpolative formula is derived for periodic functions on an interval of length \(L\) and by taking the limits \(N\rightarrow\infty\) and \(L\rightarrow\infty\) such that \(\Delta=L/N\) is kept fixed, the approximation assumes the sine-DAF form \[ f_{\text{DAF}}(x)={\Delta \over 2\pi} \sum_{k=-\infty}^\infty \omega(x_k-x){\sin(2\pi(x_k-x)/\Delta) \over x_k-x} f(x_k), \] where \(f\) is known at the points \(x_k\) and the weight function \(\omega(x)=\exp(-x^2/2\sigma^2)\) is a Gaussian with parameter \(\sigma\) with units of length. The references list 14 joint papers on DAF which have been co-authored by one or more of the authors and three known books on wavelets.