Non-separable splines and numerical computation of evolution equations by the Galerkin methods (Q953381)

The authors define a non-separable spline to be a linear combination of tensor-product cardinal splines \(s\). They construct non-separable splines satisfying the following Strang-Fix condition (SF) and moment condition (M) up to order \(r\), \[ \begin{cases} \hat{s}(0,0)=1, \\ \frac{\partial^{\alpha_1+\alpha_2}}{\partial\xi_1^{\alpha_1}\partial\xi_2^{\alpha_2}} \hat{s}(\xi_1,\xi_2)\Big|_{\substack{ \xi_2=2\pi k_2\\ \xi_1=2 \pi k_1}}=0, &\forall (k_1,k_2) \in {\mathbb Z}^2\backslash(0,0), \;0 \leq \alpha_1+\alpha_2 \leq r, \end{cases} \tag{SF} \] where \(\hat{s}\) denotes the Fourier transform of \(s\) and \[ \frac{\partial^{\alpha_1+\alpha_2}}{\partial\xi_1^{\alpha_1}\partial\xi_2^{\alpha_2}} \hat{s}(\xi_1,\xi_2)\Big|_{\xi_1 \xi_2=0}= \delta_{\alpha_1+\alpha_2,0}, \quad 0 \leq \alpha_1+\alpha_2 \leq r. \tag{M} \] Let \(p \in [1,\infty]\). For a function \(f\) in the Sobolev space \(W^{N,p}(\mathbb R^2)\) a sampling approximation \(S(f)\) is defined by \[ S(f)(x,y):= \sum_{k,l=-\infty}^{\infty} f(2^{-j}k,2^{-j}l)s(2^jx-k,2^jy-l) \quad\text{for } j=1,2,\dots \text{ and } (x,y) \in \mathbb R^2. \] Assuming (SF) and (M) to hold for some \(r \geq N\) the error can be estimated by \[ \|S(f)-f\|_{L^p(\mathbb R^2)} \leq 2^{-jN}\sum_{n=0}^{N} \bigg\|\frac{\partial^N f}{\partial x^n\partial y^{N-n}}\bigg\|_{L^p(\mathbb R^2)}. \] The coefficients of various non-separable splines are calculated. Numerical experiments are reported for the Burgers and the Kadomtsev-Petviashvili equation, respectively, in one space dimension, where the space variable is discretized using the Galerkin method with non-separable splines as basis functions and the time integration is done with a fourth order Runge-Kutta method. A comparision is made with the results obtained by using Coifman scaling functions as basis functions. Reviewer's remark: Apparently, there is a condition for \(f \in W^{N,p}(\mathbb R^2)\) missing to allow pointwise evaluation when forming \(S(f)\).