In this paper, we highlight a design of Gaussian kernels for online model selection by the multikernel adaptive filtering approach. In the typical multikernel adaptive filtering, the maximum value that each kernel function can take is one. This means that, if one employs multiple Gaussian kernels with multiple variances, the one with the largest variance would become dominant in the kernelized input vector (or matrix). This makes the autocorrelation matrix of the the kernelized input vector be ill-conditioned, causing significant deterioration in convergence speed. To avoid this ill-conditioned problem, we consider the normalization of the Gaussian kernels. Because of the normalization, the condition number of the autocorrelation matrix is improved, and hence the convergence behavior is improved considerably. As a possible alternative to the original multikernel-based online model selection approach using the Moreau-envelope approximation, we also study an adaptive extension of the generalized forward-backward splitting (GFBS) method to suppress the cost function without any approximation. Numerical examples show that the original approximate method tends to select the correct center points of the Gaussian kernels and thus outperforms the exact method.