We study a use of Gaussian kernels with a wide range of scales for nonlinear function estimation. The estimation task can then be split into two sub-tasks: (i) model selection and (ii) learning (parameter estimation) under the selected model. We propose a fully-adaptive and all-in-one scheme that jointly carries out the two sub-tasks based on the multikernel adaptive filtering framework. The task is cast as an asymptotic minimization problem of an instantaneous fidelity function penalized by two types of block l₁-norm regularizers. Those regularizers enhance the sparsity of the solution in two different block structures, leading to efficient model selection and dictionary refinement. The adaptive generalized forward-backward splitting method is derived to deal with the asymptotic minimization problem. Numerical examples show that the scheme achieves the model selection and learning simultaneously, and demonstrate its striking advantages over the multiple kernel learning (MKL) method called SimpleMKL.