Title: Approximating any Function via Coreset for Radial Basis Functions: Towards Provable Data Subset Selection For Efficient Neural Networks training. Abstract: Radial basis function neural networks (\emph{RBFNN}) are notoriously known for their capability to approximate any continuous function on a closed bounded set with arbitrary precision given enough hidden neurons. Coreset is a small weighted subset of an input set of items, that provably approximates their loss function for a given set of queries (models, classifiers, etc.). In this paper, we suggest the first coreset construction algorithm for \emph{RBFNNs}, i.e., a small weighted subset which approximates the loss of the input data on any radial basis function network and thus approximates any function defined by an \emph{RBFNN} on the big input data. This is done by constructing coresets for radial basis and Laplacian loss functions. We use our coreset to suggest a provable data subset selection algorithm for training deep neural networks, since our coreset approximates every function, it should approximate the gradient of each weight in a neural network as it is defined as a function on the input. Experimental results on function approximation and dataset subset selection on popular network architectures and data sets are presented, demonstrating the efficacy and accuracy of our coreset construction.