The Euclidean version of the Yang-Mills theory is studied in four dimensions. The field is expressed non-linearly in terms of the basic variables. The field is developed inductively, adding one excitation at a time. A given excitation is added into the ``background field'' of the excitations already added, the background field expressed in a radially axial gauge about the point where the excitation is centered. The linearization of the resultant expression for the field is an expansion A_\mu(x) \ \cong \ \sum_\alpha \; c_\alpha \; \psi_\mu^\alpha(x) where \psi^\alpha_\mu(x) is a divergence-free wavelet and c_\alpha is the associated basic variable, a Lie Algebra element of the gauge group. One is working in a particular gauge, regularization is simply cutoff regularization realized by omitting wavelet excitations below a certain length scale. We will prove in a later paper that only the usual gauge-invariant counterterms are required to renormalize perturbation theory. Using related ideas, but essentially independent of the rest of paper, we find an expression for the determinant of a gauged boson or fermion field in a fixed ``small'' external gauge field. This determinant is expressed in terms of explicitly gauge invariant quantities, and again may be regularized by a cutoff regularization. We leave to later work relating these regularizations to the usual dimensional regularization.
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