Two-index analytic coefficient space

Fix $R,\rho>0$ and a compact nondegenerate interval $I$ . For complex-valued coefficient functions $F_\alpha(\eta)$ , define

w_{\alpha\beta}= \frac{R^{-\alpha}\rho^{-\beta}\beta!\binom{\alpha+\beta}{\beta}} {(\alpha+1)^2(\beta+1)^2}, \qquad \|F\|_{R,\rho}=\sup_{\alpha,\beta\ge0}\sup_{\eta\in I} \frac{|\partial_\eta^\beta F_\alpha(\eta)|}{w_{\alpha\beta}}.

Here the factorial and binomial coefficient are ordinary nonnegative-integer quantities.

The two-index analytic coefficient space consists of sequences with finite norm, whose coefficient derivatives are continuous on $I$ at every order, with one-sided endpoint derivatives. The formal radial series is $F(Y,\eta)=\sum_{\alpha\ge0}F_\alpha(\eta)Y^\alpha$ .

Completeness

A norm-Cauchy sequence converges uniformly at each coefficient and derivative order. The uniform derivative-limit theorem identifies successive derivatives. Passing the uniform weighted Cauchy bound to the limit gives convergence in the original norm. Thus this is a Banach space.

Meaning of the weights

The binomial factor allows higher parameter derivatives as radial degree increases. This coupling permits radial integration to compensate for a parameter derivative. The polynomial denominator in each index also makes coefficient convolution bounded; both assertions require the estimates in the algebra bound and the radial operator bounds.