The role of input dimension <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> is studied in approximating, in various norms, target sets of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex></formula> -variable functions using linear combinations of adjustable computational units. Results from the literature, which emphasize the number <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n$</tex></formula> of terms in the linear combination, are reformulated, and in some cases improved, with particular attention to dependence on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> . For worst-case error, upper bounds are given in the factorized form <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\xi(d)\kappa(n)$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\kappa$</tex> </formula> is nonincreasing (typically <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\kappa(n) \sim n^{-1/2}$</tex></formula> ). Target sets of functions are described for which the function <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\xi$</tex></formula> is a polynomial. Some important cases are highlighted where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\xi$</tex> </formula> decreases to zero as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d \to \infty$</tex></formula> . For target functions, extent (e.g., the size of domains in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBR}^d$</tex> </formula> where they are defined), scale (e.g., maximum norms of target functions), and smoothness (e.g., the order of square-integrable partial derivatives) may depend on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> , and the influence of such dimension-dependent parameters on model complexity is considered. Results are applied to approximation and solution of optimization problems by neural networks with perceptron and Gaussian radial computational units.