Some mathematical limitations of the general-purpose analog computer
The general-purpose analog computer (GPAC) is the mathematical abstraction of several actual computing machines, of which the Bush differential analyzer is the prototype. A principal result about GPAC's (the Shannon-Pour-El-Lipshitz-Rubel theorem) essentially says that their outputs are exactly the differentially algebraic (DA) functions, i.e. those functions y of a real variable that satisfy some algebraic differential equation \(P(x,\quad y(x),\quad y'(x),...,y^{(n)}(x))=0,\) where P is a non-trivial polynomial in its \(n+2\) variables. Thus, to show that a given function is transcendentally transcendental (TT), which means that it is not DA, is to show that it cannot be computed by a GPAC. This paper exhibits several TT functions that arise from DA data. For example, a DA cosine series \(\sum a_ n \cos nx\) is exhibited for which the associated sine series \(\sum a_ n \sin nx\) is TT. A DA function \(u_ 0(e^{i\theta})\) is explicitly produced such that if u(z) is the solution of the Dirichlet problem for Laplace's equation on the unit disc, with boundary values \(u_ 0\), then u(z) is TT on some line segment inside the disc. This is to be interpreted as saying that this Dirichlet problem cannot be solved on GPAC. The main approach is to produce a DA two-sided Fourier series \(\sum^{\infty}_{n=-\infty}A_ ne^{in\theta}\) for which the corresponding one-sided series \(\sum^{\infty}_{n=0}A_ ne^{in\theta}\) is TT. In this paper, this is done by transferring the problem to a problem about the Riesz-Herglotz factorization of a bounded analytic function in the unit disc. Such a DA function is written down whose inner factor is TT. This is proved by invoking a theorem of S. Bank that \(\Gamma\) (z)\(\Gamma\) (az)\(\Gamma\) (bz) is TT if ab\(\neq 0\), where \(\Gamma\) is Euler's gamma function. In a subsequent paper [``An unsolvable cousin problem'', Proc. Am. Math. Soc. 104, 410-412 (1988)], the author used a simpler construction of \(\sum A_ ne^{in\theta}\) with the desired properties, and where the \(| A_ n|\) are small enough to make \(\sum A_ nz^ n\) analytic in a neighborhood of the unit circle \(\{| z| =1\}\), thus answering a problem left open in the present paper.
scientific article; zbMATH DE number 4112087
scientific article