L p -Computability in Recursive Analysis

Marian Boykan Pour-El, Ian Richards
1984 Proceedings of the American Mathematical Society  
Lp-computability is defined in terms of effective approximation; e.g. a function / G Lp[0,1] is called Lp-computable if / is the effective limit in Lp-norm of a computable sequence of polynomials. Other families of functions can replace the polynomials; see below. In this paper we investigate conditions which are not based on approximation. For p > 1, we show that / is Lp-computable if and only if (a) the sequence of Fourier coefficients of / is computable, and (b) the Lp-norm of / is a
more » ... m of / is a computable real. We show that this fails for p = 1. In [9] the authors gave a definition of Lp-computability for functions on [0,1] (where 1 < p < oo and p is a computable real). Various equivalent formulations were given, based on Weierstrass approximation (polynomials), Fourier series (trigonometric polynomials) and integration theory (step functions). Thus a function / E LP[0,1] is called IP-computable if / is the effective limit in Lp-norm of any of the following: (i) a computable sequence of polynomials; (ii) a computable sequence of trigonometric polynomials; (iii) a computable sequence of step functions (i.e. a sequence in which the jump points and heights of the steps are computable). Each of these definitions is based on a process of generation of the function / from simpler functions. From the point of view of an analyst, it is useful to have a definition based on the properties of the function / itself. In this paper we prove: For 1 < p < oo, a function f E Lp[0,1] is IP-computable if and only if: (a) the sequence of Fourier coefficients of f is computable, and (b) the IP-norm of f is computable.
doi:10.2307/2045162 fatcat:ckazkps7wzgiljbw3cc5bwix3q