$L^{p}$-computability in recursive analysis

Abstract:

${L^p}$-computability is defined in terms of effective approximation; e.g. a function $f \in {L^p}[0,1]$ is called ${L^P}$-computable if $f$ is the effective limit in ${L^p}$-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 $f$ is ${L^p}$-computable if and only if (a) the sequence of Fourier coefficients of $f$ is computable, and (b) the ${L^p}$-norm of $f$ is a computable real. We show that this fails for $p = 1$.