Affine synthesis and coefficient norms for Lebesgue, Hardy and Sobolev spaces

TL;DR

This paper establishes norm equivalences for functions in Lebesgue, Hardy, and Sobolev spaces via affine synthesis operators, connecting function norms with minimal coefficient norms in affine systems.

Contribution

It proves the affine synthesis operator maps sequence spaces onto function spaces and establishes norm equivalences for Lebesgue, Hardy, and Sobolev spaces.

Findings

Affine synthesis operator maps $oldsymbol{ ext{ell}}^1(oldsymbol{ ext{ell}}^p)$ onto $L^p(oldsymbol{ ext{Rd}})$.

Synthesis operator maps discrete Hardy space onto $H^1(oldsymbol{ ext{Rd}})$.

Coefficient norm equivalences for Sobolev spaces using difference operators.

Abstract

The affine synthesis operator is shown to map the mixed-norm sequence space $\ell^1(\ell^p)$ surjectively onto $L^p(\Rd), 1 \leq p < \infty$, assuming the Fourier transform of the synthesizer does not vanish at the origin and the synthesizer has some decay near infinity. Hence the standard norm on $f \in L^p(\Rd)$ is equivalent to the minimal coefficient norm of realizations of $f$ in terms of the affine system. We further show the synthesis operator maps a discrete Hardy space onto $H^1(\Rd)$, which yields a norm equivalence for Hardy space involving convolution with a discrete Riesz kernel sequence. Coefficient norm equivalences are established also for Sobolev spaces, by applying difference operators to the coefficient sequences.