Schauder and Riesz Bases for Higher-Dimensional Lebesgue Spaces Inspired by ReLU Neural Networks

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1{]}^d)$, $q\in (1,\mathrm{\infty })$. For a sequence $\tau \in {\mathrm{\ell }}_1$ we consider the operator $T_{\tau }g(x)={\sum }_{k=1}^{\mathrm{\infty }}{\tau }_{k}g(kx)$ and the function system ${\mathcal{Z}}^{\tau }$ obtained by applying it to the trigonometric system. We show that $\tau \mapsto T_{\tau }$ is an isomorphism of algebras from ${\mathrm{\ell }}_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_{\tau }$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system ${\mathcal{Z}}^{\tau }$ is a Schauder basis of $L_q([0,1{]}^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2\mathrm{/}3$, independently of the dimension.