  In this paper we formulate a concrete method for determining whether a system
of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method
relies on a general framework developed by Hedenmalm, Lindqvist and Seip about
20 years ago, which turns the basis question into one about the localisation of
the zeros and poles of a corresponding analytic multiplier. Our results improve
upon various criteria formulated previously, which give sufficient conditions
for invertibility of the multiplier in terms of sharp estimates on the Fourier
coefficients. Our focus is on the concrete verification of the hypotheses by
means of analytical or accurate numerical approximations. We then examine the
basis question for profiles in a neighbourhood of a non-basis family generated
by periodic jump functions. For one of these profiles, the $p$-sine functions,
we determine a threshold for positive answer to the basis question which
improves upon those found recently.
