The functional linear regression model is a common tool to determine the
relationship between a scalar outcome and a functional predictor seen as a
function of time. This paper focuses on the Bayesian estimation of the support
of the coefficient function. To this aim we propose a parsimonious and adaptive
decomposition of the coefficient function as a step function, and a model
including a prior distribution that we name Bayesian functional Linear
regression with Sparse Step functions (Bliss). The aim of the method is to
recover areas of time which influences the most the outcome. A Bayes estimator
of the support is built with a specific loss function, as well as two Bayes
estimators of the coefficient function, a first one which is smooth and a
second one which is a step function. The performance of the proposed
methodology is analysed on various synthetic datasets and is illustrated on a
black P\'erigord truffle dataset to study the influence of rainfall on the
production
