  We develop an analytic approach to Boundary Conformal Field Theory (BCFT),
focussing on the two-point function of a general pair of scalar primary
operators. The resulting crossing equation can be thought of as a vector
equation in an infinite-dimensional space ${\cal V}$ of analytic functions of a
single complex variable. We argue that in a unitary theory, functions in ${\cal
V}$ satisfy a boundedness condition in the Regge limit. We identify a useful
basis for ${\cal V}$, consisting of bulk and boundary conformal blocks with
scaling dimensions which appear in OPEs of the mean field theory correlator.
Our main achievement is an explicit expression for the action of the dual basis
(the basis of liner functionals on ${\cal V}$) on an arbitrary conformal block.
The practical merit of our basis is that it trivializes the study of
perturbations around mean field theory. Our results are equivalent to a BCFT
version of the Polyakov bootstrap. Our derivation of the expressions for the
functionals relies on the identification of the Polyakov blocks with (suitably
improved) boundary and bulk Witten exchange diagrams in $AdS_{d+1}$. We also
provide another conceptual perspective on the Polyakov block expansion and the
associated functionals, by deriving a new Lorentzian OPE inversion formula for
BCFT.
