  The shape of the vector and scalar K_{l3} form factors is investigated by
exploiting analyticity and unitarity in a model-independent formalism. The
method uses as input dispersion relations for certain correlators computed in
perturbative QCD in the deep Euclidean region, soft-meson theorems, and
experimental information on the phase and modulus of the form factors along the
elastic part of the unitarity cut. We derive constraints on the coefficients of
the parameterizations valid in the semileptonic range and on the truncation
error. The method also predicts low-energy domains in the complex $t$-plane
where zeros of the form factors are excluded. The results are useful for K_{l3}
data analyses and provide theoretical underpinning for recent phenomenological
dispersive representations for the form factors.
