We study some approximation problems by functions in the Hardy space H 2 of the upper half-plane or by their real or imaginary parts, with constraint on their real or imaginary parts on the boundary. Situations where the criterion acts on subsets of the boundary or of horizontal lines inside the half-plane are considered. Existence and uniqueness results are established, together with novel solution formulas and techniques. As a by-product, we devise a regularized inversion scheme for Poisson and conjugate Poisson integral transforms.