Abstract. This paper presents a theoretical study of stellar coronagraphy with apodized entrance apertures. The study is restricted to a perfect telescope operating in space, and a monochromatic on-axis unresolved star. It is shown that linear prolate functions are the optimal apodizers for rectangular apertures in stellar coronagraphy. With the phase mask technique (Roddier & Roddier 1997), prolate functions can produce a total extinction of the star light. For Lyot's coronagraphy, the extinction is not complete, but prolate apodizations lead to an optimal star residual intensity with surprising interesting properties: the residual star light and the planet enjoy the same apodized intensity pattern (but different dynamic) with the optimal light concentration. With this technique, very high rejection rates can be obtained for Lyot's coronagraphy, with smaller mask sizes.

This paper is concerned with the theoretical properties of high-contrast coronagraphic images in the context of exoplanet searches. We derive and analyze the statistical properties of the residual starlight in coronagraphic images and describe the effect of a coronagraph on the speckle and photon noise. Current observations with coronagraphic instruments have shown that the main limitations to high-contrast imaging are due to residual quasi-static speckles. We tackle this problem in this paper and propose a generalization of our statistical model to include the description of static, quasi-static, and fast residual atmospheric speckles. The results provide insight into the effects on the dynamic range of wave front control, coronagraphy, active speckle reduction, and differential speckle calibration. The study is focused on ground-based imaging with extreme adaptive optics, but the approach is general enough to be applicable to space, with different parameters.

SPHERE (which stands for Spectro-Polarimetric High-contrast Exoplanet REsearch) is a second-generation Very Large Telescope (VLT) instrument dedicated to high-contrast direct imaging of exoplanets which firstlight is scheduled for 2011. Within this complex instrument one of the central components is the apodized Lyot coronagraph (ALC). The present paper reports on the most interesting aspects and results of the whole numerical study made during the design of the ALC for SPHERE/VLT. The method followed for this study is purely numerical, but with an end-to-end approach which is largely fed by a number of instrumental feedbacks. The results obtained and presented in this paper firstly permit to finalize the optical design before Exp Astron (2011) 30:39-58 laboratory performance testing of the ALC being built for SPHERE/VLT (see paper II "Laboratory tests and performances"), but will also hopefully help conceiving future other instruments alike, for example within the very promising extremely large telescope perspective.

We consider distributed multitask learning problems over a network of agents where each agent is interested in estimating its own parameter vector, also called task, and where the tasks at neighboring agents are related according to a set of linear equality constraints. Each agent possesses its own convex cost function of its parameter vector and a set of linear equality constraints involving its own parameter vector and the parameter vectors of its neighboring agents. We propose an adaptive stochastic algorithm based on the projection gradient method and diffusion strategies in order to allow the network to optimize the individual costs subject to all constraints. Although the derivation is carried out for linear equality constraints, the technique can be applied to other forms of convex constraints. We conduct a detailed mean-square-error analysis of the proposed algorithm and derive closed-form expressions to predict its learning behavior. We provide simulations to illustrate the theoretical findings. Finally, the algorithm is employed for solving two problems in a distributed manner: a minimum-cost flow problem over a network and a space-time varying field reconstruction problem.subject to ∈Ip D p w + b p = 0, p = 1, . . . , P. (1b) Each agent k in the network seeks to estimate its own M k × 1 parameter vector w k , and has knowledge of its cost function J k (·) and the set of linear equality constraints that agent k is involved in. Each constraint is indexed by p, and defined by the L p × M matrices D p , the L p × 1 vector b p , and the set