Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes

where f is an unknown function, n > 1, σ > 0 is known and W is a standard Brownian sheet in [0, 1]d. We wish to estimate the function f given a realization y = { Yt, t ∈ [0, 1]d } . This is known as the Gaussian white noise problem and has been studied in several papers starting with Ibragimov and Hasminskii (1981). We suppose that f belongs to a d-dimensional anisotropic Hölder class Σ(β̃, L) for β = (β1, . . . , βd) ∈ (0, 1]d and L = (L1, . . . , Ld) such that 0 < Li < ∞. This class is defined by :

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