An optimal approximation of Rosenblatt sheet by multiple Wiener integrals

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Let Zα,β be the Rosenblatt sheet with the representation Zα,β(t,s)=∫ᵗ₀∫ˢ₀∫ᵗ₀∫ˢ₀Qα(t,y₁,y₂)Qβ(s,u₁,u₂)B(dy₁,du₁)B(dy₂,du₂) where B is a Brownian sheet, 12<α,β<1, Qα and Qβ are the given kernel. In this paper, we contruct multiple Wiener integrals of the form align* ∫ᵗ₀∫ˢ₀∫ᵗ₀∫ˢ₀&[k₁(y₁,y₂)-12α(u₁,u₂)-12β+k₂(y₁ y₂)12α(y₁ y₂)-12α|y₁-y₂|α⁻¹ &·(u₁ u₂)12β(u₁ u₂)-12β|u₁-u₂|β⁻¹]B(dy₁,du₁)B(dy₂,du₂), k₁,k₂≥0, align* and obtain an optimal approximation of Zα,β(t,s).