Matching minimax prediction rates for discretely observed functional linear regression are n^{-ν/(ν+1)}+(nm)^{-ν/κ} under independent design, and those two terms plus m^{-ν}+m^{-4α} under common design.
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The Cost of Discretization in Functional Linear Regression: Minimax Rates and Adaptation
Matching minimax prediction rates for discretely observed functional linear regression are n^{-ν/(ν+1)}+(nm)^{-ν/κ} under independent design, and those two terms plus m^{-ν}+m^{-4α} under common design.