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Explore a 52-minute mathematics lecture that delves into Fourier multipliers and their applications to functions on the discrete n-torus ℤ_K^n. Learn about functional inequalities across different mathematical spaces, focusing on the challenges encountered with intermediate K values between the hypercube (K=2) and n-torus (K=∞). Discover a specific class of Fourier multipliers that maintain bounded properties independent of dimension when applied to functions with low-degree Fourier expansions on ℤ_K^n. Examine Figiel's inequalities for Rademacher projections on level-l characters and understand how transcendental number theory, including Baker's theorem, enables bounded projections onto more refined character sets. Conclude with an exploration of a dimension-free Bernstein-type discretization inequality using ℤ_K^n as the sampling set.
