What the study found
The study found exact orders for the average Nikolskii factor of random trigonometric polynomials. For 1 ≤ p < q < ∞, the average factor is constant, while for 1 ≤ p < q = ∞, it grows like the square root of log n.
Why the authors say this matters
The authors compare their averages with worst-case bounds and show that the typical behavior is much smaller than the worst-case behavior. The study suggests this gives a sharper picture of how these polynomials behave on average.
What the researchers tested
The researchers studied the Nikolskii factor N_{p,q}(T_a), defined as the ratio of the q-norm to the p-norm of a trigonometric polynomial. They considered random trigonometric polynomials with independent Gaussian coefficients and also gave a generalization to random multivariate trigonometric polynomials.
What worked and what didn't
For 1 ≤ p < q < ∞, the average Nikolskii factor was of order n^0, meaning it stayed constant. For 1 ≤ p < q = ∞, it was of order (ln n)^{1/2}. The worst-case bounds mentioned in the abstract were larger: n^{1/p-1/q} for finite q and n^{1/p} for q = ∞.
What to keep in mind
The abstract does not describe limitations, data sources, or proof details. The results are stated for random trigonometric polynomials with independent N(0, σ^2) coefficients, and the multivariate extension is only mentioned briefly in the abstract.
Key points
- Average Nikolskii factors were determined exactly for random trigonometric polynomials.
- For finite q, the average factor is constant rather than growing with n.
- For q = infinity, the average factor grows like the square root of log n.
- The average behavior is smaller than the worst-case bounds stated in the abstract.
- The paper also extends the result to random multivariate trigonometric polynomials.
Disclosure
- Research title:
- Average Nikolskii factors are constant or logarithmic for random trigonometric polynomials
- Authors:
- Yun Ling, Jiaxin Geng, Jiansong Li, Heping Wang
- Institutions:
- Capital Normal University, Capital Normal University, Capital Normal University, Capital Normal University
- Publication date:
- 2026-04-27
- OpenAlex record:
- View
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