What the study found
The study found that average Nikolskii factors for random trigonometric polynomials with independent Gaussian coefficients have exact orders that are smaller than worst-case bounds. For 1 ≤ p < q < ∞, the average factor is constant, and for 1 ≤ p < q = ∞, it grows like the square root of log n.
Why the authors say this matters
The authors present these average-case orders as a comparison with known worst-case bounds, which are much larger. They also state that the result generalizes to random multivariate trigonometric polynomials.
What the researchers tested
The researchers studied the Nikolskii factor, defined as the ratio of the q-norm to the p-norm of a trigonometric polynomial. They considered random trigonometric polynomials with independent N(0, σ²) coefficients, and they extended the analysis 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 abstract contrasts these with worst-case bounds of order n^{1/p-1/q} and n^{1/p}, respectively.
What to keep in mind
The summary only states results for random trigonometric polynomials with independent Gaussian coefficients. It does not describe further limitations, assumptions beyond the coefficient model, or detailed proof methods.
Key points
- Average Nikolskii factors were found to be constant for 1 ≤ p < q < ∞.
- When q = ∞, the average Nikolskii factor grew like (ln n)^{1/2}.
- These average-case orders are smaller than the stated worst-case bounds.
- The paper also extends the result to random multivariate trigonometric polynomials.
- The coefficients were independent Gaussian random variables with variance σ².
Disclosure
- Research title:
- Random trigonometric polynomials have smaller average Nikolskii factors
- 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
Get the weekly research newsletter
Stay current with peer-reviewed research without reading academic papers — one filtered digest, every Friday.