GoKawiilI’ve just uploaded to the arXiv my paper “Local Bernstein theory, and lower bounds for Lebesgue constants“. This paper was initially motivated by a problem of Erdős} on Lagrange interpolation, but in the course of solving that problem, I ended up modifying some very classical arguments of Bernstein and his contemporaries (Boas, Duffin, Schaeffer, Riesz, etc.) to obtain “local” versions of these classical “Bernstein-type inequalities” that may be of independent interest.
Bernstein proved many estimates concerning the derivatives of polynomials, trigonometric polynomials, and entire functions of exponential type, but perhaps his most famous inequality in this direction is:
Lemma 1 (Bernstein’s inequality for trigonometric polynomials) Let be a trigonometric polynomial of degree at most , with for all . Then for all .
Similar inequalities concerning norms of derivatives of Littlewood-Paley components of functions are now ubiquitious in the modern theory of nonlinear dispersive PDE (where they are also called Bernstein estimates), but this will not be the focus of this current post.
A trigonometric polynomial of degree is of exponential type in the sense that for complex . Bernstein in fact proved a more general result:
Lemma 2 (Bernstein’s inequality for functions of exponential type) Let be an entire function of exponential type at most , with for all . Then for all .
There are several proofs of this lemma – see for instance this survey of Queffélec and Zarouf. In the case that is real-valued on , there is a nice proof by Duffin and Schaeffer, which we sketch as follows. Suppose we normalize , and adjust by a suitable damping factor so that actually decays slower than as . Then, for any and , one can use Rouche’s theorem to show that the function has the same number of zeroes as in a suitable large rectangle; but on the other hand one can use the intermediate value theorem to show that has at least as many zeroes than in the same rectangle. Among other things, this prevents double zeroes from occuring, which turns out to give the desired claim after some routine calculations (in fact one obtains the stronger bound for all real ).
The first main result of the paper is to obtain localized versions of Lemma 2 (as well as some related estimates). Roughly speaking, these estimates assert that if is holomorphic on a wide thin rectangle passing through the real axis, is bounded by on the intersection of the real axis with this rectangle, and is “locally of exponential type” in the sense that it is bounded by on the upper and lower edges of this rectangle (and obeys some very mild growth conditions on the remaining sides of this rectangle), then can be bounded by plus small errors on the real line, with some additional estimates away from the real line also available. The proof proceeds by a modification of the Duffin–Schaeffer argument, together with the two-constant theorem of Nevanlinna (and some standard estimates of harmonic measures on rectangles) to deal with the effect of the localization. (As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.)
Once one localizes this “Bernstein theory”, it becomes suitable for the analysis of (real-rooted, monic) polynomials of a high degree , which are not bounded globally on (and grow polynomially rather than exponentially at infinity), but which can exhibit “local exponential type” behavior on various intervals, particularly in regions where the logarithmic potential
behaves like a smooth function (hereis the empirical measure of the rootsof). A key example is the (monic) Chebyshev polynomials, which locally behave like sinusoids on the interval(and are locally of exponential type above and below this interval):
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