Dimension independent Bernstein–Markov inequalities
in Gauss space
We obtain the following dimension independent Bernstein–Markov inequality in Gauss space: for each there exists a constant such that for any and all polynomials on we have
where is the standard Gaussian measure on . We also show that under some mild growth assumptions on any function with we have
where is the generator of the Ornstein–Uhlenbeck semigroup and
2010 Mathematics Subject Classification. Primary: 41A17; Secondary: 41A63, 42C10, 28C20.
Key words. Gaussian measure, Bernstein-Markov inequality, Freud’s inequality, weighted approximation.
Let be the standard Gaussian measure on , given by
where is the Euclidean length of . Here and throughout, we will denote by be the density of the Gaussian measure with respect to the Lebesgue measure on . For , define to be the space of those measurable functions on for which
As usual, is defined by the condition . For convenience of notation, we will abbreviate as .
1.1. Freud’s inequality in high dimensions
In his seminal paper , Freud obtained the following weighted Bernstein–Markov type inequality on the real line.
Theorem 1 (Freud’s inequality, ).
There exists a universal constant such that for any and all polynomials on , we have
After making a change of variables in (1), Freud’s inequality can be rewritten in terms of norms as
After proving Theorem 1, Freud  extended his Gaussian estimates (1) to more general weights on the real line, nowadays known as Freud weights, where the function satisfies certain growth and convexity assumptions. In this case, the bound in (1) is replaced by a certain quantity which depends on the so-called Mhaskar–Rakhmanov–Saff numbers of the weight . Since the works [5, 6] of Freud, several different proofs of such one-dimensional weighted Bernstein–Markov inequalities have been found (see, e.g., [7, 22, 14, 15, 16, 13]), in part due to important implications of such estimates in approximation theory (see, e.g., [5, Theorem 2] and [6, Theorems 4.1 and 5.1]). We refer the reader to the beautiful survey  of Lubinsky for a detailed exposition of results on this subject.
From the point of view of high dimensional probability, it is natural to ask whether inequality (2) admits a dimension independent extension for the Gaussian measure on . Throughout the ensuing discussion, for a smooth function and , we will denote
There exists a universal constant such that for any , and all polynomials on we have
For finite values of , the following question naturally arises, in analogy to (2).
Question 3 (Bernstein–Markov inequality in Gauss space).
Is it true that for each there exists a constant such that for any integer , and all polynomials on the the dimension independent Gaussian Bernstein–Markov inequality
There exists a universal constant such that for any , any and all polynomials on , we have
The main result of the present paper is that the linear bound on in (6) can be improved.
For each there exists a constant such that for any , and all polynomials on , we have
Notice that for each we have , therefore (7) is worse than (5) but improves upon (6). Also, notice that for , inequality (7) recovers (5). Our proof of Theorem 6, relies on a similar Bernstein–Markov type inequality for the generator of the Ornstein–Uhlenbeck semigroup (see Theorem 8 below) and Meyer’s dimension-free Riesz transform inequalities in Gauss space from .
1.2. Reverse Bernstein–Markov inequality in Gauss space
Our initial motivation to study Question 3 comes from a dual question that Mendel and Naor [19, Remark 5.5 (2)] asked on the Hamming cube. A positive answer to their question would, by standard considerations, imply its continuous counterpart in Gauss space, namely a reverse Bernstein–Markov inequality. To state the latter question precisely, let be the probabilists’ Hermite polynomial of degree on , i.e.,
For and a multiindex , where , we consider the multivariate Hermite polynomial on , given by
The family forms the orthonormal system on . Denote by and let be the generator of the Ornstein–Uhlenbeck semigroup. Then, one has
for every multiindex . The operator should be understood as the Laplacian in Gauss space. Now consider any polynomial on which lives on frequencies greater than , i.e., of the form
Question 7 (Mendel–Naor, ).
Is it true that for each there exists a constant such that for any , any , and all polynomials of the form (10) on , living on frequencies greater than , we have
In our upcoming manuscript , we show that for every there exists some such that for all polynomials which live on frequencies , i.e. are of the form
For small values of , (12) improves upon previously known bounds in Question 7 which follow from works of Meyer [19, Lemma 5.4] and Mendel and Naor [19, Theorem 5.10] on the Hamming cube for this smaller subclass of polynomials. In particular, when , (12) positively answers a special case of Question 7. We refer to  for further results on reverse Bernstein–Markov inequalities along with extensions for vector-valued functions on the Hamming cube.
1.3. Bernstein–Markov inequality with respect to
The best result that we could obtain in this direction is the following theorem.
For any integer , any , and any polynomial on , we have
1.4. General function estimates
Our techniques for proving Theorem 8 allow us to replace -th powers in norms in (14) by an arbitrary convex increasing function in the spirit of the works [1, 2] of Arestov. We recall that Arestov’s theorem asserts that if is nondecreasing convex function on , and
is a trigonometric polynomial of degree at most , then for every , the sharp inequality
holds true. In fact, Arestov’s result holds true for a somewhat larger class of functions which contains for every (instead of just ), thus implying the usual Bernstein–Markov inequality for trigonometric polynomials.
One straightforward way to obtain an analog of (15) in Gauss space is to invoke the rotational invariance of the Gaussian measure. Indeed, we will shortly show the following estimates.
Let be an increasing convex function. For any , and all polynomials on we have
Our main result of this section is that under mild assumptions on , one can further improve (17).
For any and all polynomials on , we have
for any function with , such that for every
for each fixed and some . Here
The rest of the paper is structured as follows. In Section 2, we present the proof of Theorem 9 and its consequence, Proposition 5. In Section 3, we prove our main result, Theorem 10 from which we also deduce Theorem 8. Finally, Section 4 contains the deduction of Theorem 6 from Theorem 8 and Section 5 contains the proof of Proposition 2.
We would like to thank Volodymyr Andriyevskyy for discussions that helped us to simplify the proof of Lemma 15.
2. Proof of Theorem 9
Proof of Theorem 9. Let
be two independent multivariate standard Gaussian random variables on . Take any polynomial on of degree , and consider the trigonometric polynomial
Clearly . It follows from Arestov’s inequality (15) that
Since , and the random variables
are also independent multivariate standard Gaussians, we see that after taking the expectation of (20) with respect to , we obtain
where denotes inner product in . Therefore following the same steps as before and using Arestov’s inequality (15), we obtain
Finally, it remains to use convexity of the map and Jensen’s inequality to get
which completes the proof of (17).
Deriving Proposition 5 is now straightforward.
3. Proof of Theorem 10
3.1. Step 1. A general complex hypercontractivity
Any polynomial on admits a representation of the form
for some coefficients . Next, given , we define the action of the second quantization operator (or Mehler transform) on as
Clearly , and .
In what follows we will be working with a real-valued function (and sometimes we will further require ) such that
for some constants and every . These assumptions are sufficient to avoid integrability issues.
Fix , and let be a real-valued function such that . Assume that
Then, for all , and for all polynomials on we have
Denote the scaled Gaussian measure on of variance by
Take any polynomial . We will denote partial derivatives by lower indices, for example
Fix a complex number satisfying (25), and consider the map
First notice that if the map is increasing then (26) follows. Indeed, consider any polynomial on of the form and define . Then,
and, similarly, also
Therefore, (26) can be rewritten as .
Indeed by making change of variables we obtain . Therefore
Notice that if we denote then (29) simply means that . The latter follows from integration by parts. Therefore, we have
To compute the first term, one differentiation gives
which implies that
For the second term, we have
Thus we get that where
where is the -th partial derivative of and we denote
and similarly means that we first differentiate the polynomial twice in the -th coordinate and then we apply the flow (27). Similarly, we have
Next, further abusing the notation, we will denote . We have
Let , be such that , and satisfies (24) for each fixed . Take any , such that the inequality
holds for all , and all . Then for all polynomials on