ON DISTRIBUTIONS OF TRIGONOMETRIC POLYNOMIALS IN GAUSSIAN RANDOM VARIABLES Текст научной статьи по специальности «Математика»

% 1И..1..й11?

Серия «Математика»

2021. Т. 37. С. 77—92

Онлайн-доступ к журналу: http://mathizv.isu.ru

УДК 519.2

MSC 60E05, 60E015, 28C20, 60F99

DOI https://doi.org/10.26516/1997-7670.2021.37.77

On Distributions of Trigonometric Polynomials in Gaussian Random Variables *

G. I. Zelenov1'2

1 Lomonosov Moscow State University, Moscow, Russian Federation

2 Higher School of Economics, Moscow, Russian Federation

Abstract. We prove new results about the inclusion of distributions of trigonometric polynomials in Gaussian random variables to Nikolskii-Besov classes. In addition, we estimate the total variance distances between distributions of trigonometric polynomials via the L9-distances between the polynomials themselves.

Keywords: Nikolskii—Besov class, Gaussian measure, distribution of a trigonometric polynomial.

We study the images of the standard Gaussian measure under trigonometric polynomials. We prove an estimate for the total variation distance between such images in terms of the Lq-distance between the polynomials themselves. Our result is a generalization of the result obtained in [9]. We also discuss the densities of such images and their properties in terms of fractional Sobolev spaces. It was proved in [16] that for any non-const trigonometric polynomial f the image measure o f-1 has a density from the Nikolski-Besov class Ba. However, the proof in [16] had some gaps. Here we explain how to correct the reasoning in [16], moreover, we prove that jn o f-1 has a density from the Nikolski-Besov class Ba with 5 greater

1. Introduction

* This research is supported by the Russian Science Foundation Grant 17-11-01058 (at Lomonosov Moscow State University).

than a obtained in [16], which is a stronger result. Concerning distributions of algebraic polynomials, see [3-7; 10; 11; 14; 15].

2. Measures and mappings

The standard Gaussian measure jn on Rra is a probability measure with density pln (x) with respect to Lebesgue measure on Rra, where

1 n

pln (x) = e-^/2, (x,y) = (2.1)

Let f:X ^ R be a measurable function on a measure space (x,v) and V o f-1 the image of v under f defined by

V o f-l(B) = f-l(B)) where B c R is a Borel set.

In terms of probability theory, if f is a random variable and v is a probability measure, then v o f-1 is the distribution of f. In this paper, we study measures v = In o f-l, v = 7„ o g-1. The total variation distance drv between v and v on the R can be defined as follows:

dTv (v,v) := sup - v), p GC6°°(Rl), ||p|U < ^.

For v = ln o /-l, v = 7„ o g-1 the change of variables rule implies that

dTv(v, v):=supj^ (p of-p o g)d~fn, p G C6°°(Rl), ||p||^ < ^.

3. Trigonometric polynomials

For a function on Rra we use the notation dkf := f-

Definition 1. A function f is a trigonometric polynomial of order d if

d

f(x) = ao + ^2(ак • cos(vk,x) + bk sin(vk, x)), (3.1)

k= 1

where ai, h £ R, each = (Vkt1, - - -, Vk,n) £ Zn is a non-zero vector with |Vk| = |Vki \ + - - - + |Vk,n\ < d. The set of all trigonometric polynomials of order d on Rra will be denoted by T(d,n).

Remark 1. The sum in (3.1) has only d terms. This fact has many implications. For example, T(d, n) is a linear space only when n = 1. Each f G T(d, n) has at most d nonzero terms in (3.1). If n = 1, then there are exactly d different non-zero elements Vk > 0. We can ignore Vk < 0 in (3.1), as cos is even and sin is odd. Thus (3.1) provides the decomposition of f G T(d, 1) with respect to a basis consisting of 2d + 1 functions. For n > 1, there are more than d possible non-zero vectors Vk even with all Vk,i, ■ ■ ■, Vk,n > 0. So one can take f,g G T(d, n) such that f + g G T(d,n).

Any f G T(d, n) can be represented by using complex exponents. Proposition 1. Every f G T(d,n) has a representation of the form:

In this representation of f, the coefficients Ck and dk belong to C, while the real number a0 and the vectors Vk are the same as in formula (3.1).

From formula (3.2) one can easily derive the following two results.

Corollary 1. Let f G T(d, n) and a1,...,an G R. Then the functions Fk defined by the formulas

Fi(t) = f(t,a2,a3,...,an), F2(t) = f(ai,t,a3,...,an),■■■ Fn (t) = f(ai, a2, aa,^^ t)

belong to the set T(d, 1) with respect to the variable t G R.

Corollary 2. Let f G T(d, 1) be non-constant. Then for any a G R the equation f(t) = 0 has at most 2d solutions t G [a, a + 2^].

We now discuss some properties of functions from T(d, 1) related to the Gaussian measure jn (see (2.1)). Formula (3.1) guarantees that all functions from the class T(d,n) are bounded, thus, for any p G [1, œ) we have

We will use y ■ ||p to denote the norm y ■ \\Lv(7n) of the Lp(7„)-space.

Proposition 2. Let f G T(d,n). Then 1) All partial derivatives dif also belong to T( d,n). 2) There is a number C (d) suchthat \\dif ||2 <C (d)||/||2. 3) There is a number C(d) such that Wdif + dig||2 < C(d)||/ + gW2 for all f,g G T(d, n).

Proof. The fact that dif G T(d,n) follows directly from (3.1).

The definition of T(d,n) also implies that there are 1 ■ ((2d + 1)n — 1) distinct non-zero vectors Vk satisfying (3.2) and (3.1). Thus, the definition

(3.2)

T(d,n) CL™(ln) cLP(ln).

of T(d, n) implies that T(d,n) is a subset of a linear space of dimension (2d+1)ra — 1. The function \\f\\2 is a norm on this linear space and \\f \\2д = \\dif \\2 is a seminorm on the same space. We deal with a finite-dimensional linear space, hence there is a number С(d, n) such that

\\дгf\\2 = \\/\\2A <С(d,n)\\/\\2, i = 1,...,n. (3.3)

\\дгf + дгд\\2 = \\f + ghA <С(d,n)\\f + g\\2, i = 1,...,n. (3.4)

To complete the proof, we need to replace С(d, n) with some С(d) independent of n. Any vk = (vkt\,.. ■, Vk,n) in Definition 1 has at most d nonzero coordinates vk,j. Hence, any term in (3.1) and (3.2) depends on at most d variables Xj. So f in (3.1) depends on at most d2 variables, because the sum in (3.1) has at most d nonzero terms. This enables us to consider only d2 variables when dealing with one f e T(d,n), and 2d2 variables when dealing with two functions f,g e T(d,n). Hence in (3.3) one has С(d, n) < С(d, d2) and in (3.4) one has С(d, n) < С(d, 2d2). □

Proposition 2 implies that not only T(d, n) e Lp(^n), but also T(d,n) is contained in the Sobolev space Wp,1(^n) of functions <p e Lp(^n) such that their partial derivatives dip belong to Lp(jn). The proposition 2 and the definition 1 imply that T(d,n) С Сь°°(Кга). Let F,G e Сь°°(Кга). From formula (2.1) for jn, we can easily derive the integration by parts formula

f dkF ■Gdln = —f (F ■ dkG — xk -F ■G)dln. (3.5)

jRn JR"

As T(d, n) С Сь°°(Кга), the formula (3.5) holds true for F,G e T(d,n). The following theorem is proved in [12].

Theorem 1 (Turan's Lemma). Let d e N and let I С R be an interval. Then there exists a number A0 such that for each f e T(d, 1) and each set E С I with Lebesgue measure y(E) > 0 one has

sup lp(t)| < С(d, |/|) sup |p(i)|(Ao|/|)2d(ME))-2d, tei teE

where |/| is the length of I and С(d, |/|) depends only on d e N and |/|.

Let f e T(d,n) and f = const. If 0 <p < 1, we define \\f\\p = \\f\\LP(7n) by the same formula as in the case p > 1. In the case p = 0 we set \\f\\0 = limp^0 \\f\\p. Note that \\ ■ \\p are not norms in the case p e [0,1).

There is a number M(/) such that jn(x e Rra | f(x) > M(/)) = e-1. Turan's Lemma yields the following bounds (see [13, Section 2]).

Proposition 3. For every f e T(d,n) one has

7n(x e Rra | f(x) < (AoX)-2dM(/)) < X-1, (3.6)

\\ f\\P < (3Ao max(1,2dp))2d ■ M(f), (3.7)

(eAo)-2dM(/) < \\/\o < (3Ao)2d ■ M(f). (3.8)

Corollary 3. If p,q e [0, and d e N, then there exist numbers m(d, p, q) > 0 and M(d, p, q) > 0 such that for every f e T(d, n)

m(d,p, q) ■ ||/||p < ||/< M(d,p, q) ■ ||/||p. (3.9)

Proof. If 0 < p < q < to, we have ||/||P < HfHq (Holder's inequality applied to |/\P and 1). Using (3.7) and (3.8) completes our proof. □

Remark 2. Note that for a fixed number d the numbers m = m(d,p, q) and M = M(d,p, q) in (3.9) do not depend on n.

The following proposition is an analog of the Carbery-Wright inequality (see [8] and [13]) for functions from T(d,n).

Proposition 4. For every d e N there is a number c(d) such that for all n e N and f e T(d, n) one has

ln(\i\< t) -||/H\/2d < c(d)t1/2d, t> 0. (3.10)

Proof. In (3.6) we take A = M 1/2d(f) ■ A01 -t~1/2d, t > 0, apply (3.7) with p = 1 and get 7(\/\ < t) -||/|| 1/2d < c(A0,d) ■ t1/2d. □

Corollary 4. For every d e N and p > 1 there is a number c(d, p) such that for all n e N and f e T(d, n) one has

djn <e~P+1/4dc(d,p)||/||-1/2d. (3.11)

V ( f2 + z)p

Proof. The function £ = (f2 + e)-1 is a random variable on Rn with the measure jn- We know that f2 > 0, so £ e (0, e ). Hence

/ (f2 + e)-Pd-n = E(C)= f p ■ tP-1-fn{(f2 + e)-1 > t)dt JRn JO

= p r-f2<') * = s. r MM^U.

1 jo ( s + e )p+1 eP jo (u + 1)p+1 This chain of equalities along with (3.10) implies that

ff a-p^ £)1/4d' c(d,p) ^ . 0 ... u1/4d .

I( f2 +e)Pd-fn < w Vf', c(d, p):=2c(d)p/ -——du.

■n £p- f1/2d Jo (u + 1)1+p

JRn £P ■ _ M1

Estimate (3.11) is proved. □

Remark 3. In order to derive the presented results from Turan's Lemma (Theorem 1), it is crucial that we define our set T(d,n) by formula (3.1). One might expect that in place of f e T(d, n) we could consider

N

f (x) = ao + ^2(ak ■ cos(vk, x) + bk sin(vk,x)) (3.12)

k=1

with arbitrary N € N and vectors vk with integer coordinates similar to the ones in Definition 1. However, Theorem 1 cannot be used without the condition N < d in (3.12). This fact is evident from Section 2 of [13], especially from the remark made there. Note that any function f € T(d, n) satisfies the definition of an exponential polynomial given in the aforementioned remark in [13]. This follows from formula (3.2). Thus, the proofs of the presented results heavily employ the condition N < d in (3.12). Notice, however, that f (x) from (3.12) still belongs to T(d1,n) with di = max(d, N). Even in the case N > d we have f € T(max(d, N),n).

Remark 4. Let us mention the following error made in our paper [16]: there we defined trigonometric polynomials in such a way that we actually allowed N > d in (3.12), while still basing our proofs on Turan's Lemma and its corollaries. As explained above, this reasoning is incorrect. The main results of [16] (Theorem 3 and Theorem 4) are not made completely invalid by this error, however. The numbers a in Theorem 3 and Theorem 4 depend not only on d but also on N from (3.12). In theorem 3 it is enough to replace a = 1/(4d + r) with a = 1/(4 max(d, N) + r). This fact follows from Remark 3 above and Remark 7 below.

Moreover, replacing all T~Rd(Rn) in [16] with T(d,n) defined as in Definition 1 of this paper, we can fix most of the errors occurred in [16]. Nevertheless, even after this substitution, some other corrections should be made, which is discussed in Remarks 7 and 8 below.

To formulate our next theorem, we set H^Hd = sup^=i... n \\dkg||2• We follow the convention that for g = const one has H^H-1 =

Theorem 2. For each d € N there is a number C(d) such that, for each n and every pair of trigonometric polynomials f,g € T(d,n), one has

Proof. Fix a function <p € C0°(R) with < 1. Consider the function

(<p( /) - р(д))дкд = дк(u(f) - u(g)) - <p(f)(dkf - dkg), (3.13)

where dk denotes the partial derivative with respect to the variable xk. Note that the hypotheses of our theorem say that f(x) = /(x1,..., xn) and g(x) = <jf(x1, • • •, xn). Thus, for each e > 0 we have

||7n о Г1 - о g-1hv < c(d){\\g\\-dmd) + 1) II/- g\\

2

1/(2d+1)

For any к = 1,... ,n one has дк(u(g)) = <p(g) ■ dkg. Thus,

I

f W) -4>(9))dln = f ((dkg)2 + dln (3.14)

JRP JRn (дкУ) + ^

r5

r5

We apply (3.13) to dkg in (3.14) and get

f _ f 9kgdk(u(f) -u(g)) , f dkg • <p(f)(dkf — dkg) +

L (dkg )2 + ^ d<n h (dkg )2 d<n+

+ e i (<p(f) — <p(g))№g)2 +e)-1d<n _ T\ +T2 + T3. (3.15)

JRn

We now estimate each term in (3.15) separately. First, we shall prove that Ti in (3.15) is bounded by C(d)e-i/2\\f — g\\2. To this end, we first consider the one-dimensional case n _ 1. In this case x G R and therefore dkg _ g'(x), d^g _ g"(x), xk _ x. The integration by parts formula (3.5) and the formula (2.1) allow us to write

rp f dk9dk(u(f) -U(g)) f g'(u(f) - u(g))'

Ti _ L (dk9)2 + S d<n = Jr (g>)2 + e d<1 _

_- i(u( —u(4 wi— K (3-16) < 1 ^(x) -9(x)l |x| <i(dx) + 3JR1 )^ + g 1 Pl1 (x) dx-

Observe that T(d, 1) is a (2d + 1)-dimensional linear space (see Remark 1) and \\/\\rc> _ sup^eK |/(x)| is a norm on it. Hence there is a number C(d) depending only on d such that |f(x) — <jf(x)| < C(d)\\f — g\\2. Thus,

i^W+T'071 (x) dx <C(d) \\f — 9hJK Pl1 (x)dx- (3-17)

We take the intervals Ij _ [2k j; 2kj + 2-] and write

i la''(x)l f la''(x)l

J* faW^^(x) dx < <*>/, * (3'18)

j — — <Xl J J

Note that both g' and g" belong to T(d, 1) (see Proposition 2). Let's now consider only x G Ij. Each interval Ij is of length 2k. Corollary 2 implies that g" G T(d, 1) has m < 2d zeros ^ < ... < rm in the interval Ij. Consider the intervals (Ti, Ti+i) with i _ 0,.. .,m and t0 _ 2k j, Tm+i _ 2kj + 2k. On each of the intervals (^, Ti+i) the function g" has a constant sign. Thus,

rTi+1 lg''l rTi+1 1 '

L dx _slg<9"(x) Jn WfTe d(9'(x)) <Te •

There are (m + 1) < (2d + 1) intervals (ri, Ti+i) C Ij. Therefore,

f , dx < £ ± <

Ji, (g'(x))2 +e ^

Applying this result to (3.18), we have

f \9 "(g)! d7<*(2d +1)

R №))2 + e dji < А

dll < n( ) sup Pl1 (x) < C(d)e-l/2. (3.19)

The series in (3.19) converges (we substitute p11 (x) using (2.1)):

1 2 2

V supp7i (x) = V sup —= е-T" = -7=V e-2^ 2 = Si < +то.

Combining (3.19), (3.16) and (3.17), we estimate Ti from (3.15):

Ti <Ci(d) i^2 + -—= / \ f(x) - g(x)\\x\1i(dx). yje Vе Jr

By the Cauchy-Bunyakovskii inequality the integral on the right is estimated by Ц/ — g\\2 ■ C with C = ||x||2. Hence we proved the bound

Ti = f ))—+'Ш dji < CW)Hf — S\\2 —= (3.20)

JR (9) +£ Vе

for the first term in (3.15) in the case n = 1.

We now proceed to the case n > 1. The space Rra can be decomposed into the sum Rra = (ek) ф Ek with Ek = (ek)±. If Rra = (ek) ф Ek, then the measure 7n can be represented as 7n = 7i ®jn-i, where 7i is the standard Gaussian measure on (ek) ~ R and jn-i is the standard Gaussian measure on Ek = (ek~ Rra-i. By Fubini's theorem

Ti = f f i. (')("(/;('));+^W))'d7id7„-i. (3.21)

jEkj r (9x(t!) +£

In (3.21) we have x £ Ek = (ek)x, t £ R, (■)' is the derivative with respect to the variable t, and fx(t) := f(x + tek), gx(t) := g(x + tek). Due to Corollary 1, for any fixed x £ Ek, the functions f(t) = fx(t) and g(t) = gx(t) belong to T(d, 1). We can use (3.20) to write

Ir dji <C (d)\\ Д. — Л\Ь —,

where fx = fx(t) and gx = gx(t) are regarded as functions of one variable t. Substituting this into (3.21) we get

Ti <CC(d) —= \\fx — gx\^7n-i(dx)j / = cs(d) —= \\f — g\\2, (3.22)

where also the Cauchy-Bunyakovskii inequality \\ F\\bi(p) < \\ F\\ ^(p) for the probability measure P = jn-i and F(x) = \\fx — gx\\2 is used.

The estimate (3.22) for the term T\ in (3.15) is valid for any n G N. Now we will consider the term T2 in (3.15). It is easily estimated as follows (here we need the Proposition 2 and the Proposition 3):

rp i dkgip(f)(dkf -dkg) , f \dkg\\dkf - dkg\ T2 = -L-(M2T~e-d7n < L (dkg)2 + e d7n

. f \9kf -9kg\ \\dkf -dkgh . . -1/2,,, H

</ —^— dln =-7-U2,-< C2(d)e /\\f-g\\

(3.23)

'r'

n

2^ In 2e1/2

2-

To estimate the last term in (3.15), we apply (3.11) to the trigonometric polynomial dkg and obtain the bound

T = £ f ^ < i lA^**

jr™ {dkg)2 + e Jr™ dkg2 +e (3.24)

< 2e ■ C(d, 1)\\dkg\\-1/2iie-1+1/4d = Cl(d)\\dkg\\-1/2de1/4d. Using (3.22), (3.23) and (3.24) together with (3.15) we get

I w) - <p(g)) din < O(d)\\f — g\\2— + Ci(d)\dfc<7\\-1/(2dh1/(4d)

J r« ve

with some new constants. Taking e = \\f — g\2:d"l/2d+1 we obtain

I (t(f) — V(g))din < (O(d)+ C1 (d)\\dkg\^1/(d-1))y — g\1/2d+1.

J r« v 7

Finally, we replace \\dkg\\2 with \\<jf\a = supfc=1,...,n \\dkg\\2 and take the supremum over all smooth p with \\p|| ^ < 1, completing our proof. □

Remark 5. In case g = const, Theorem 2 generalizes the result of Davydov (see [9, Section 4]) to the case of Rra. Davydov's estimate for a pair of non-constant functions F,G e T(d, 1) is

1 1

-1 _ ^ ^ n-1W^r ^ r<„(r!\\I rr _ n II 2d+1 <- r< „MMI rr _ n II 2d+1

n o F-1 - 71 o G-1\tv < CFiG(d)\\F - g\\f+1 < CF,G(d)\\F - g\\

00

where Of,g(*) depends on the number d and can depend on some Besov-Nikolskii norms of F and G (for a discussion of Besov-Nikolskii spaces, see the next section). Our Theorem 2 provides a similar result, but for T(d,n) instead of T(d, 1): along with (3.9) it implies that for any pair of non-constant functions f,g e T(d,n) with g = const one has

1

\\7n ° f-1 — in ° g-1 \\TV < Ofl(d)\\f — g\\ld+1,

where Oa(d) depends on the number d and can depend on \\<jf\a.

In the next section we show that the Besov-Nikolskii norm of g e T(d, 1) can be bounded by a number depending on \\<jf\a. Thus, the result of Theorem 2 has even more similarity with the result of [9, Section 4].

4. Fractional Sobolev spaces

Let Vh be the shift of a measure v by a vector h: vh(A) = v(A — h). Let 0 < a < 1. Let Ba(Rk) be the Besov-Nikolskii class (see [1]) of all functions g £ L1(Rk) such that, for some number C(g), one has

||g(- + h) — g||li <C(g)|h|a Vh £ Rk.

If 0 < a < 1, the class Ba(Rk) coincides with the set of all densities pu of bounded Borel measures v on R k for which with some Cv one has

||Vh — ^|tv <Cv|h|a Vh £ Rk. (4.1)

A measure v belongs to Ba(Rk) if its density is in Ba(Rk); (4.1) implies that v has a density (see [2, Proposition 3.4.3].) For measures (and functions) from the linear space Ba(Rk) we use the following norm || ■ |B:

|MB := inf{C: Hv — Vh\|tv <C|h|a}.

Note that Ba(Rk) is not a Banach space with this norm.

To check that a measure v belongs to the class Ba(Rk) we employ the following proposition (see [7, Proposition 3.1], [6, Theorem 1], [5, §2]).

Proposition 5. Let a £ (0,1) and let v be a Borel measure on R1 such that for each function p £ C£°(R) one has

<p'(x)v(dx) <C\MU\<p'\|1

I Sol \ \ \ l-a.

1r

Then v £ Ba(R) and ||v\\B<* < 2l-aC.

Remark 6. It is enough in Proposition 5 to use only <p £ C^°(R) with , < 1. For such <p the condition in the proposition can be written as

<p'(x) v(dx) < С\У\\

l—a

.

We now prove that non-constant trigonometric polynomials f £ T(d, n) have distributions from the class v £ Ba(Rk). We show that the measure Jn ° /-1 with f £ T(d, n) belongs to the class Ba(Rk) for some a.

Theorem 3. For every d £ N, there is a number C(d) such that, for every f £ T(d,n) and every <p £ C^°(R) with \\p\\œ < 1, one has

f (f)dln <C(d)\\f\\—1/2d\w\w-1^1)

JR"

Therefore, jn of 1 belongs to the Besov-Nikolskii class B1/(2d+1)(R) provided that f is not a constant.

r

Proof. We assume that f = const, since for f = const we use the convention \\f\\-1 = and the inequality in our theorem is trivial in that case. Consider an arbitrary function * e O^°(R) with \\ip\\0 < 1. Then

/ * (f)di =f \ * (f )}dln + / (d ■ ty^ din. (4.2)

Jr« Jr« v(dkf)2 + e J Jru (dkf)2 +e

Let us write the first term in (4.2) as

/ (Mrs*U)il" = I9*^(tST-e dl".

Integrating by parts, we obtain

>d2f + xk dkf 2(dkf)2dlf n f 3 Idg f |

_ Wdkf + x^f _ 2Wr"kf \*i < 3 |dfcf| di + Jr« (dkf )2 + e ((dkf )2 + e)2) din < Jr« (dkf )2 + e din+

+ f ^fl2IXfc 1 din <O((6d + 3)V^/2+1)e-1/2. (4.3) Jr« (dkf)2 + £

The last step in (4.3) follows from the following two inequalities:

dkf •-£< (dkf )2 + s, [ TTT^^ din <Oe-1/2(3d^).

R

(dk f)2 + e

The second inequality here is proved similarly to (3.19) . We now use (3.11) with = 1 to write

f (iff((fI din < \\*\\o ■ e-1+1/4d ■ C(d, 1)\\dfcf\\-1/2(i. (4.4) 7r« (dkf)2 + e

Using (4.2), (4.3) and (4.4), and taking e = \\*\\00'+2/(2d+l) we obtain

j*(f)di < C1(d)\dfcf\\-1/2dW\\oe1/4d + C2(d)e-1/2 < < (C1(d)\\dfc f\\-1/M + C2(d))\\*WO-1^11. This result holds true for every k e N. Thus, we have

f * (f)di < (c1(d)\f\-1/2d + C2(d))\\*\\10-1/2d+1. (4.5)

Since f = const and \\f\\a > 0, the function f ■ \\f\\- belongs to T(d, 1). Applying (4.5) tog = f ■ H/N-1 e T(d, 1), we obtain

f *(f \\f \-1) *i < (c 1(d) + Ck(dj)H*\\i-1/(2d+1). (4.6)

Let p e Cft°°(R), \M x < 1, m = p(i|| f\\d). Then p(i) = f(t ■ \ \f\ I-1) and P(/) = \\/W-1 ' f (/\\/W-1)- Hence, using (4.6) we get

ftf (f )d7 < C (d)\\f\\-1\\f\\L"1/(2d+1) = C (d)\\f \\-1/(2d+1)\\p\\L"1/(2d+1),

with C(d) = (c1(d) + c2(d)). Having this estimate, we complete the proof by applying Proposition 5 and Remark 6. □

Remark 7. Let us complete our discussion of the error in [16], explained in Remark 4. As noted in Remark 4, replacing T'Rd(Rn) in [16] with T(d, n) we correct some errors in [16], but for some statements this replacement is not sufficient. In some proofs in [16] we used the following fact:

If f,g emd(Rn), then f-geTK2d(Rn)- (4.7)

This is true for f,g e T~Rd(Rn), but it is not true for f,g e T(d, n) if n > 1. One can easily find f,g e T(d, n) such that for h = f ■ g the decomposition

h(x) =ao + Efc=1(£k ■ *•x> + dk ■ e-i<vk'x>) will have more than N > 2d vectors vk, which is not allowed for T(2d,n) (see Definition 1 and Remark 3). So the inclusion f,g e T(d,n) does not guarantee that /■ge T(2d, n). We now correct the reasoning in [16] relying on (4.7). Both Theorem 3 and Theorem 4 in [16] use (4.7). Let us first discuss Theorem 3. In its proof we use (4.7) when we state that

(V f; V f} = TTl=1(dif )2 belongs to T^2d(Rn). With T~Rd(Rn) replaced by T(d,n), Theorem 3 from [16] now states:

For every d e N and any number a > ^d, there is a number C(d,a) that depends only on d and a such that, for every f e T(d, n) and every function p e C£°(R) with \\p||^ < 1, we have

f p(f)djn < C(d,a)a-1/a\M 1/°\\<SWL"1/a• (4.8)

JR"

Note that the hypotheses in Theorem 3 from [16] are the same as in Theorem 3 above. Combined with Remark 6, Theorem 3 above yields

(f)dln < С(d)\\/||-1/2d|\<p\\ 1/°4<S\a

1

2d + 1

So our present paper provides a result similar to (4.8), but with a larger exponent a. Here and in [16] we employ such inequalities to establish the membership of jn o f-1 in Ba(R). Therefore, a larger value for a is preferable. Thus, Theorem 3 above gives a better result than Theorem 3 in [16], so in principle there is no need to explain in detail how to correct

r

n

the reasoning there, nevertheless, this can be done. Indeed, in Theorem 3 in [16] the condition (4.7) is used to prove the first inequality in

f 1 < c(d) -1+1/4d < c(d) -1+1/4d

L (vf, v f) +8ain - y(vL V ^v^ - ay2d •

But a similar result can be derived from inequality (3.11) of this paper. Note that by (3.11) for all k we have

f * dln -f 12 + ^ dln - e-1+1/4dc(d, 1mf\\-1/2d.

J r" (v f, v J) + £ J r" (9k J)2 + £

Now observe that max, \\dk f \\2 > Vl^/\\l+^+\\dnf \\| = \(v f, v f)\\{/2.

Taking into account that f £ T(d,n) has at most d2 distinct variables Xi, we can assume that n — d2. Thus, we have

C 1 c(d, 1)e-1+1/4d c(d, 1) -d1/2de-1+1/4d

dl'n — i______ HQ ill M/2A —

J R" (v f, v f) +s ,n- (max, \\dkf \\2)1/2d - \\(v f, v f)\\{/4d '

This inequality enables us to correct the proof of Theorem 3 in [16]. However, as we have already noted earlier in Remark 7, that theorem can be now replaced with a stronger result: Theorem 3 of this paper.

Remark 8. Let us now discuss Theorem 4 in [16], which also relies on (4.7). We used (4.7) in [16] when stating that A/ = det Mf belongs to T~R2kd(Rn)- To correct this reasoning in [16], we add the condition

Af = det Mf £ T(N,n). Observe that this condition is satisfied for N = k! ■ (2d)2k. This follows from the fact that every element of the matrix Mf has the form m^- = (vfi, vfj) with fi, f) £ T(N,n). The decomposition

m

= (Vh, Vf3) = aa + £ck • e^*<x> + dk • e-^*<x>)

IJ — (VI, V]) — "U -r z_,k

involves at most (2d)2 distinct vectors vk with |vk| < 2d < (2d)2. Hence

det Mf = au + Y.k=i(ck • ei(^*'x> + dk • e-i(?*>*>) involves at most k! • ((2d)2)k = k! • (2d)2k distinct vectors vk and for all of them one has \vk| < 2kd < k! • (2d)2k. Thus, we always have Af = det Mf £ T(d,N) with N = k! • (2d)2k, as required by Definition 1. After all these adjustments, Theorem 4 in [16] will take the following form.

Theorem 4': Let k,d £ N, a > 0, b > 0, t > 0. Then there is a number C(d, k, a, b, t) > 0 such that for every map f = ( f1,..., fk) : Ra ^ Rk with fi from T(d,n) satisfying the conditions ||Af||1 > a, maxj<kafi < b and Af £ T(N,n) and for all functions <p £ C£°(Rk) and vectors e £ Rfc with |e| = 1 there holds the inequality

i de<p(f(x)) jn (dx) < C (d, k, a, b, rmiUdeAI1—, « = (2N + r)-1.

JRn

Consequently, jn of 1 g Ba(Rk ) for any a < 1/(2NV).

As noted above, one can find N such that Af G T(N, n). The condition fi G T(d, n), i < k, guarantees that Af G T(N0,n) with N0 = k! • (2d)2k.

5. Conclusion

The estimate obtained by Davydov in [9] for trigonometric polynomials on R1 is generalized to the case of trigonometric polynomials on Rn. We also obtain new results on the inclusion of the images of Gaussian measures under trigonometric polynomials to Nikolskii-Besov classes. In addition, some inaccuracies made in [16] are corrected.

References

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5. Bogachev V.I., Kosov E.D., Popova S.N. A new approach to Nikolskii-Besov classes. Moscow Math. J., 2019, vol. 19, no. 4. pp. 619-654. https://doi.org/10.17323/1609-4514-2019-19-4-619-654

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8. Carbery A., Wright J. Distributional and L4 norm inequalities for polynomials over convex bodies in IRn. Math. Research Letters., 2001, vol. 8, no. 3. pp. 233-248. https://doi.org/10.4310/MRL.2001.v8.n3.a1

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Georgii Zelenov, Candidate of Science (Physics and Mathematics), Junior Research Fellow, Department of Mechanics and Mathematics, Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation; Associate Professor, Faculty of Computer Science, National Research University Higher School of Economics, 20, Myasnitskaya st., Moscow, 101000, Russian Federation, email: [email protected]

Received 05.06.2021

О распределениях тригонометрических полиномов от гауссовских случайных величин

Г. И. Зеленов1'2

1 МГУ им. М. В. Ломоносова, Москва, Российская Федерация

2 НИУ ВШЭ, Москва, Российская Федерация

Аннотация. В статье доказаны новые результаты о вложении распределений тригонометрических полиномов от гауссовских случайных величин в классы Бесова - Никольского. Также получена оценка расстояния по вариации между распределениями тригонометрических полиномов через расстояние в Ьч-метрике между самими полиномами.

Ключевые слова: Класс Никольского - Бесова, гауссовская мера, распределение тригонометрического полинома.

Список литературы

1. Бесов О. В., Ильин В. П., Никольский С. М. Интегральные представления функций и теоремы вложения. Т. 1,2. 2-е изд. М. : Наука, 1996. 480 с.

2. Bogachev V. I. Differentiable measures and the Malliavin calculus. Amer. Math. Soc., Rhode Island, Providence, 2010. 510 p.

3. Bogachev V. I. Distributions of polynomials on multidimensional and infinite-dimensional spaces with measures // Успехи математических наук. 2016. Т. 71, № 4. С. 107-154.

4. Bogachev V. I. Distributions of polynomials in many variables and Nikolskii-Besov spaces // Real Anal. Exchange. 2019. Vol. 44, N 1. P. 49-63.

5. Bogachev V. I., Kosov E. D., Popova S. N. A new approach to Nikolskii-Besov classes // Moscow Math. J. 2019. Vol. 19, N 4. P. 619-654.

6. Bogachev V., Kosov E., Zelenov G. Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy-Landau-Littlewood inequality // Trans. Amer. Math. Soc. 2018. Vol. 370, N 6. P. 4401-4432.

7. Богачев В. И., Зеленов Г. И., Косов Е. Д. Принадлежность распределений многочленов к классам Никольского - Бесова // Доклады Академии наук. 2016. Т. 469, №6. С. 651-655.

8. Carbery A., Wright J. Distributional and L4 norm inequalities for polynomials over convex bodies in IRn // Math. Research Lett. 2001. Vol. 8, N 3. P. 233-248.

9. Davydov Y. A. On distance in total variation between image measures // Statistics & Probability Letters. 2017. Vol. 129. P. 393-400.

10. Kosov E. D. Fractional smoothness of images of logarithmically concave measures under polynomials// J. Math. Anal. Appl. 2018. Vol. 462, N 1. P. 390-406.

11. Косов Е. Д. Классы Бесова на конечномерных и бесконечномерных пространствах // Математический сборник. 2019. Т. 210, № 5. С. 41-71.

12. Назаров Ф. Л. Локальные оценки экспоненциальных полиномов и их приложения к неравенствам типа принципа неопределенности // Алгебра и анализ. 1993. Т. 5, № 4. C. 3-66.

13. Назаров Ф. Л., Содин М. Л., Вольберг А. Л. Геометрическая лемма Каннана - Ловаса - Шимоновича, не зависящие от размерности оценки распределения значений полиномов и распределение нулей случайных аналитических функций // Алгебра и анализ. 2002. Т. 14, № 2. С. 214-234.

14. Nourdin I., Poly G. Convergence in total variation on Wiener chaos // Stochastic Process. Appl. 2013. Vol. 123, N 2. P. 651-674.

15. Zelenov G. I. On distances between distributions of polynomials // Theory Stoch. Processes. 2017. Vol. 22, N 2. P. 79-85.

16. Зеленов Г. И. Дробная гладкость распределений тригонометрических полиномов на пространстве с гауссовской мерой // Известия Иркутского государственного университета. Серия Математика. 2020. Т. 31. С. 78-95.

Георгий Ильич Зеленов, кандидат физико-математических наук, младший научный сотрудник, механико-математический факультет, Московский государственный университет им. М. В. Ломоносова, Российская Федерация, 119991, г. Москва, ГСП-1, Ленинские горы, 1; доцент, факультет компьютерных наук, Национальный исследовательский университет Высшая школа экономики, Российская Федерация, 101000, г. Москва, Мясницкая ул., 20, email: [email protected]

Поступила в 'редакцию 05.06.2021