Approximation of polynomials from Walsh tail spaces

We derive various bounds for the $L_p$ distance of polynomials on the hypercube from Walsh tail spaces, extending some of Oleszkiewicz's results (2017) for Rademacher sums.

Similarly, we say that f belongs on the k-th tail space, where k ∈ {1, . . ., n}, if f (S) = 0 for every subset S with |S| ≤ k.More generally, given a nonempty subset I ⊆ {0, 1, . . ., n}, we denote by We shall also adopt the natural notations P n >k = P n {k+1,...,n} , P n ≤k = P n {0,1,...,k} , P n =k = P n {k} and so on.Many modern developments in discrete analysis (see [18]) are centered around quantitative properties of functions whose spectrum is bounded above or below, in analogy with estimates established for polynomials in classical approximation theory on the torus T n or on R n .One of the first results of this nature, going back at least to [3,4], is the important fact that all finite moments of low-degree Walsh polynomials are equivalent to each other up to dimension-free factors.Namely, given any 1 ≤ p ≤ q < ∞ and k ∈ N, there exists a (sharp) constant M p,q (k) such that for any n ≥ k, every polynomial f : {−1, 1} n → R of degree at most k satisfies ∥f ∥ q ≤ M p,q (k)∥f ∥ p , (3) where ∥ • ∥ r always denotes the L r norm on {−1, 1} n with respect to the uniform probability measure.Note that the reverse of (3) holds trivially with constant 1 by Hölder's inequality.We refer to [8,13,16] for the best known bounds on the implicit constant M p,q (k).In the special case k = 1, (3) is the celebrated Khintchine inequality [15] for Rademacher sums.Our starting point is the simple observation that the moment comparison estimates (3) have the following (equivalent) dual formulation in terms of distances from tail spaces.Proposition 1.For every 1 ≤ p ≤ q < ∞ and k ∈ N, the constant M p,q (k) in inequality (3) is also the least constant for which every function where the conjugate exponent This material is based upon work supported by the NSF grant DMS-1929284 while the authors were in residence at ICERM for the Harmonic Analysis and Convexity program.
Again, the reverse of (4) holds with constant 1.In the special case k = 0, inequality (4) becomes trivial with M p,q (0) = 1 as both sides are equal to |Ef |.When k = 1, which corresponds to the dual of the classical Khintchine inequality, we derive the following more precise formula for the distance from the tail space P n >1 .
Theorem 2. For every 1 < r ≤ ∞ and n ∈ N, every f : This is the dual to a well-known result of Hitczenko [10] (see also [17,11]), obtaining pindependent upper and lower bounds for the L p -norms of Rademacher sums, where p ∈ [1, ∞).
At this point, we should point out that in both Proposition 1 and Theorem 2, the exponents of the norms are always strictly greater than 1.For instance, choosing On the other hand, it follows from a result of Oleszkiewicz [19], which is the main precursor to this work, that the L 1 -distance of f 1 from the k-th tail space satisfies inf and thus exhibits a starkly different behavior as n → ∞ from the L r norms with r > 1.
More generally, it is shown in [19] that for every where for a a a = (a 1 , . . ., a n ) we denote f a a a (x) = n i=1 a i x i and we make the convention that a n+1 = 0.The quantity appearing on the right hand side of (8) can be rephrased in terms of the Kfunctional of real interpolation (see [1,Chapter 3]).Recall that if (A 0 , A 1 ) is an interpolation pair, then the Lions-Peetre K-functional is defined for every t ≥ 0 and a ∈ A 0 + A 1 as It is elementary to check (see [12]), that if Note that the right-hand side is invariant under permutations of the entries of a a a.The main result of this work is an appropriate extension of the upper bound in Oleszkiewicz's result (8) to polynomials of arbitrary degree on the discrete hypercube.
Theorem 3.For every d ∈ N, there exists where f is the vector of Fourier coefficients of f , viewed as an element of As was already pointed out by Oleszkiewicz, the method of [19] does not appear to extend beyond Rademacher sums.Instead, in our proof we shall employ the discrete Bohnenblust-Hille inequality from approximation theory (see [2,7,6,5]) along with a classical bound of 1 Throughout the paper we shall use standard asymptotic notation.For instance, ξ ≲ η (or η ≳ ξ) means that there exists a universal constant c > 0 such that ξ ≤ cη and ξ ≍ η stands for (ξ ≲ η)∧(η ≲ ξ).We shall use subscripts of the form ≲ t , ≳ t , ≍ t when the implicit constant c depends on some prespecified parameter t.
Figiel on the Rademacher projection of polynomials.A discussion concerning the size of the implicit constant C d appearing in (11) is postponed to Section 2 (see Remark 8 there).
Unlike the two-sided inequality (8), our bound ( 11) is only one-sided and as a matter of fact there are examples in which it is far from optimal.In particular, for functions which are permutationally symmetric, we obtain a more accurate estimate.In what follows, we shall denote by T k (x) = k ℓ=0 c(k, ℓ)x ℓ the k-th Chebyshev polynomial of the first kind characterized by the property T k (cos θ) = cos(kθ), where θ ∈ R.Moreover, we shall use the ad hoc notation c(k, ℓ) For ℓ ∈ {1, . . ., n}, let f ℓ be the ℓ-th elementary symmetric multilinear polynomial We have the following bound on the distance of symmetric polynomials from tail spaces.
This bound can sometimes be reversed and, in particular, it gives a sharp estimate as n → ∞ for the L 1 -distance of the elementary symmetric polynomial f d from the k-th tail space. and The main motivation behind the work [19] was a question of Bogucki, Nayar and Wojciechowski, asking to estimate the L 1 -distance of the Rademacher sum f 1 from the k-th tail space.Corollary 5 extends (at least asymptotically in n) the answer given by Oleszkiewicz to all symmetric homogeneous polynomials.We point out though that for k = 1, ( 16) is sharper than Oleszkiewicz's bound (7) as n → ∞, as ( 7) is tight only up to a multiplicative constant.
Acknowledgements.We are grateful to Krzysztof Oleszkiewicz for valuable discussions.H. Z. is grateful to Institut de Mathématiques de Jussieu for the hospitality during a visit in 2023.

Proofs
We proceed to the proofs of our results.We start with the simple duality argument leading to Proposition 1, variants of which will be used throughout the paper.
Proof of Proposition 1.Consider the identity operator acting as id(h) = h on a function of the form h : {−1, 1} n → R.Then, the optimal constant M p,q (k) can be expressed as by duality.Moreover, observe that since (P n ≤k , ∥ • ∥ r ) is a subspace of L r , its dual is isometric to where A ⊥ is the annihilator of A. Since it is also clear that id * = id, (17) concludes the proof.□ Using a theorem of Hitczenko [10] as input and the same duality, we deduce Theorem 2.
Proof of Theorem 2. The result of [10] asserts that if a a a = (a 0 , a 1 , . . ., a n ) and f a a a (x) = a 0 + n i=1 a i x i , where x 0 , x 1 , . . ., x n are independent Bernoulli random variables, and the first equality holds due to symmetry.In other words, the linear operator given by T a a a = f a a a is an isomorphism, and thus the same holds for its adjoint.Recalling that and the duality between sums and intersections of normed spaces [1, Theorem 2.7.1], we see that the dual space of (R n+1 , K(•, ) can be identified with By Parseval's identity, the action of the adjoint is given by and thus the conclusion is equivalent to fact that T * is an isomorphism.□ We now proceed to the proof of the general upper bound for polynomials given in Theorem 3. The first ingredient for the proof is a discrete version of the classical Bohnenblust-Hille inequality from approximation theory (see the survey [7]) proven in [2,6].This asserts that for every d ∈ N, there exists a (sharp) constant B d ∈ (0, ∞) such that for any n ≥ d, every polynomial f : Moreover, 2d d+1 is the least exponent for which the implicit constant becomes independent of the ambient dimension n.The best known upper bound B d ≤ exp(C d log d) for the constant B d is due to the work of Defant, Mastyło and Pérez [6].
The level ℓ-Rademacher projection of a function f : {−1, 1} n → R is defined as Moreover, we write Rad ≤d = ℓ≤d Rad ℓ .Apart from the discrete Bohnenblust-Hille inequality (25), we will also use a standard bound on the norm of the ℓ-Rademacher projections which is usually attributed to Figiel (see also [9,Section 3] for a short proof).
Combining the above with Parseval's identity, we deduce the following bound. where Equipped with Lemma 7, we can complete the proof of Theorem 3.
Proof of Theorem 3. Consider the normed spaces and Moreover, let Z = P n >d ∩ P n ≤k ⊂ X, viewed as a normed subspace of X. Lemma 7 asserts that the linear operator A : X/Z → Y given by has norm ∥A∥ ≤ 1.Therefore, the same holds for its adjoint A * : Y * → (X/Z) * .By the usual duality between sums and intersections of normed spaces [1, Theorem 2.7.1], we see that the space Y * is isometric to Moreover, as X/Z is a quotient of X, its dual is the subspace of X * = L 1 /P n >k which is identified with the annihilator of Z inside X * .In other words, it is the set (X/Z) * = f + P n >k : E[f g] = 0 for every g ∈ Z = f + P n >k : f ∈ P n ≤d = span(P n ≤d ∪ P n >k ) P n >k (34) equipped with the L 1 quotient norm.Finally, for a sequence a a a = (a S ) |S|≤d ∈ Y * and an equivalence class f + Z ∈ X/Z, we have where the first brackets ⟨•, •⟩ denote the duality in Y and the following brackets denote the duality in X/Z.Therefore, we conclude that and thus, the condition ∥A * ∥ ≤ 1 means that for any f : Finally, since This can be proven using similar ideas as in the proof of Theorem 3.
A slight variant of the arguments above also yields Theorem 4 for symmetric functions.
Proof of Theorem 4. Let f be a symmetric function of the form f = d ℓ=0 α ℓ f ℓ where f ℓ is the ℓ-th elementary symmetric polynomial.Then, the Hahn-Banach theorem gives inf Observe now that we can write Thus, by Figiel's bound (27), and the desired inequality follows from (41).□ Equipped with Theorem 4, we present the proof of Corollary 5.
Proof of Corollary 5.The upper bound in (16) follows immediately from Theorem 4. For the lower bound, consider the auxiliary symmetric function H k,n : {−1, 1} n → R given by where f ℓ is the ℓ-th elementary symmetric polynomial, and notice that H k,n has degree at most k.As T k (x) is odd or even when k is odd or even respectively, it follows that β ℓ,k,n = 0 if k − ℓ is odd.We distinguish two cases depending on the parity of k − d.
• Suppose that k − d is even and consider the function that is also symmetric and of degree at most d.Then, on one hand we know that inf On the other hand, we have the following lower estimate, inf The rest of the argument is identical.□ Remark 11.In this paper, we studied dual versions of moment comparison estimates on the hypercube (3) and investigated the endpoint case of their duals (4) for polynomials.By formal reasoning similar to the proof of Proposition 1, one can derive dual versions of various other polynomial inequalities, including Bernstein-Markov inequalities and their reverses and bounds for the action of the heat semigroup.We refer to [8] for a systematic treatment of such estimates.