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On an Inequality of Sidon Type for Trigonometric Polynomials A. O. Radomskii Moscow State University Abstract We establish a lower bound for the uniform norm of the trigonometric polynomial of special form via the sum of the L1 -norm of its summands. This result generalizes a theorem due to Kashin and Temlyakov, which, in turn, generalizes the classical Sidon inequality. We set, as usual, by kf kp the norm in Lp (0, 2π), 1 ≤ p ≤ ∞: p1 Z 2π 1 kf kp = |f (x)|p dx , 1 ≤ p < ∞, 2π 0 kf k∞ = ess sup |f (x)|. [0,2π] By the classical Sidon theorem (see [6]; pp. 393-395) if the {nk }∞ k=1 , is lacunary with lacunarity index λ (i.e., nk+1 /nk ≥ λ > 1 for any k), then there exists a positive constant c(λ) depending only on the lacunarity index sequence of natural numbers such that, for any trigonometric polynomial of the form T (x) = N X ak ∈ R, ak cos nk x, k=1 the following inequality holds: kT k∞ ≥ c(λ) N X |ak |. k=1 Kashin and Temlyakov obtained a more general result in the special case. Namely, they proved (see [1] and also [2]) Theorem A (B. S. Kashin, V. N. Temlyakov). trigonometric polynomial of the form f (x) = 2l X k=l+1 pk (x) cos 4k x, 1 For any 2 where the pk are real trigonometric polynomials of deg(pk ) ≤ 2l , where k = l + 1, . . . , 2l and l = 1, 2, . . . , the following inequality holds kf k∞ ≥ c 2l X kpk k1 , k=l+1 where c > 0 is an absolute constant. Kashin and Temlyakov also introduced QC -norm (see [1] and [2]) and proved that for this norm the previous result is true in a more general case: for a function f ∈ L1 (0, 2π) with Fourier series f ∼ P∞ s=0 δs (f, x), where 1 δ0 = 2π 2π Z QC -norm f (x) dx, 0 of X δs = 2s−1 ≤|k|<2s f is dened as Z 1 ∞ X rs (ω)δs (f, ·) ≡ 0 kf kQC s=0 where fˆ(k)eikx , s = 1, 2, . . . , {rk (ω)}∞ k=0 dω, L∞ is the system of Rademacher. For any real function f ∈ L (0, 2π) the following inequality holds Theorem B (B. S. Kashin, V. N. Temlyakov). 1 ∞ kf kQC 1 X ≥ kδs (f )k1 . 16 s=0 A similar result to the theorem A is proved by the author in the case of an arbitrary lacunarity index and under weaker constraints on the degrees of the polynomials pk (see [3] and [5]). Suppose that a sequence of natural numbers satises the condition nk+1 /nk ≥ λ > 1, k = 1, 2, . . . . There exist the constants c = c(λ) > 0 and γ = γ(λ) ∈ (0, 1) depending only on λ such that, for any trigonometric Theorem 1. {nk }∞ k=1 3 polynomial of the form f (x) = N X pk (x) cos nk x, k=l where pk are real trigonometric polynomials of deg(pk ) ≤ [γnl ], k = l, . . . , N, N ≥ l, l = 1, 2, . . . , the following inequality holds kf k∞ ≥ c N X kpk k1 . k=l In connection with research of QC -norm Grigor'ev proved the following result (see [4]). There exists the sequence of trigonometric such that Theorem (Grigor'ev). polynomials σk (x) = {σk (x)}∞ k=1 X cj eijx , kσk k1 ≥ 2k ≤|j|<2k+1 and n X σk k=1 ∞ π , kσk k∞ ≤ 6, k = 1, 2, . . . , 4 √ ≤ A n, n = 1, 2, . . . , where A > 0 is an absolute constant. Using some ideas of Grigor`ev`s work we have proved (see [5]) the following result which shows that in theorem 1 (in nk = 2k ) the condition deg(pk ) ≤ [γ2l ] can`t k−k ε substituted for deg(pk ) ≤ [2 ] neither any ε ∈ (0, 1). the case be Suppose that ε, εe are real numbers such that < ε < εe < 1. For any W ∈ N there exist the real trigonometric pk (x), k = 1, . . . , W , such that polynomials Theorem 2. 1 2 ε deg pk ≤ 2k−k , kpk k1 ≥ 13 , kpk k∞ ≤ 70, k = 1, . . . , W, and n X k max pk (x) cos 2 x ≤ CW εe, 1≤n≤W k=1 ∞ where C = C(ε, εe) > 0 is a constant depending only on ε and εe. 4 REFERENCES [1] B. S. Kashin, V. N. Temlyakov, On a norm and approximation characteristics of classes of functions of several variables, in Metric Theory of Functions and Related Problems in Analysis, Collection of articles dedicated to P. L. Ul`yanov on his seventieth birthday (AFTs, Moscow, 1999), pp. 6999 [in Russian]. [2] B. S. Kashin, V. N. Temlyakov, On a certain norm and related applications, Math. Notes, Vol. 64, No. 4 (1998), pp. 551554. [3] A. O. Radomskii, On an Inequality of Sidon Type for Trigonometric Polynomials, Math. Notes, 2011, Vol. 89, No. 4, pp. 555-561. [4] P. G. Grigor'ev, On a sequence of trigonometric polynomials, Math. Notes, 1997, Vol. 61, No. 6, pp. 780-783. [5] A. O. Radomskii, On the possibility of an extension of inequalities of Sidon type, Mat. Zametki, (in press). [6] A. Zygmund Trigonometric Series. V. 1. Ì.: Mir, 1965 [in Russian].