Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 444–447. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 444–447. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Yudin. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

On Trigonometric Series with Positive Partial Sums

V. A. Yudin

Received February, 1997

In this paper, we study trigonometric series (TS) with nonnegative partial sums whose spectrum is A ∪ (−A), where A is the set of natural numbers free of sums which is well-known in number theory, i.e., the equation x + y = z has no solutions in A. We state that TS of this type,

1 + ∑_{\begin{matrix}ν∈ℤ \\ |ν|∈A \end{matrix}} c_{ν}e^{iνx},  c_{ν} = \overline{c_{−ν}},  (1)

are Fourier series of functions from L_{3}. It was stated earlier [1] that these series belong to L_{2}.

We shall begin with considering polynomial mappings of a unit circle in a complex plane. Let f(z), z ∈ ℂ, be a polynomial and

f(z) = ∑_{k≥1} c_{k}z^{k}, ∥f∥_{p} = (\frac{1}{2π} ∫_{0}^{2π}|f(e^{it})|^{p} dt)^{p}, p > 0;

let

f(Γ) = {z ∈ ℂ : z = f(e^{it}), 0 ≤ t < 2π}

be the image of the unit circle. Let

P (α, R) = {z = x + iy ∈ ℂ : x cos α + y sin α ≥ R}.

We denote by A^{∗} = {z : z = x − y, x ≠ y, x, y ∈ A} the difference set A. In order to carry out the estimation, we shall need the following assertion which is a corollary of the Riesz theorem [2]: for any 1 < p < ∞, m ∈ ℕ, we have

∥ ∑_{1≤k≤m} c_{k}z^{k}∥_{p} ≤ λ_{p}∥f∥_{p}.

For the constant λ_{p} we have the inequality

λ_{p} ≤ p,  p ≥ 2,  (2)

whose elementary proof can be found in [3, 4]. Some arguments connected with the orthogonality of the degrees of trigonometric polynomials will be used in this paper.

Since f(0) = 0, it follows from the argument principle that the convex hull of the curve f(Γ) in the complex plane contains a certain neighborhood of zero. Using the structure of the set A, we shall show that this hull contain a disk of a “sufficiently large” radius.

We have the following

ON TRIGONOMETRIC SERIES WITH POSITIVE PARTIAL SUMS 445

Theorem 1. Assume that the following conditions hold:

(1) all coefficients c_{k} of the polynomial f(z) with numbers that do not belong to A vanish,

(2) A^{∗} ∩ A = ∅.

Then the curve f(Γ) intersects any half-plane P (α, R), where

R = \frac{∥f∥_{3}^{3}}{216∥f∥_{2}^{2}} .  (3)

Proof. We set

g(t) = f(e^{it})e^{iα}, G(t) = Re g(t) = \frac{1}{2}(g(t) + \overline{g}(t)), M = min_{t} G(t).

Since M + G(t) ≥ 0, we have

∥G∥_{3}^{3} = \frac{1}{2π} ∫_{0}^{2π}|G(t)|^{3} dt = \frac{1}{2π} ∫_{0}^{2π}|G(t) + M − M| × |G(t)|^{2} dt

≤ \frac{1}{2π} ∫_{0}^{2π}(G(t) + 2M)|G(t)|^{2} dt = 2M∥G∥_{2}^{2} + \frac{1}{16π} ∫_{0}^{2π}(g(t) + \overline{g}(t))^{3} dt

= 2M∥G∥_{2}^{2} + \frac{1}{16π} ∫_{0}^{2π}(g^{3}(t) + \overline{g}^{3}(t)) dt + \frac{3}{16π} ∫_{0}^{2π}|g(t)|^{2}(g(t) + \overline{g}(t)) dt.

The second term vanishes by the multiplicative properties of the trigonometric system. The polynomial g^{3}(t) has a positive spectrum. Since

|g(t)|^{2} = |f(e^{it})|^{2} = ∑_{k,l∈A} c_{k}\overline{c}_{l}e^{i(k−l)t},

the nonzero Fourier coefficients of |g(t)|^{2} are contained in the difference set A^{∗}. By the hypothesis of the theorem, this set does not intersects A. Hence, the third term also vanishes. Thus

∥G∥_{3}^{3} ≤ 2M∥G∥_{2}^{2}.  (4)

By the orthogonality of g and \overline{g}, we have

∥G∥_{2}^{2} = ∥\frac{1}{2}(g + \overline{g})∥ = \frac{1}{4}(∥g∥_{2}^{2} + ∥\overline{g}∥_{2}^{2}) = \frac{1}{2}∥f∥_{2}^{2}.

By the Riesz theorem and by inequality (2), we have

∥G∥_{3}^{3} ≥ (2λ_{3})^{−3}∥g∥_{3}^{3} = (2λ_{3})^{−3}∥f∥_{3}^{3} ≥ \frac{1}{216}∥f∥_{3}^{3}.

Consequently, we find from (4) that

∥f∥_{3}^{3} ≤ 216M∥f∥_{2}^{2},

and this implies (3). Theorem 1 is proved.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

Corollary. Under the conditions of Theorem 1 imposed on the polynomial f(z), the following assertion holds: for any α, f(Γ) intersects P (α, R) when R = ∥f∥_{3}/216.

Indeed, by the Hölder inequality, we have

∥f∥_{3} ≥ ∥f∥_{2},  (5)

and therefore

R = \frac{∥f∥_{3}^{3}}{216∥f∥_{2}^{2}} ≥ \frac{∥f∥_{3}}{216} .

If we apply (5) and inequality (2) once again, then, by the Parseval identity we find that f(Γ) intersects any half-plane P (α, R) for

R = \frac{1}{216}(∑_{k∈A} |c_{k}|^{2})^{1/2}.

Theorem 2. Assume that the following conditions hold for the TS (1):

(1) there exists a nonnegative subsequence of partial sums,

(2) A^{∗} ∩ A = ∅.

Then the TS (1) is a Fourier series of the function from L_{3}.

Proof. Let

S_{n_{k}}(t) = 1 + ∑_{\begin{matrix}1≤|ν|≤n_{k} \\ |ν|∈A \end{matrix}} c_{ν}e^{iνt} ≥ 0.

We set α = 0, R = 1/2, and

f(e^{it}) = ∑_{1≤ν≤n_{k}} c_{ν}e^{iνt}

in (3); then ∥f∥_{3} ≤ 108. By the Minkowski inequality,

∥S_{n_{k}}(t)∥_{3} = ∥1 + f(e^{it}) + \overline{f(e^{it})}∥_{3} ≤ 1 + 2∥f∥_{3} ≤ 217.

For any n ∈ ℕ, we choose a number n_{k} such that n ≤ n_{k}. By the Riesz theorem, we find the upper bound of the norm of the partial Fourier sum, namely,

∥S_{n}(t)∥_{3} = ∥S_{n}(S_{n_{k}}(t))∥_{3} ≤ 3∥S_{n_{k}}(t)∥_{3} ≤ 651.

We find that the norms of partial sums of the TS are uniformly bounded. Hence (see [2]), the TS (1) is a Fourier series of the function from L_{3}. Theorem 2 is proved

In conclusion, we shall consider one special case where A = {k^{3} : k ∈ ℕ}. According to Euler’s theorem, A^{∗} ∩ A = ∅. We find from Theorems 1 and 2 that

(i) the curve

z = {∑_{k≥1} c_{k}e^{ik^{3}t}}_{t∈ℝ}

intersects any straight line of the form

x cos α + y sin α = \frac{1}{216}(∑_{k≥1} |c_{k}|^{2})^{1/2},

(ii) if the TS “with respect to the cubes”

∑_{k∈ℤ}c_{k}e^{ik^{3}t}

has a nonnegative sequence of partial sums, then it is a Fourier series of the function from L_{3}.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Fundamental Research (project no. 96-01-00212).

REFERENCES

1. Yudin, V. A., On Trigonometrical Series with Positive Partial Sums, Mat. Zametki, 1993, vol. 53, no. 3, pp. 149–152.

2. Edwards, R., A Modern Introduction, Heidelberg: Springer, 1982.

3. Yudin, V. A., On Fourier Sums in L_{p}, Trudy Mat. Inst. Akad. Nauk SSSR, 1987, vol. 180, pp. 235–236.

4. Garling, D. J. H., Hardy Martingales and the Unconditional Convergence of Martingales, Bull. London Math. Soc., 1991, vol. 23, pp. 190–192.

Translated by S. Vakhromeev