Erdős: An inequality for the maximum of trigonometric polynomials

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P. Erdős, "An inequality for the maximum of trigonometric polynomials," Annales Polonici Mathematici, 12(2), 151-154, 1962. https://doi.org/10.4064/ap-12-2-151-154

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Bears on. E1150.

Overview

Erdős studies the sup norm

$$M=\max_{0\le \vartheta<2\pi}|f_n(\vartheta)|, \qquad f_n(\vartheta)=\sum_{k=1}^n(a_k\cos k\vartheta+b_k\sin k\vartheta),$$

relative to the coefficient energy. Parseval gives the baseline $M\ge 2^{-1/2}(\sum_{k=1}^n(a_k^2+b_k^2))^{1/2}$ (p. 151). The paper conjectures the uniform improvement (1), with an absolute $c>0$, and notes that $c\le \sqrt2-1$ by $f(\vartheta)=\cos\vartheta$; the suggestion that equality may be optimal is explicitly conjectural (p. 151).

The theorem following (3) assumes

$$\max_k\max(|a_k|,|b_k|)=1,\qquad \sum_{k=1}^n(a_k^2+b_k^2)=An,$$

and asserts the existence of $c_A>0$, depending only on $A$ and tending to zero as $A\to0$, such that

$$M>\frac{1+c_A}{\sqrt2} \left(\sum_{k=1}^n(a_k^2+b_k^2)\right)^{1/2}$$

(pp. 151–152). Since (2) forces $0<A\le2$, this is a density-dependent improvement over Parseval, not the absolute-constant conjecture (1). The final line of the proof states the quantitative choice $c_A\ge A^4/10{,}000$ (equation (13), p. 154), while observing that the constant was not optimized.

The proof is by excluding the near-Parseval upper bound (6). Lemma 1 says that under (3) and (6), the set on which $|f_n|$ is below the threshold (7) has small measure (p. 152). Its printed statement gives $<20\varepsilon^{1/2}$, whereas the displayed calculation actually yields $U<3\pi\varepsilon^{1/2}<10\varepsilon^{1/2}$ (p. 153), and the sharper constant is the one used in (13). Lemma 2 applies Bernstein's derivative inequality $\|f_n'\|_\infty\le n\|f_n\|_\infty$ to obtain an upper bound of order $A^{1/2}n^{3/2}$ (p. 153). Lemma 3 minimizes the weighted coefficient energy by filling the lowest frequencies first and obtains

$$\int_0^{2\pi}f_n'(\vartheta)^2\,d\vartheta>A^3n^3/4$$

(p. 153). Combining Lemmas 2 and 3 gives the total-variation lower bound (8). Because a degree-$n$ trigonometric polynomial has at most $2n$ monotonic arcs, its variation while its values lie in the two narrow intervals (9) is bounded by (10). Equations (11)–(12) therefore force the complementary low-amplitude set to have measure $U>A^2/10$, contradicting Lemma 1 unless $\varepsilon>A^4/10{,}000$ (pp. 153–154).

For complex analytic polynomials the paper separately proposes (4):

$$\max_{|z|=1}\left|\sum_{k=1}^n\varepsilon_kz^{m_k}\right|>(1+c_1)\sqrt n, \qquad |\varepsilon_k|=1.$$

This is stated only as a conjecture, and Erdős explicitly says that he cannot prove it even for $m_k=k$ (p. 152). He attributes to Newman only the weaker additive improvement $\sqrt n+c_1/\sqrt n$; the footnote records an associated integral estimate, but neither assertion is proved in this paper (p. 152). Equation (5), concerning analytic polynomials whose squared coefficient norm appreciably exceeds their largest coefficient, is likewise introduced only as a likely conjecture (p. 152).

Relation to E1150

Write an E1150 polynomial as

$$P(z)=\sum_{j=0}^{n}\epsilon_jz^j, \qquad \epsilon_j\in\{-1,1\},$$

and put $N=n+1$ and $Q(z)=zP(z)$. Then

$$f_N(t)=\Re Q(e^{it})=\sum_{k=1}^{N}\epsilon_{k-1}\cos(kt)$$

has $a_k=\epsilon_{k-1}$, $b_k=0$, and satisfies (2)–(3) with $A=1$. Thus the theorem gives

$$\max_t|P(e^{it})|=\max_t|Q(e^{it})| \ge \max_t|f_N(t)| >\frac{1+c_1}{\sqrt2}\sqrt N,$$

where the paper's explicit conclusion supplies only $c_1\ge10^{-4}$. This is not progress toward the required fixed improvement over $\sqrt n$: the displayed factor $(1+10^{-4})/\sqrt2$ is below $1$, so ordinary complex Parseval, $\|P\|_\infty\ge\sqrt{n+1}$, is stronger. To reach E1150 by this real-part reduction one would need a proved constant exceeding $\sqrt2-1$, which the theorem does not furnish.

Equation (4) is the paper's direct counterpart of E1150. Taking $m_k=k$, real $\varepsilon_k$, and $N=n+1$ makes its polynomial exactly $zP(z)$; hence (4), if proved with an absolute $c_1>0$, would imply E1150 (indeed with $\sqrt{n+1}$ in place of $\sqrt n$). But (4) is explicitly a conjecture, and the paper stresses that even this consecutive-exponent case is unproved. Newman's attributed $\sqrt N+O(N^{-1/2})$ improvement has vanishing relative size and therefore cannot supply E1150's fixed $c>0$. Equation (5) is also conjectural and, when specialized to $N$ Littlewood coefficients, gives a parameter $B=N-1$ without any stated uniform lower bound for $c_B$. Consequently the paper is useful mainly as the original formulation and as a total-variation method for improving the real Parseval constant under coefficient-density assumptions; it neither proves nor disproves E1150.