### A curious property of series involving terms of generalized sequences.

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We prove that $|{\sum}_{k=1}^{n}sin\left((2k-1)x\right)/k|<Si\left(\pi \right)=1.8519...$ for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

We prove: If $$\frac{1}{2}+\sum _{k=1}^{n}{a}_{k}\left(n\right)cos\left(kx\right)\ge 0\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}x\in [0,2\pi ),$$ then $$1-{a}_{k}\left(n\right)\ge \frac{1}{2}\frac{{k}^{2}}{{n}^{2}}\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}k=1,\cdots ,n.$$ The constant $1/2$ is the best possible.

The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.

Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, [...] Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain...