# Questions tagged [trigonometric-polynomials]

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### Sum of absolute values of trigonometric polynomials

I am trying to tackle the following problem: Let $A_{f},A_{g} \in \mathbb{R}^{3 \times 3}$ be symmetric matrices and let $f: [-\pi,\pi)^{2} \to \mathbb{R}$ and $g: [-\pi,\pi)^{2} \to \mathbb{R}$ ...
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### Closed formula for iterated Fourier series

I'm trying to obtain a closed formula for the following integral. \begin{align} I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt] & {}+ b_{1,...
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### Rigorous estimates on roots of function

We consider the function $$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$ The arguments of the two sines differ by a factor ...
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### Majorizing $|\{\alpha\}-1/2|$ by trigonometric polynomials

Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$...
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### What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there: How can $\sqrt{2x^2}$ become $4x^2$ ?
• 99
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### Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$

For any even $n$, there should be a polynomial $f(x,y)$ that vanishes on the points $(\cos^n(t),\sin^n(t))$ for all $t$ (since it is the image of the projective variety $x^2+y^2 = 1$ under the $n/2$th-...
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### Optimization problem on trigonometric polynomials

I would like to maximize $$\int_0^{2\pi} \frac{(f'(x))^2}{f(x)}dx$$ subject to $f(x)\leq 1$ for all $x$ over the space of nonnegative trigonometric polynomials of degree smaller or equal to $n$. ...
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1 vote
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### Reference request: Gaussian almost periodic functions

Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic". Almost periodic means roughly that $C$ is uniformly approximable by ...
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1 vote
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### Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...
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### Bounding expected value of maximum of dot product with random chirp

Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form \begin{equation*} \mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{...
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I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have \begin{align*} \mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
Formulation of the Conjecture Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by \psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( a_{kj}\sin(jt)+...