The trigonometric-polynomials tag has no wiki summary.

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### Can a lower-order trigonometric polynomials fit the data generated by a higher-order one? [on hold]

Let $x$ be a discrete finite-dimension vector $x$, and write $\hat{x}(\Omega) = \mathcal{F}_{\Omega}x$ where $\mathcal{F}_{\Omega}$ is the linear mapping collecting the Fourier transform of $x$ at the ...

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### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

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### Explicit constructions of fundamental systems for spherical harmonics

For definitions and notations on the theory of spherical harmonics I refer to www.cis.upenn.edu/~cis610/sharmonics.pdf
Let $n,k\geq 0$, and let $S^n$ be the unit sphere on $\mathbb{C}^{n+1}$. Let ...

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### Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$
where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...

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### trigonometric polynomial

Can anyone tell me the following statement is true or not? Thank you.
There are two polynomials:
\begin{align}
p(r,\theta)
&=\sin(n_0\theta) + \sum_{j=1}^{\ell}a_j r^{n_j}\sin(n_j\theta),
\quad ...

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### Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...

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### certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...

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### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k ...

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### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
...