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1
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0answers
39 views

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*} ...
-8
votes
0answers
56 views

New formulas for Generating sin(nx) & cos(nx) [closed]

I have proved formulas for generating $\sin (nx)$ and $\cos(nx)$ in term of only $\sin x$ and integer $n$, or $\cos x$ and integer $n$, such as: $$\sin(9x) = 9\sin x -120\sin^3 x + 432\sin^5 x – ...
3
votes
0answers
76 views

Concentration of weighted random chirp

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*} ...
7
votes
1answer
196 views

A conjecture about the measure estimates of a trigonometric polynomial

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( ...
1
vote
0answers
75 views

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 ...
0
votes
0answers
40 views

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 ...
1
vote
0answers
206 views

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, $ ...
6
votes
0answers
191 views

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 ...
2
votes
1answer
249 views

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 ...
8
votes
1answer
321 views

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 ...
2
votes
1answer
152 views

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 ...
2
votes
1answer
220 views

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})} ...