# Questions tagged [trigonometric-polynomials]

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### Is there a nullstellensatz for trigonometric polynomials?

Let $$f(x) = \sum_{j=1}^n c_j e^{2\pi i\alpha_j x}, g(x) = \sum_{k=1}^m d_k e^{2\pi i\beta_k x}$$ be two (quasi-periodic) trigonometric polynomials, where the coefficients $c_j, d_k$ are complex and ...
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### Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a ...
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### Tight upper bounds on trigonometric polynomials

According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
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### Solutions of equation $\sin \pi x_1\sin \pi x_2=\sin \pi x_3\sin \pi x_4$ [closed]

I am interested in finding all the solution $(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$ of equations: $$\sin \pi x_1\sin\pi x_2=\sin \pi x_3\sin\pi x_4.$$ I have found out a paper: Rational products of sines ...
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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 ...
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, $0<\alpha&... 0answers 37 views ### Convolution of Fejer kernel I have a question about how to prove $$|p(y)|\leq 3\int_{T^2}|p(y-z)|K_N(z)dz$$ where$p(y)=\sum_{|n|<N}a_n e^{iny}$and$K_N(y)=\sum_{|n|<N}\frac{N-|n|}{N}e^{iny}$? 0answers 118 views ### What numbers (irrational) in radicals are expressible as trigonometric rational fraction with only rational multiplies of$\pi$? What irrational expressions$A$with radicals can be expressed as trigonometric rational fraction (not a series) with only rational multiplies of$\pi$. Example:$ \frac{1}{\sqrt5} = \frac{\sin\frac{\...
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 $4x^2$ ?