# Questions tagged [trigonometric-polynomials]

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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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Prove that $$f(x, y) \equiv \arccos\left(\frac{x-y}{K}\right) - \arccos\left(\frac{x-y}{K}+y\right) - \frac{y}{x}\arccos(1-y^2) \ge 0$$ with the constraints: $K\ge 2$ is an integer, $g(x, y) = (K-1)... 0answers 132 views ### Bounding expected value of maximum of dot product with random chirp Let$\mathbf{x}\in\mathbb{C}^n$with$\|\mathbf{x}\|=1$with$n<\frac{N}{2}$. I am interested in a bound of the form \begin{equation*} \mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{... 0answers 100 views ### Lower$L^1$norm estimates of null average trigonometric polynomials depending on the order of the polynomial Let$p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$be a null average trigonometric polynomial (null average means that is$\int_\mathbb T p =0$or, equivalently, there are no$a_0$and$b_0$). ... 0answers 789 views ### Is there an infinite product like this for$\cos x$? There are infinite products of iterated square roots for$\log x$and$\arccos x$as functions of$x$. For example $$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{... 0answers 95 views ### Octahedron and System of trigonometric equations Could somebody help me to prove the following?$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0\sum_{k=1}^6 \cos (\phi_k)=0\sum_{k=... 0answers 72 views ### 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 ... 0answers 99 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 ... 0answers 286 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,$ 0<\alpha&...
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 $K_N(y)=\sum_{|n|<N}\frac{N-|n|}{N}e^{iny}$?
### 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{\...