Question

How to prove $\int_0^\pi\biggl |\sum_{i=0}^k \frac{a_i\sin(i+\frac{1}{2})x}{2\sin \frac{1}{2}x}\biggr| \, dx \le c(k+1)$ for some constant $c$?

score 103 · Answer 1 · 2026-04-15 04:06:35

103

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How to prove $\int_0^\pi\biggl |\sum_{i=0}^k \frac{a_i\sin(i+\frac{1}{2})x}{2\sin \frac{1}{2}x}\biggr| \, dx \le c(k+1)$ for some constant $c$?

Published on 15 Apr 2026 - 4:06

score 264 · Answer 2 · 2026-04-16 03:45:46

264

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138

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45

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List Question

How to prove $\int_0^\pi\biggl |\sum_{i=0}^k \frac{a_i\sin(i+\frac{1}{2})x}{2\sin \frac{1}{2}x}\biggr| \, dx \le c(k+1)$ for some constant $c$?

Is it true that $\int \hat{f}g \, dx=\int f\hat{g} \, dx$ for $f,g\in L^2$?

How to prove the following double integral is positive definite?

Is $L^2(\mathbb R)$ isometrically isomorphic with $\ell^2(\mathbb Z)?$

Books for beginners on the Fourier Transform

Is a function $f:\mathbb R/\mathbb Z\to \mathbb R$ bounded ?

How can we show that $\int_{|\alpha |\leq N}\hat f(\alpha )e^{2i\pi x\alpha }d\alpha $ converges to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$.

Show that there is $f\in \mathcal C^0(S^1)$ s.t. $\lim_{n\to \infty }\|S_nf-f\|_{L^\infty }\neq 0$.

Approximation estimates in Sobolev spaces

Convergence of Sequence of Fourier Transforms

Limit of $\lim_{t \to \infty} \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx}$

Proving that a function belongs in the space of tempered distributions

Fourier Transform: Musical instruments

Fourier series Coefficients and Wolfram Alpha

Prove the completion of the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ is not separable

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