# Tagged Questions

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### When orthogonal polynomials form an Hilbert basis?

Let $\mu$ be a probability measure on $\mathbb R$, and consider the sequence of orthonormal polynomials in $L^2(\mu)$. These polynomials are constructed by applying Gram-Schmidt to the sequence ...
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### Pulsating waves of zeta function

Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ ...
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### the multidimensional Hilbert transforms or partial Hilbert transforms

The one-dimensional Hilbert transform can be defined by the convolution $Hf:=f*\frac{1}{\pi x}$, or can be given by Fourier multiplier $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
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### The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
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### Square-summable sequence and Fourier series

Every square-summable sequence $(a_{n})_{n}$ is represented by $a_{n}=\widehat{f}(4^n)$, where $\widehat{f}(i)$ is Fourier coefficient of continuous function $f$. Where can I find proof of this ...
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### Applications of Fourier analysis for math students

Does anyone have recommendations for a good book which explains how Fourier analysis is applied in engineering and physics that does not assume any knowledge of the above topics? I'd like to see the ...
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### reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
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### reference request for proof of Gibbs phenomenon at jump discontinuities

Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to ...