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4

The answer given by Xiaoyu He uses the same arguments as usual van der Corput's theorem (see Theorem 2.2 from Graham, S. W. & Kolesnik, G. "van der Corput's method of exponential sums"): Suppose that $f$ is a real valued function with two continuous derivatives on $I$. Suppose also that there is some $\lambda > 0$ and some $a > 1$ such that ...

5

I write $e(x)=e^{2\pi i x}$. Here is a naive bound via Kuzmin's estimate, which says that if you have a sequence $c_n$ with monotonic differences $\delta_n$ all in some interval $[k+\epsilon, k+1-\epsilon]$, then the entire exponential sum is small: $$|\sum_n e(c_n)| \ll \epsilon^{-1}$$ If we write $c_n = \sqrt{nN}$ then in Elkies' notation \delta_n = ...

2

Yes. A sledge hammer with which you can hit this is Stein's Topics in Harmonic Analysis related to the Littlewood-Paley Theory, published by the Annals of Math Studies series of the PUP. The main thing you are looking for the Theorem 2, the "square function theorem" for decompositions of functions on compact Lie groups. (Note that in this ...

6

One can indeed make $\|\phi\|$ exponentially small in $A$ (though the example I have requires an exponentially large number of frequencies $\alpha_i$). We use the dual formulation, that is we find a sum of exponentials $g$ whose $L^\infty$ norm is small on $[0.9A, A]$. Firstly, if we consider the sum of just two exponentials $g_0(t) = 1 + e^{\pi i/A}$, ...

0

Thinking about this a bit more, here is a guess — I haven't got time right now to check the details, so comments and corrections are welcome from others. The Euclidean motion group $E(2)$ is the group of all isometries of ${\bf R}^2$ with its usual Euclidean metric. Since $E(2)$ acts on ${\bf R}^2$ in a measure-preserving way, this action gives a ...

4

Maybe: isotypical decomposition of the representation $L^2(\mathbb{R}^2,\mathbb{C})$ of the group $U(1)$ might do.

5

On any abelian Lie group $G = \mathbf R^n\times\mathbf T^p\times\mathbf Z^q\times \Phi$ ($\Phi$ finite), one defines as usual $C_0^\infty(G)=\{$smooth functions on $G$ with compact support$\}$. (The notion makes sense on any manifold.) As for $\mathscr S(G)$, Bruhat (1956, p.138) defines it as the space of all smooth $f$ on $G$ such that, for every ...

4

In the language of Marcus-Spielman-Srivastava, Corollary 1.5, identify $l^2(S)$ with ${\mathbb C}^d$ where $d = |S|$ and let the vectors $u_i \in {\mathbb C}^d$ be the orthogonal projections into $l^2(S)$ of the Fourier transforms of the standard basis vectors of $l^2(\{-1,1\}^n)$. By Corollary 1.5 with $r = 2$, we can find a subset $T$ of $\{-1,1\}^n$ such ...

-1

First I should discuss a little about the reason for such notations that I am looking for. I wish to decompose the 3D operation into a bunch of 1D operations so I can do something in parallel. The Fast Fourier Transformation (FFT) of three-dimensional (3D) data is of particular importance for many numerical simulations used in High Performance Computing ...

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