# Tagged Questions

Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

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### uniform one-sided van der Corput inequality

Is the following true (and if yes, where the best proof is written?)? For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$? Hm,...
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### The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
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### Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
The following result is well-known (I consider the 3-dimensional case only): Theorem: if $f \in H^s(\mathbb{R}^3)$ with $s > 3/2$ is compactly supported, then $$\left| \int_{\mathbb{R}^3} f - \... 1answer 93 views ### Smooth dependence on the initial condition of the integral of an ODE I am considering an ODE \dot{x}=f(x), with x\in\mathbb{R}^d and d<\infty. f is a C^k function and I denote by \Phi_t x be the value of the solution at time t. I assume that my ODE ... 0answers 24 views ### diffusion and potentials in several dimensions In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ... 3answers 3k views ### What is the standard notation for a multiplicative integral? If f: [a,b] \to V is a (nice) function taking values in a vector space, one can define the definite integral \int_a^b f(t)\ dt \in V as the limit of Riemann sums \sum_{i=1}^n f(t_i^*) dt_i, or ... 1answer 179 views ### Is there a matrix that converts the gradient of every possible function to gradient of other function? I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ... 1answer 214 views ### Is the exponent 2 sharp in the Balog-Szemerédi-Gowers Theorem? The Balog-Szemerédi-Gowers theorem can be stated in the following form: let A,B be subsets of \mathbb{Z}/n\mathbb{Z} (say) with equal cardinality, such that$$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|...
Could you show that \sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(...