Nicolas Essis-Breton
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 Nov 8 accepted Applying Ito formula to the Brownian bridge Nov 7 accepted Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$ Nov 5 accepted Show smooth functions of compact support are dense in the Schwartz space Nov 2 accepted Show the usual Schwartz semi-norm is a norm on the Schwartz space Oct 29 accepted Show $|\left(e^{-ixt}-1 \right)/t| \le |x|$, $x ,t\ne 0$ Oct 29 accepted Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$ Oct 29 accepted Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$ Oct 25 accepted Probabilistic calulation of the Fourier transform of the Cantor function Oct 25 accepted Fourier transform of the Cantor function Oct 23 accepted $E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$ Oct 22 accepted $f \in L^1(\mathbb R), f>0$ then $|\hat f(y)| < \hat f(0), y \ne 0$ Oct 21 accepted If the Fourier series of $f$ is absolutely convergent does it implies that it converges to f Oct 20 accepted Brownian motion: changing the order of expectation and integration in $E \left( \int_s^t B_x dx \mid F_s \right)$ Oct 18 accepted $f$ absolutely continuous, $\hat f(n) \downarrow 0$, $\hat f(n)$ positive and even, then $\sum \hat f_n <\infty$ Oct 18 accepted If $f$ is absolutely continuous then $\sum|\hat f (n) | < \infty$ Oct 16 accepted Find the rate of growth for $\sum_{n=1}^N 1/n^p$ in term of big $O$ notation Oct 16 accepted Show from every positive sequence decreasing to zero, we can build a sequence satisfying a concavity condition Oct 16 accepted Derivating $f(t)=\int_0^t x dx$ using measure theory Oct 16 accepted Convergence of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ Oct 15 accepted Inequality for term of a positive sequence : show $\frac{1}{n} \ge c_n - c_{n+1} \ge \frac{1}{n+1}$