grandes-ecoles 2020 Q26

grandes-ecoles · France · centrale-maths2__mp Indefinite & Definite Integrals Definite Integral Evaluation (Computational)

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t)\,\mathrm{d}t$$ Show that $U$ is the identity map on $E_1$.

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We set, for all $f \in E_1$,
$$U(f)(s) = \int_0^1 k_s'(t) f'(t)\,\mathrm{d}t$$
Show that $U$ is the identity map on $E_1$.

Paper Questions

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