grandes-ecoles 2023 Q47

grandes-ecoles · France · centrale-maths1__official Sequences and Series Properties and Manipulation of Power Series or Formal Series

Let $L$ be the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$, and let $P = W \circ L \circ W^{-1}$ with $W : p \mapsto p(\alpha X)$.
Verify that $D = L \circ (L-I)^{-1}$ then that $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.

Let $L$ be the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$, and let $P = W \circ L \circ W^{-1}$ with $W : p \mapsto p(\alpha X)$.

Verify that $D = L \circ (L-I)^{-1}$ then that $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.

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