grandes-ecoles 2022 Q9

grandes-ecoles · France · mines-ponts-maths2__pc Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

Let $(c_{j})_{j \in \mathbf{N}}$ be a sequence of complex numbers such that the series $\sum c_{j}$ converges absolutely. We set $$\forall t \in \mathbf{R}, \quad u(t) = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$
Justify the existence and continuity of the function $u$. For $k \in \mathbf{N}$, show that $$\frac{1}{2\pi} \int_{-\pi}^{\pi} u(t) e^{i(k+1)t} \mathrm{d}t = c_{k}$$

Let $(c_{j})_{j \in \mathbf{N}}$ be a sequence of complex numbers such that the series $\sum c_{j}$ converges absolutely. We set
$$\forall t \in \mathbf{R}, \quad u(t) = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$

Justify the existence and continuity of the function $u$.\\
For $k \in \mathbf{N}$, show that
$$\frac{1}{2\pi} \int_{-\pi}^{\pi} u(t) e^{i(k+1)t} \mathrm{d}t = c_{k}$$

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