grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2012 centrale-maths1__mp

36 maths questions

QI.A.1 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$. a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$; b) $f, g \in L^{2}(\mathbb{R})$.
QI.A.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $f, g \in C(\mathbb{R})$ be such that $f * g(x)$ is defined for every real $x$. Show that $f * g = g * f$.
QI.A.3 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that if $f$ and $g$ have compact support, then $f * g$ has compact support.
QI.B.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that a function $h$ is uniformly continuous on $\mathbb{R}$ if and only if $\lim_{\alpha \rightarrow 0} \|T_{\alpha}(h) - h\|_{\infty} = 0$.
QI.B.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $T_{\alpha}(f * g) = \left(T_{\alpha}(f)\right) * g$.
QI.B.3 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $\left\|T_{\alpha}(f * g) - f * g\right\|_{\infty} \leqslant \left\|T_{\alpha}(f) - f\right\|_{2} \times \|g\|_{2}$.
QI.B.4 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Deduce that $f * g$ is uniformly continuous on $\mathbb{R}$ in the case where $f$ has compact support.
QI.B.5 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.
QI.C.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Assume that $f \in L^{1}(\mathbb{R})$ and $g \in C_{b}(\mathbb{R})$. a) Show that $f * g$ is continuous. b) Show that if $g$ is uniformly continuous on $\mathbb{R}$, then $f * g$ is uniformly continuous on $\mathbb{R}$.
QI.C.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $k$ be a non-zero natural number. Assume that $g$ is of class $C^{k}$ on $\mathbb{R}$ and that all its derivative functions, up to order $k$, are bounded on $\mathbb{R}$. Show that $f * g$ is of class $C^{k}$ on $\mathbb{R}$ and specify its derivative function of order $k$.
QI.C.3 Sequences and Series Functional Equations and Identities via Series View
In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise. a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise. b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.
QI.D.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges pointwise to $f$ on $\mathbb{R}$.
QI.D.2 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.
QI.D.3 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by $$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$ and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula $$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$ a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity. b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$. c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.
QI.D.4 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Does there exist a function $g \in C_{b}(\mathbb{R})$ such that for every function $f$ in $L^{1}(\mathbb{R})$, we have $f * g = f$?
QII.A Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by $$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$ For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.
QII.B.1 Sequences and Series Functional Equations and Identities via Series View
Let $f, g \in L^{1}(\mathbb{R})$. Assume that $g$ is bounded. a) Show that $f * g$ is integrable on $\mathbb{R}$ and determine $\int_{\mathbb{R}} f * g$ in terms of $\int_{\mathbb{R}} f$ and $\int_{\mathbb{R}} g$. b) Show that $\widehat{f * g} = \hat{f} \times \hat{g}$.
QII.B.2 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Show that there exist two functions $f$ and $g$ in $L^{1}(\mathbb{R})$ such that $f * g(0)$ is not defined.
QII.C.1 Sequences and Series Evaluation of a Finite or Infinite Sum View
We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ Express the Fourier transform $\hat{k}_{n}(x)$ using the function defined by $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
QII.C.2 Sequences and Series Convergence/Divergence Determination of Numerical Series View
We define $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Justify that $\varphi \in L^{1}(\mathbb{R})$.
QII.C.3 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ We admit that $\int_{\mathbb{R}} \varphi = \pi$ where $\varphi(x) = \left(\frac{\sin x}{x}\right)^2$ for $x\neq 0$ and $\varphi(0)=1$. We set $K_{n} = \frac{1}{2\pi} \hat{k}_{n}$. Show that the sequence of functions $\left(K_{n}\right)_{n \geqslant 1}$ is an approximate identity.
QII.D.1 Sequences and Series Functional Equations and Identities via Series View
Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ For every real $t$ and every non-zero natural number $n$, show that $I_{n}(t) = \left(f * K_{n}\right)(t)$.
QII.D.2 Sequences and Series Functional Equations and Identities via Series View
Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ Deduce, for every real $t$: $$f(t) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(x) \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}x$$
QIII.A.1 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
To any function $g$ in $C(\mathbb{R})$, we associate the linear form $\varphi_{g}$ on $L^{1}(\mathbb{R})$ defined by $$\varphi_{g}(f) = \int_{\mathbb{R}} f(t) g(-t) \mathrm{d}t$$ Let $(g_{1}, \ldots, g_{p})$ be a family of elements of $C_{b}(\mathbb{R})$. Show that the family $(g_{1}, \ldots, g_{p})$ is free if and only if the family $(\varphi_{g_{1}}, \ldots, \varphi_{g_{p}})$ is free.
QIII.A.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $E$ be an infinite-dimensional vector space and $\left(f_{n}\right)_{n \in \mathbb{N}}$ a family of linear forms on $E$. We denote $$K = \bigcap_{n \in \mathbb{N}} \operatorname{Ker}\left(f_{n}\right)$$ Show that the codimension of $K$ in $E$ is equal to the rank of the family $\left(f_{n}\right)_{n \in \mathbb{N}}$ in the dual space $E^{*}$ (begin with the case where this rank is finite).
QIII.A.3 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We assume that $g \in C_{b}(\mathbb{R})$. We consider the vector subspace $$N_{g} = \left\{f \in L^{1}(\mathbb{R}) \mid f * g = 0\right\}$$ and the vector space $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ where $T_{\alpha}(g)(x) = g(x-\alpha)$. Show that the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$ is equal to the dimension of $V_{g}$.
QIII.A.4 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
a) Let $\beta \in \mathbb{R}$ and let $g$ be the function defined by $g(t) = \mathrm{e}^{\mathrm{i}\beta t}$ for all $t \in \mathbb{R}$. Determine the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$. b) Let $n$ be a natural number. Show that there exists a function $g$ in $C_{b}(\mathbb{R})$ such that $N_{g}$ has codimension $n$ in $L^{1}(\mathbb{R})$.
QIII.B.1 Second order differential equations Structure of the solution space View
Let $g \in C_{b}(\mathbb{R})$. We say that $g$ satisfies hypothesis A if $g$ is a function of class $C^{\infty}$ on $\mathbb{R}$, bounded and whose derivative functions of all orders are bounded on $\mathbb{R}$. Show that if $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$ and if $g$ satisfies hypothesis A, then $g$ is a solution of a linear differential equation with constant coefficients.
QIII.B.2 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $g \in C_{b}(\mathbb{R})$ satisfying hypothesis A (i.e., $g$ is of class $C^{\infty}$ on $\mathbb{R}$, bounded, and all its derivative functions of all orders are bounded on $\mathbb{R}$). Deduce the set of functions $g$ satisfying hypothesis A and such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.
QIII.C.1 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $g \in C_{b}(\mathbb{R})$. We assume that $N_{g}$ has finite codimension $n$ in $L^{1}(\mathbb{R})$, and that $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ has dimension $n$. Show that there exist real numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ and functions $m_{1}, \ldots, m_{n}$ of a real variable such that, for every real $\alpha$, $$T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$$
QIII.C.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $F$ be a finite-dimensional subspace of $C(\mathbb{R})$, with dimension denoted $p$. For any function $f \in C(\mathbb{R})$ and for any real $x$, we denote $e_{x}(f) = f(x)$. a) Show that there exist real numbers $a_{1}, \ldots, a_{p}$ such that $(e_{a_{1}}, \ldots, e_{a_{p}})$ is a basis of the dual space $F^{*}$. b) If $\left(f_{1}, \ldots, f_{p}\right)$ is a family of elements of $F$, show that $\operatorname{Det}\left(f_{i}\left(a_{j}\right)\right)_{1 \leqslant i,j \leqslant p}$ is non-zero if and only if $\left(f_{1}, \ldots, f_{p}\right)$ is a basis of $F$.
QIII.C.3 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $\alpha_{1}, \ldots, \alpha_{n}$, $m_{1}, \ldots, m_{n}$ be as in III.C.1 such that $T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$ for every real $\alpha$. By applying question III.C.2) to $V_{g}$, show that if $g$ is of class $C^{k}$ then the functions $m_{1}, \ldots, m_{n}$ are of class $C^{k}$.
QIII.C.4 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3, defined on $[-1,1]$ by $h_{r}(t) = \frac{(1-t^2)^r}{\lambda_r}$ and zero outside $[-1,1]$. Show that for every non-zero natural number $r$, $V_{h_{r} * g}$ is finite-dimensional.
QIII.C.5 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3. Show that for $r$ sufficiently large the dimension of $V_{h_{r} * g}$ is equal to that of $V_{g}$.
QIII.C.6 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $m_{1}, \ldots, m_{n}$ be as in III.C.1. Deduce that the functions $m_{1}, \ldots, m_{n}$ are of class $C^{\infty}$.
QIII.C.7 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Determine the set of functions $g \in C_{b}(\mathbb{R})$ such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.