Centrale-Supélec 2026 — Mathématiques 2 (PC)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A — Construction d'une fonction test (Q1 à Q7)

   φ(x) = e^{−1/x} for x > 0, and φ(x) = 0 for x ⩽ 0.

   • Q1 — φ is of class C^∞ on R^*.
   • Q2 — the existence of a polynomial P_n such that φ^{(n)}(x) = P_n(1/x)e^{−1/x}.
   • Q3 — deduce that φ ∈ C^∞(R).
   • Q4 — construct ψ_{a,b}, smooth, strictly positive on ]a, b[, zero elsewhere.
   • Q5, Q6 — the function ρ_ε is well defined, belongs to D(R), and its integral equals 1.
   • Q7 — ρ_ε concentrates the integral at a point.
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2. Exercise 2 — Partie B — Distributions sur R (Q8 à Q13)

   A distribution is a linear form on D(R) satisfying, on each segment K, a bound in terms of the sup norms of the first derivatives.

   • Q8 — D′(R) is a vector space.
   • Q9 — the Dirac mass δ_a is a distribution.
   • Q10 — the distribution T_f associated with a piecewise continuous function.
   • Q11 — two continuous functions that give the same distribution are equal.
   • Q12 — the difference quotient ψ of a test function is smooth.
   • Q13 — the principal value of 1/x is a distribution.
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3. Exercise 3 — Partie C · I — Dériver une distribution (Q14 à Q17)

   The derivative of a distribution is defined by T′(φ) = −T(φ′).

   • Q14 — T′ is indeed a distribution.
   • Q15 — for f of class C^1, T_f′ = T_{f′}: the definition extends ordinary differentiation.
   • Q16 — the absolute value: not differentiable, but differentiable in the sense of distributions.
   • Q17 — the Heaviside function has derivative δ_0.
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4. Exercise 4 — Partie C · II — Les distributions de dérivée nulle (Q18 à Q23)

   • Q18 — a distribution with zero derivative is associated with a constant function; the proof goes through test functions with zero integral.
   • Q19, Q20 — translation for continuous functions: zero derivative gives constant, zero second derivative gives affine.
   • Q21 to Q23 — if T_f′ = T_g with f and g continuous, then f is of class C^1 and f′ = g.
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5. Exercise 5 — Partie D — Suites de distributions (Q24 à Q27)

   Convergence of a sequence of distributions is defined pointwise on test functions.

   • Q24 — differentiation is continuous for this convergence.
   • Q25 — an example of a sequence of functions that does not converge pointwise, while the sequence of associated distributions converges.
   • Q26, Q27 — uniform convergence, then convergence in the 2-norm, imply convergence of the distributions.
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6. Exercise 6 — Partie E — Le théorème d'unicité de Cantor (Q28 à Q34)

   Assume that the trigonometric series Σ (a_n cos(nx) + b_n sin(nx)) converges for every real x, and that its sum is zero. We want to deduce that all the coefficients are zero.

   • Q28 — the sequences (a_n) and (b_n) tend to 0.
   • Q29 — the case where the series Σ|a_n| and Σ|b_n| converge.
   • Q30 to Q33 — divide by n²: the resulting series converges normally, its sum F is affine, bounded, constant, then zero.
   • Q34 — conclusion.
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