Questions: Distribution Theory and Generalized Functions

Question 1

In distribution theory, the derivative of a distribution T is defined by:

• A. ⟨T', φ⟩ = -⟨T, φ'⟩ for all test functions φ
• B. ⟨T', φ⟩ = ⟨T, φ'⟩ for all test functions φ
• C. T'(x) = lim_{h→0} (T(x+h) - T(x))/h
• D. ⟨T', φ⟩ = d/dx ⟨T, φ⟩

Answer: (A)

The derivative is defined by transferring the derivative to the test function with a sign change, generalizing integration by parts: ∫f'φ dx = -∫fφ' dx. This definition ensures that every distribution has derivatives of all orders.

Question 2

The Dirac delta distribution δ is defined as a function that is infinite at the origin and zero elsewhere.

• T. True
• F. False

Answer: False

The delta distribution is NOT a function at all. It is a continuous linear functional defined by ⟨δ, φ⟩ = φ(0) for all test functions φ. The informal description 'infinite at the origin' is a useful heuristic but mathematically incorrect—no function (even in the Lebesgue sense) has this property while integrating to 1.

Question 3

What does the Malgrange-Ehrenpreis theorem guarantee?

Model answer: Every constant-coefficient linear partial differential operator has a fundamental solution (distribution solving LΦ = δ)

This is a foundational existence theorem in distribution theory. While the fundamental solution may not be a classical function (it could be a genuine distribution), its existence means every constant-coefficient PDE Lu = f with f a distribution has a solution u = Φ * f.

Question 4

The space of test functions D(ℝⁿ) consists of:

• A. C^∞ functions with compact support
• B. All continuous functions
• C. L² functions
• D. Schwartz class functions (rapid decrease)

Answer: (A)

Test functions are infinitely differentiable functions that vanish outside a bounded set. The Schwartz space S(ℝⁿ) of rapidly decreasing functions is a larger space that also supports a distribution theory (tempered distributions), which is better suited for Fourier analysis.