Discussion Overview
The discussion revolves around the differentiability of the convolution of two continuous functions defined on the right half-line, specifically exploring conditions under which the convolution is differentiable. The scope includes theoretical aspects of calculus and functional analysis.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant defines the convolution of two continuous functions f and g on [0,∞] and questions the general differentiability of the convolution f✶g.
- Another participant asserts that if either function f or g is differentiable, then the convolution is also differentiable.
- A subsequent reply challenges this by asking what occurs if both functions f and g are nowhere differentiable.
- One participant references a 1951 paper that constructs a continuous function for which the convolution is only differentiable at a single point (0), suggesting that there are exceptions to the earlier claim.
- Another participant expresses gratitude for the reference and shares their personal struggle with the topic, indicating a connection to the work of Mikusinski.
Areas of Agreement / Disagreement
Participants express differing views on the conditions for differentiability of the convolution, with some asserting that differentiability of either function guarantees the differentiability of the convolution, while others highlight cases where this may not hold true.
Contextual Notes
The discussion includes references to specific mathematical constructs and literature, indicating that assumptions about differentiability may depend on the nature of the functions involved. The implications of the referenced paper suggest limitations in the generality of the earlier claims.