Questions tagged [convolution]
Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.
3,089 questions
0 votes
0 answers
7 views
On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.
Let $\sigma_I=\{e^{2i\pi t}:t\in I\}$ and $\sigma_J=\{e^{2\pi i t}:t\in J\}$ be two disjoint subarcs on the the first quadrant of unit circle of arc length $\theta$, where $I,J\subseteq [0,1/4]$ of ...
9 votes
1 answer
105 views
If all convolutions of a tempered distribution T with any compactly supported function are compactly supported, is T compactly supported?
What I mean is that we have $T \in \mathcal{S}'(\mathbb{R}^n)$ such that for all $f \in C^\infty_c (\mathbb{R}^n)$, we have that $\mathrm{supp}(T \ast f)$ is compact, and we're asking whether $\mathrm{...
- 113
1 vote
0 answers
63 views
Can Young's inequality give a pointwise bound for a convolution?
I’m trying to understand a step in Appendix A.1 of Bejenaru and Herr, The cubic Dirac equation: small initial data in $H^1(\mathbb{R}^3)$. The paper proves global well-posedness for the cubic Dirac ...
- 479
1 vote
1 answer
102 views
Evaluation of a Broken-up Convolution Integral
I would like a comment on the correctness of my derivation below: I have the following Duhamel Convolution integral $$ U(t)=\int_0^t e^{-p(t-y)}f(y)\text{d}y $$ For reasons that have to do with the ...
- 1,298
2 votes
1 answer
157 views
Difficult Convolution Problem -- I Am Stuck with the Integration
I have the following function which is a probability density function on the $-1 < t < 1$ interval: $$f(t) = \frac{2\sqrt {1-t^2}}{\pi}$$ I wish to convolve this function with itself to get a ...
0 votes
0 answers
66 views
Circular convolution modulo $3$
I am working with a convolution sum of the form $$ h(j) = \sum_{k=0}^2 f\!\big((j-k) \bmod 3\big)\, g(k), $$ where $f, g : \{0,1,2\} \to \mathbb{C}$. Because of the modulo $3$ structure in the index ...
- 27
3 votes
1 answer
114 views
How to recover differentiator approximation from RC circuit solution?
I'm reading The Art of Electronics, in 1.4.3, the section on passive differentiators. The math of the situation is given by $$\frac{d}{dt}(V_{in}(t)-V_{out}(t))=\frac{1}{RC}V_{out}(t)\tag{1}$$ In the ...
- 35
1 vote
1 answer
70 views
Expansion of Series Issue from a Textbook
Sorry for any issues with this post - it is my first. Jones, P. W. & Smith, P. (2018). Stochastic Processes: An Introduction. Chapman & Hall In Chapter 3.3, an expansion of a series for a ...
- 21