2015年9月6日 · The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions give? Why are is it being integrated on negative infinity to infinity? What is the physical significance of the convolution?

Based on your connection, it seems to me that convolution therefore defines a different "natural multiplication" between functions if we consider functions $\mathbb{R} \to \mathbb{R}$ as generalized power series in which the exponents can take …

2023年8月2日 · $\begingroup$ I am sorry for posting my question in the wrong forum. I am new to math forums :) My question was aimed at understanding why convolution can be interpreted as the mean of f using a weight function g (g >= 0, ||g|| = …

Definition of Convolution. Ask Question Asked 6 years, 7 months ago. Modified 6 years, 7 months ago ...

2010年10月26日 · I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give me a good explanation. I can't seem to grasp other than the fact that it is just a particular integral of two functions.

2014年12月27日 · Convolution corresponds via Fourier transform to pointwise multiplication. You can multiply a tempered distribution by a test function and get a tempered distribution, but in general you can't multiply two tempered distributions and get a tempered distribution.

Intuition for Convolution. A convolution is an amount of overlap of one function f as it is shifted over another function g at a given time offset. Let’s say we are transforming a certain function f(t) by passing it through a filter g(t) to get the output h(t): f(t) -> [ g(t) ] -> h(t)

But I am having trouble showing 1; I tried to use $\epsilon, \delta$ definition of continuity instead of the limit definition. I guess this is my 2nd question, is it true that in general if the metric space is first countable, $\lim x_n = x (\Rightarrow) \lim f(x_n) = f(x)$ is equivalent with the $\epsilon,\delta$ definition of continuity?

2015年7月10日 · This definition logically becomes the common definitions (that have a flip as what OP described) defining how each output element can be calculated, no matter whether for convolution between two continuous-domain functions, two discrete-domain functions or …

The natural generalisation of that formula is $(f*g*h)(z)=\int_T\int_Tf(x)g(y)h(z-x-y)\mathrm dx\mathrm d y$, which shows that convolution is associative (you can easily equate this formula with those obtained from repeated two-argument convolution using either of …