Undergrad Generalizing the translation operator

Click For Summary
The discussion focuses on the effects of generalized translation operators, specifically ##e^{g(p)\partial_p}## and ##e^{g(p)\partial_p^2}##, on a function ##f(p)##. It establishes that the operator ##e^{a\partial_p}## translates the function by a constant shift, while ##e^{a\partial_p^2}## corresponds to the Weierstrass transform. The challenge arises in determining the action of the more complex operators involving a function ##g(p)##. Participants suggest expressing these operators in a Taylor series to analyze their effects on ##f(p)##. Understanding these generalizations is crucial for advancing the application of translation operators in various mathematical contexts.
BlackHole213
Messages
33
Reaction score
0
If I have the operator, ##e^{a\partial_p}## acting on ##f(p)##, I know that $$e^{a\partial_p}f(p)=f(p+a)\,.$$
If I have ##e^{a\partial_p^2}f(p)##, this is just the Weierstrass transform of ##f(p)##. However, what happens if I have a general operator, ##e^{g(p)\partial_p}## or ##e^{g(p)\partial_p^2}##. How would I know what ##e^{g(p)\partial_p}## or ##e^{g(p)\partial_p^2}## does to ##f(p)##?
 
Physics news on Phys.org
Write it out in a Taylor series.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

Similar threads

High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Is the Expression Valid in Demonstrating a Poincare Algebra Commutator?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Inverse Laplace transform of a rational function
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Am I using the right limits on this triple integral?
  • · Replies 1 ·
Replies
1
Views
2K
What is the Symmetry of the Ricci Tensor?
  • · Replies 7 ·
Replies
7
Views
3K
High School Question about a limit definition
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Canonical transformation from canonical to kinetic momentum
  • · Replies 7 ·
Replies
7
Views
2K