Variational methods - conjugate of function

Does that mean just the linear functional on H that defines the function F*? Or is it the complex conjugate of that linear functional?In summary, the conversation discusses a function F, its conjugate F*, and a new function G(u)=F(u-a). The task is to verify that the conjugate of G, denoted as G*, is equal to F*(u*)+<a,u>. The specific case of F:R^2->R and a=(2,-1) is also mentioned.
  • #1
braindead101
162
0
Let F:H->R bar be a function and F*:H->R bar its conjugate. Fix aEH and show that the conjugate of the new function G(u)=F(u-a) is G*(u*)=F*(u*)+<a,u>
Verify the case where F:R^2->R, F(x)=1/2(x)^2 and a = (2,-1)

I don't really know how to show this. please help
 
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  • #2
First, define your terms. What kind of objects are H and R? Euclidean spaces? General vector spaces? Hilbert spaces? Is "R", at least, the set of real numbers? The fact that you then use R2 as a specific case implies that it is. Is <a, u> the inner product in H? Finally, what, precisely, is your definition of "conjugate"?
 

1. What is a variational method?

A variational method is a mathematical approach used to find the best approximation to a solution of a given problem. It involves finding a function that minimizes a certain functional, which is a mathematical expression that takes in a function as its input and outputs a scalar value.

2. What is a conjugate of a function?

The conjugate of a function is a mathematical concept that is closely related to the original function. It involves taking the Fourier transform of the original function and then taking the complex conjugate of the resulting function. The conjugate function can provide important information about the original function, such as its symmetry and decay properties.

3. How are variational methods and conjugate functions related?

Variational methods and conjugate functions are closely related in the context of optimization problems. In particular, the conjugate function can be used to construct a dual problem, which is a different formulation of the original problem that is often easier to solve using variational methods. Additionally, the conjugate function can provide useful information about the optimal solution of the original problem.

4. What are some applications of variational methods and conjugate functions?

Variational methods and conjugate functions have a wide range of applications in mathematics, physics, and engineering. They are commonly used in optimization problems, as well as in the study of partial differential equations and functional analysis. They also have practical applications in areas such as image processing, signal processing, and machine learning.

5. What are the advantages of using variational methods and conjugate functions?

Variational methods and conjugate functions offer several advantages over other numerical methods for solving optimization problems. They can handle a wide range of function spaces and are particularly well-suited for problems with constraints. They also provide useful information about the optimal solution, such as the minimum value of the objective function. Additionally, they can often provide more accurate solutions compared to other numerical methods.

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