How to compute inner product in the Hardy space

In summary, the inner product in the Hardy space is defined as the integral of the product of two functions, with respect to the normalized Lebesgue measure on the unit circle. It is linear, positive definite, and conjugate symmetric, and can be used to measure the similarity between two functions by calculating the angle between them. The inner product can also be used to determine the norm of a function, which is the square root of the inner product of the function with itself. In practice, the inner product can be calculated using numerical integration methods or software packages with built-in functions for computing inner products.
  • #1
LikeMath
62
0
Hi,
Let [itex]H^2[/itex] be the Hardy space on the open unit disk.
I am wondering how can I compute the following inner product

[itex]<\frac{1}{\left(1-\overline{\alpha_1} z\right)^2}\frac{z-\alpha_2}{1-\overline{\alpha_2} z},\frac{z}{\left(1-\overline{\alpha_1} z\right)^2}>,[/itex]

where [itex]\alpha_1,\alpha_2[/itex] in the unit disk.

I tried to expand the functions but it became complicated. Also it did not work with the integration.

Is there an idea to be tried?

Thanks in advanced
Likemath
 
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  • #2
Any idea?
 

Related to How to compute inner product in the Hardy space

1. How is the inner product defined in the Hardy space?

The inner product in the Hardy space is defined as the integral of the product of two functions, with respect to the normalized Lebesgue measure on the unit circle. In other words, it is the sum of the products of the two functions evaluated at each point on the unit circle, multiplied by the corresponding point's measure.

2. What are the properties of the inner product in the Hardy space?

The inner product in the Hardy space is linear, positive definite, and conjugate symmetric. This means that it follows the properties of a standard inner product, such as being additive, having a positive definite inner product space, and satisfying the conjugate symmetry property (i.e. <f,g> = <g,f>).

3. How is the inner product used to measure the similarity between two functions in the Hardy space?

The inner product can be used to measure the similarity between two functions in the Hardy space by calculating the angle between them. The closer the angle is to 0, the more similar the functions are. This is because the inner product is larger when the two functions are more aligned and smaller when they are less aligned.

4. Can the inner product be used to determine the norm of a function in the Hardy space?

Yes, the inner product can be used to determine the norm of a function in the Hardy space. The norm of a function is defined as the square root of the inner product of the function with itself. In other words, the norm is the square root of the integral of the function squared over the unit circle.

5. How can the inner product be calculated in practice?

The inner product in the Hardy space can be calculated using various numerical integration methods, such as the trapezoidal rule or Simpson's rule. Alternatively, it can also be calculated using software packages that have built-in functions for computing inner products in function spaces.

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