Question on orthogonal function with respect to weight.

Thread starter yungman
Start date Nov 25, 2009
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Function Orthogonal Weight

In summary, orthogonal functions with respect to weight are a set of mutually perpendicular functions when weighted by a specific weight function. They have many important applications in mathematics, including in solving differential equations, approximating functions, and constructing basis functions. A well-known example of such functions is the Legendre polynomials, which are used in various mathematical applications. These functions differ from other types of orthogonal functions because of the weight factor, which can result in different sets of orthogonal functions. In signal processing, they are significant in decomposing signals into simpler components, allowing for easier analysis and manipulation.

Nov 25, 2009

yungman

If F(x) and G(x) is orthogonal with respect to weight W(x), does this mean F(x) and G(x) are not necessary orthogonal by themselves?

[tex]\int[/tex]F(x)G(x)W(x)dx=0 do not mean [tex]\int[/tex]F(x)G(x)dx=0

If [tex]\int[/tex]F(x)G(x)dx=0 then W(x)=1

Thanks

Alan

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Nov 25, 2009

marcusl

Science Advisor

Gold Member

Yes, correct on both items.

Nov 25, 2009

yungman

Thanks.
Alan

1. What does it mean for a function to be orthogonal with respect to weight?

Orthogonal functions with respect to weight refer to a set of functions that are mutually perpendicular when weighted by a specific weight function. This means that when these functions are multiplied by the weight function and integrated over a specific interval, the result is equal to zero. This property is useful in many mathematical applications, particularly in the fields of signal processing and functional analysis.

2. How are orthogonal functions with respect to weight used in mathematics?

Orthogonal functions with respect to weight have many important applications in mathematics, including in the solution of differential equations, approximation of functions, and in the construction of basis functions for vector spaces. They are also used in the numerical solution of integrals and in the analysis of physical systems.

3. Can you give an example of orthogonal functions with respect to weight?

One of the most well-known examples of orthogonal functions with respect to weight are the Legendre polynomials, which are used to solve certain types of differential equations and to approximate functions. These polynomials are orthogonal with respect to the weight function w(x) = 1, meaning that when multiplied by this weight and integrated over the interval [-1, 1], the result is equal to zero.

4. How do orthogonal functions with respect to weight differ from other types of orthogonal functions?

Orthogonal functions with respect to weight are different from other types of orthogonal functions because they are weighted by a specific weight function. This weight function can vary depending on the application and can result in different sets of orthogonal functions. Other types of orthogonal functions, such as Fourier series or Legendre polynomials, do not have this weight factor.

5. What is the significance of orthogonal functions with respect to weight in signal processing?

In signal processing, orthogonal functions with respect to weight are used to decompose signals into simpler components. This allows for easier analysis and manipulation of the signal. For example, the discrete Fourier transform, which is used in many signal processing applications, relies on the orthogonality of complex exponential functions with respect to a specific weight function.