Completeness of orthonormal functions

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In summary, the completeness of a set of orthonormal functions in a function space is determined by the condition that the summation of the complex conjugates of these functions at two different points in the space is equal to the Kronecker delta. This means that the orthogonal functions must span the entire space in order for this condition to be satisfied. This concept can also be extended to vector spaces, as seen in theorem 1 in the provided source.
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Karthiksrao
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In many areas (say, electrodynamics) we come across expansions of any function in terms of a series of orthonormal functions that span the space. Now the condition for completeness of a set of orthonormal functions in that space is given by

[itex]\sum_{n=1}^\infty U_n^*(x') U_n(x) = \delta(x'-x) [/itex]

where x and x' are two points in the function space.

I am not able to understand intuitively what this is due to.

Why should the summation for the functions at two different points go to zero ? Why would the orthogonal functions not span the entire space if the summation does not go to zero ?

Finally, is there a corresponding relation in vector space ? That will probably give me a better understanding of what is happening, if we extend it to function space..

Thanks a ton!
 
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1. What is the concept of completeness in orthonormal functions?

The concept of completeness in orthonormal functions refers to the ability of a set of functions to approximate any other function within a given domain. In other words, the set of orthonormal functions can represent any function as a linear combination of these functions.

2. How do you prove the completeness of orthonormal functions?

The completeness of orthonormal functions can be proven through the use of the Gram-Schmidt process, which involves orthogonalizing a set of functions to create a set of orthonormal functions. Then, the completeness of the orthonormal functions can be shown by demonstrating that any function in the domain can be represented as a linear combination of these orthogonalized functions.

3. What is the significance of completeness in orthonormal functions?

The completeness of orthonormal functions is significant because it allows for the use of Fourier series and transforms, which have many applications in mathematical and physical sciences. It also provides a powerful tool for representing and analyzing functions and signals in a concise and efficient manner.

4. Can a set of orthonormal functions be complete in one domain but not in another?

Yes, a set of orthonormal functions can be complete in one domain but not in another. This is because the completeness of orthonormal functions is dependent on the choice of inner product, which can vary between different domains. For example, a set of orthonormal polynomials may be complete in the domain of real numbers, but not in the domain of complex numbers.

5. Are there any conditions for a set of orthonormal functions to be complete?

Yes, there are certain conditions that must be met for a set of orthonormal functions to be complete. These include the functions being linearly independent, having finite energy, and satisfying the Parseval's theorem, which states that the total energy of a function can be calculated as the sum of the squared coefficients of the orthonormal functions in its representation.

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