MHB Mean square convergence of Fourier series

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Mean square convergence of Fourier series refers to the condition where the sequence of partial sums of the Fourier series converges to the function in the mean square sense. Specifically, this means that the integral of the square of the difference between the function and its Fourier series representation approaches zero as the number of terms increases. The discussion emphasizes the importance of this convergence in ensuring that the Fourier series accurately represents the original function in terms of energy. Additionally, it highlights that mean square convergence is a stronger form of convergence than pointwise convergence. Understanding this concept is crucial for applications in signal processing and harmonic analysis.
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What is the statement of the mean square convergence of Fourier series?
 
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dwsmith said:
What is the statement of the mean square convergence of Fourier series?

You can find the definition of mean square convergence >>here<<.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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