Series Approximation of Functions: Taylor/McLaurin, Fourier & More

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SUMMARY

The discussion centers on the approximation of functions using series beyond Taylor/McLaurin and Fourier series. It highlights the potential of using Gaussian functions, specifically the form Aexp(-bx^2), for function expansion. The conversation emphasizes that as long as the individual functions in the series form a basis for the function space, such expansions are valid. The sine and cosine functions are cited as a basis for Fourier expansions, reinforcing the importance of functional analysis in this context.

PREREQUISITES
  • Understanding of Taylor and McLaurin series
  • Familiarity with Fourier series and their applications
  • Knowledge of Gaussian functions and their properties
  • Basic concepts of functional analysis and function spaces
NEXT STEPS
  • Research Gaussian function expansions and their applications
  • Explore advanced topics in functional analysis
  • Learn about alternative series approximations, such as Chebyshev polynomials
  • Investigate the convergence properties of various series expansions
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Mathematicians, physicists, and engineers interested in advanced function approximation techniques and series analysis.

gulsen
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I wonder if there's any other series that can be used to approximate a function, other that Taylor/McLaurin and Fourier. For instance, can we expand a function in terms of Gaussians (Aexp(-bx^2))? Maybe something else?
 
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From what I remember of functional analysis (it was a while ago), as long as the individual functions in the series form a basis for the function space, then yes, you can expand it in terms of that series. For example, the sine/cosine functions form a basis and are used in Fourier expansions.
 

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