If we are going to approximate a smooth function with legendre polynomials then is there relation that guarantees that the expansion coefficients decay ,Can we calculate the error bound of the approximation? If any one know about the problem I will be thankful if you let me know
I don't know the answer directly, but the following papers may be helpful:
SM Jung (2009) Approximation of analytic functions by Legendre functions, NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 71(12) 103-108.
I Bashmakova and S Rafalson (1996) On the approximation of continuous functions by Fourier-Legendre sums, JOURNAL OF APPROXIMATION THEORY 86(2) 197-215.
You may use the following result, which hlods for all expansion into orthogonal polynomials:
energy of f(x) = energy of f_n(x) + energy of error
with
f(x) = your funtcion to decompose
f_n(x) = approximation with n legendre polynomials
energy of a function = integral of the square of the function (in this case) between -1 and 1
this will give you an estimation of the energy of the error
I hope this makes sense to you
Dear Giuseppe Papari,
I am very thankfull to you for your answer.Can you gave me some hints that how can I guarantee that the coefficients of the series ( series of legendre approximation ) decay .How can I calculate the decay rate of the coefficients.
I don't know the answer directly, but the following papers may be helpful:
SM Jung (2009) Approximation of analytic functions by Legendre functions, NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 71(12) 103-108.
I Bashmakova and S Rafalson (1996) On the approximation of continuous functions by Fourier-Legendre sums, JOURNAL OF APPROXIMATION THEORY 86(2) 197-215.
Thanks you so much Mr Michael Wendl and Saeed Kazim .I found the work interesting and helpfull
You may also want to check the book Trefethen's book on "Approximation Theory and Approximation Practice" where he talks about these issues. Start taking a look at chapter 15 & 17.
Dear Everen Yarman,I am thank full for your suggestion Unfortunately I dont have access to the book can you please send it for me if you have a soft copy.
Dear @Hammad, approximating using the Legendre polynomials as basis has the same rules of convergence like any series expansion; it has only the advantage of simplicity and direct computation of the series coefficients because of orthogonality w.r.t. weight function 1.
Legendre polynomials can be built from the usual basis {1,x,x2,x3,...,xn} by using the common Gram-Schmidt orthogonalization process.
Dear Demetris Christopoulos and Abedallah M Rababah,Many thanks for usefull suggestion.
Please check the following paper on the convergence rates of Legendre approximation.
.
http://www.ams.org/journals/mcom/2012-81-278/S0025-5718-2011-02549-4/S0025-5718-2011-02549-4.pdf
Dear H. E Lehtihet,
Many thanks ,I got the thing which I needed.I am thank full to all contributors for their scientific answers.
Dear Hammad Khalil,
If you are satisfactory of the paper given by H.E. Lehtihet for ur questions. Can u share it with me by sending the paper to the following email: [email protected] I have tried to download unfortunately fail. Tq
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