How to construct an orthogonal, complete set of functions on the interval $[0, L]$ with given conditions at $x=0,L$?

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The functions must be at least $C^2$, preferably $C^infty$. Completeness for square-integrable functions is sufficient.
The conditions at the points $x=0,L$ prescribe the values of the function and its derivatives.



In my case I need simply $f(0)=f(L) = 0$ and $f'(0)=f'(L) = 0$, but I wonder whether one may consider the general case of arbitrarily prescribed values $f^(n)(0)$ and $f^(n)(L)$?



Does it get easier if I prescribe the values at $x=0$ and demand periodicity, i.e. $f^(n)(L)=f^(n)(0)$ up to some order $n$?
Literature recommendations are welcome. Thank you in advance.




functional-analysis orthogonality orthogonal-polynomials




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    The functions must be at least $C^2$, preferably $C^infty$. Completeness for square-integrable functions is sufficient.
    The conditions at the points $x=0,L$ prescribe the values of the function and its derivatives.



    In my case I need simply $f(0)=f(L) = 0$ and $f'(0)=f'(L) = 0$, but I wonder whether one may consider the general case of arbitrarily prescribed values $f^(n)(0)$ and $f^(n)(L)$?



    Does it get easier if I prescribe the values at $x=0$ and demand periodicity, i.e. $f^(n)(L)=f^(n)(0)$ up to some order $n$?
    Literature recommendations are welcome. Thank you in advance.




    functional-analysis orthogonality orthogonal-polynomials




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      The functions must be at least $C^2$, preferably $C^infty$. Completeness for square-integrable functions is sufficient.
      The conditions at the points $x=0,L$ prescribe the values of the function and its derivatives.



      In my case I need simply $f(0)=f(L) = 0$ and $f'(0)=f'(L) = 0$, but I wonder whether one may consider the general case of arbitrarily prescribed values $f^(n)(0)$ and $f^(n)(L)$?



      Does it get easier if I prescribe the values at $x=0$ and demand periodicity, i.e. $f^(n)(L)=f^(n)(0)$ up to some order $n$?
      Literature recommendations are welcome. Thank you in advance.




      functional-analysis orthogonality orthogonal-polynomials




      share|cite|improve this question













      The functions must be at least $C^2$, preferably $C^infty$. Completeness for square-integrable functions is sufficient.
      The conditions at the points $x=0,L$ prescribe the values of the function and its derivatives.



      In my case I need simply $f(0)=f(L) = 0$ and $f'(0)=f'(L) = 0$, but I wonder whether one may consider the general case of arbitrarily prescribed values $f^(n)(0)$ and $f^(n)(L)$?



      Does it get easier if I prescribe the values at $x=0$ and demand periodicity, i.e. $f^(n)(L)=f^(n)(0)$ up to some order $n$?
      Literature recommendations are welcome. Thank you in advance.





      functional-analysis orthogonality orthogonal-polynomials





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      share|cite|improve this question




      share|cite|improve this question








      edited Jul 23 at 8:30









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      asked Jul 23 at 8:27









      user578943

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