Approximation of functions
1) Algorithm of approximation:
to) Choice of the approximating function and the norm
b) Verification of the existence of the solution
c) Verification of the oneness of the solution
d) Examination of the characteristics and the property of the solution
and) Calculation of the solution
2) Error weighed between one function to approximate and approximating its:
being S(y) the function to approximate, F(f,y) approximating and w(y) the function weight which concurs in some cases to obtain the optimal one also using one different norm from that one that reputa necessary.
3) Norm p-esima:
con 1 £ p £ ¥
in the case of p = 1 to diminish the norm it corresponds to render minim the area comprised between the two functions while for p®¥ it means to diminish the maximum refuse in absolute value between the two functions, in this last case is spoken about norm of Chebyshev that can be obtained with methods iterati to you in a n° remarkable of steps. The more comfortable norm is had for p = 2 in how much is always possible the calculation of approximating in shape the sluice.
4) Approximation in the norm of Chebyshev:
It is spoken about approximation minimax in how much is proposed to us to render minimum the maximum refuse between approximate function and function approximating by means of opportune chosen of the coefficients of a polynomial or a function rations them that they constitute the polynomial approximating.
5) Better approximating in the norm of Chebyshev:
Necessary and sufficient condition so that F(f,y) continuous function in I) isthe
bestapproximating than… S(y) (is that the curve 
error has at least n alternations in that is is n 1
points in the which 
being n the n° of the
members of the carrier f.
If j it belongs to
with discreet format from n 1 it aims the best approximation is
obtained resolving the linear system 
If
instead j it belongs to with
discreet format from more than n 1 points it is necessary to
characterize with estremale composed from n 1 points on which the best
approximation corresponds to the best approximation on with of
departure.
6) polynomial Approximation minimax:
Polynomial 
the that goes away little
from an assigned continuous function is determined univocamente in how
much the n° of consecutive points in correspondence of the which Pn(y) â?"S(y) assumes
with alternated signs the maximum value is not smaller of n 2, it
verification that is the relations 
e
.
Being famous with estremale it cannot be calculated with methods iterati to you by means of the algorithm of Remel or Stiefel.
7) Algorithm of Remel:
to) Given a function to approximate
S(y) and polynomial approximating a points of the same one are chosen n 2 and the
equations are written , that it can be resolved
regarding the n 2 incognito toj characterizing in such a way the polynomial approximating
b) For every point y the function is estimated error, if alternate it at least n 1 times and does not assume greater absolute values of |And| the optimal one has been caught up otherwise is necessary to vary with of the points selects to you including the points that introduce error with greater absolute value of |And| chosen always with the criterion of the alternation of the error between consecutive points.
8) Algorithm of Stiefel:
to) given the S(functiony) assigned by means of an ended number of points, if they choose some n 2 in order to form a polynomial of order n
b) the amounts are estimated 
with 1 = 1, 2 … , n 2
c) is estimated the maximum
shunting line 
d) it is estimated for interpolation polynomial the Pn(y) using the first and last n 1 points of the entirety {ythe}
and) the error in correspondence of all the points is estimated of with prechosen and the point is chosen correspondent to the higher entirety, such point comes replaced to that more close in which the error of the same sign is had
f) itera sin when the optimal error is not obtained.
9) Approximation minimax by means of functions rations them:
The algorithm of Remes is still used that per², being approximating they rations, gives place to a not linear system that can be resolved in one of the following ways:
to) using artifices in order to eliminate the nonlinearity
The system comes rewritten in the shape 
where 
famous if is simply a
function weight, in particular is proceeded as it follows:
a1) they fix arbitrarily of the coefficients begins them
for B(y) and the correspondent
resolves itself problem of weighed approximation minimax 
a2) he proceeds himself resolving to the generic
step n the problem minimax 
a3) if the procedure converges to a generic step L ha BL-1(y) @ BL(y)
b) Resolving the not linear system
b1) in the direct methods the amount is assumed temporary famous and
b2) indirect methods
