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''of the first kind''

Polynomials that are orthogonal on the interval  $  [ - 1 , 1 ] $
with the weight function

$$ 
h _ {1} ( x)  =  
\frac{1}{\sqrt {1 - x  ^ {2} }}
 ,\ \ 
x \in ( - 1 , 1 ) .
$$

For the standardized Chebyshev polynomials one has the formula

$$ 
T _ {n} ( x)  =  \cos ( n   \mathop{\rm arc}  \cos  x ) ,\ \ 
x \in [ - 1 , 1 ] ,
$$

and the recurrence relation

$$ 
T _ {n+1} ( x)  =  2 x T _ {n} ( x) - T _ {n-1} ( x) ,
$$

by which one can determine the sequence

$$ 
T _ {0} ( x)  =  1 ,\  T _ {1} ( x)  =  x ,\ \ 
T _ {2} ( x)  =  2 x  ^ {2} - 1 ,
$$

$$ 
T _ {3} ( x)  =  4 x  ^ {3} - 3 x ,\  T _ {4} ( x)  =  8 x  ^ {4} - 8 x  ^ {2} + 1 ,
$$

$$ 
T _ {5} ( x)  =  16 x  ^ {5} - 20 x  ^ {3} + 5 x , .  .  . .
$$

The orthonormalized Chebyshev polynomials are:

$$ 
\widehat{T}  _ {0} ( x)  =  
\frac{1}{\sqrt \pi }
 T _ {0} ( x)  =  
\frac{1}{\sqrt \pi }
 ,
$$

$$ 
\widehat{T}  _ {n} ( x)  =  \sqrt {
\frac{2} \pi 
 } T _ {n} ( x)  =  \sqrt {

\frac{2} \pi 
 }  \cos ( n   \mathop{\rm arc}  \cos  x ) ,\  n \geq  1 .
$$

The leading coefficient of  $  T _ {n} ( x) $, 
for  $  n \geq  1 $, 
is  $  2  ^ {n-1} $. 
Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula

$$ 
\widetilde{T}  _ {n} ( x)  =  
\frac{1}{2  ^ {n- 1} }
 T _ {n} ( x)  = \ 

\frac{1}{2  ^ {n- 1} }
  \cos ( n  { \mathop{\rm arc}  \cos }  x ) ,\ \ 
n \geq  1 .
$$

The zeros of  $  T _ {n} ( x) $, 
given by

$$ 
x _ {k}  ^ {( n)}  =  \cos  
\frac{2 k - 1 }{2n}
 \pi ,\ \ 
k = 1 \dots n ,
$$

frequently occur as interpolation nodes in quadrature formulas. The polynomial  $  T _ {n} ( x) $
is a solution of the differential equation

$$ 
( 1 - x  ^ {2} ) y  ^ {\prime\prime} - x y  ^  \prime  +
n  ^ {2} y  =  0 .
$$

The polynomials  $  \widetilde{T}  _ {n} ( x) $
deviate as least as possible from zero on the interval  $  [ - 1 , 1 ] $, 
that is, for any other polynomial  $  \widetilde{F}  _ {n} ( x) $
of degree  $  n $
with leading coefficient 1 one has the following condition

$$ 
\max _ {x \in [ - 1 , 1 ] }  | \widetilde{F}  _ {n} ( x) |  > \ 
\max _ {x \in [ - 1 , 1 ] }  | \widetilde{T}  _ {n} ( x) |
 =  
\frac{1}{2  ^ {n- 1} }
 .
$$

On the other hand, for any polynomial  $  Q _ {n} ( x) $
of degree  $  n $
or less and satisfying

$$ 
\max _ {x \in [ - 1 , 1 ] } \ 
| Q _ {n} ( x) |  =  1 ,
$$

one has, for any  $  x _ {0} \in ( - \infty , - 1 ) \cup ( 1 , \infty ) $, 
the inequality

$$ 
| Q ( x _ {0} ) |  \leq   | T _ {n} ( x _ {0} ) | .
$$

If a function  $  f $
is continuous on the interval  $  [ - 1 , 1 ] $
and if its modulus of continuity  $  \omega ( \delta , f  ) $
satisfies the Dini condition

$$ 
\lim\limits _ {\delta \rightarrow 0 }  \omega ( \delta , f ) \ 
 \mathop{\rm ln}  
\frac{1} \delta 
  =  0 ,
$$

then this function can be expanded in a Fourier–Chebyshev series,

$$ 
f ( x)  =  \sum _{n=0} ^  \infty   a _ {n} \widehat{T}  _ {n} ( x) ,\ \ 
x \in [ - 1 , 1 ] ,
$$

which converges uniformly on  $  [ - 1 , 1 ] $. 
The coefficients in this series are defined by the formula

$$ 
a _ {n}  =  \int\limits _ { - 1} ^ { 1 }  f ( t) \widehat{T}  _ {n} ( t)
 
\frac{dt}{\sqrt {1- t  ^ {2} } }
 .
$$

If the function  $  f $
is  $  p $-
times continuously differentiable on  $  [ - 1 , 1 ] $
and if its  $  p $-
th derivative  $  f  ^ {(} p) $
satisfies a Lipschitz condition of order  $  \alpha $, 
i.e.  $  f  ^ {(} p) \in  \mathop{\rm Lip}  \alpha $, 
then one has the inequality

$$ 
\left | f ( x) - \sum _{k=0} ^ { n }  a _ {k} \widehat{T}  _ {k} ( x) \right |
 \leq   
\frac{c _ {1}   \mathop{\rm ln}  n }{n ^ {p + \alpha } }
 ,\ \ 
x \in [ - 1 , 1 ] ,
$$

where the constant  $  c _ {1} $
does not depend on  $  n $
and  $  x $.

Chebyshev polynomials of the second kind are defined by

$$ 
U _ {n} ( x)  =  
\frac{1}{n+1} T _ {n+ 1} ^ { \prime } ( x)  =  \sin  [ ( n
+ 1 )  { \mathop{\rm arc}  \cos }  x ] 
\frac{1}{\sqrt {1 - x  ^ {2} } } .
$$

These polynomials are orthogonal on the interval  $  [ - 1 , 1 ] $
with weight function

$$ 
h _ {2} ( x)  =  \sqrt {1 - x  ^ {2} } ,\ \ 
x \in [ - 1 , 1 ] .
$$

For any polynomial  $  \widetilde{Q}  _ {n} ( x) $
with leading coefficient 1 one has the inequality

$$ 

\frac{1}{2  ^ {n-} 1 }
  =  \int\limits _ { - 1} ^ { 1 } 
| \widetilde{U}  _ {n} ( x) |  dx  \leq   \int\limits _ { - 1} ^ { 1 } 
| \widetilde{Q}  _ {n} ( x) |  dx .
$$

The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. [[#References|[1]]]). Both systems of Chebyshev polynomials are special cases of [[Ultraspherical polynomials|ultraspherical polynomials]] and [[Jacobi polynomials|Jacobi polynomials]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 23–51  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
</table>