In this paper we study two families of functions F e {F_e} and F o {F_o} , and show how to approximate the functions in the interval [ − 1 , 1 ] [ - 1,1] . The functions are assumed to be real when the argument is real. We define \[ F e = { f : f ( − x ) = f ( x ) , f ( 1 ) = 0 , f ∈ L 2 [ − 1 , 1 ] } {F_e} = \{ f:f( - x) = f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} \] and \[ F o = { f : f ( − x ) = − f ( x ) , f ( 1 ) = 0 , f ∈ L 2 [ − 1 , 1 ] } . {F_o} = \{ f:f( - x) = - f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} . \] Let further P m {\mathcal {P}_m} be the set of all real polynomials of degree not higher than m such that the polynomials belong to the set F e {F_e} if m is even and to the set F o {F_o} if m is odd. We determine the least squares approximation for the function f ∈ F e f \in {F_e} (or F o {F_o} ) in the class P 2 n {\mathcal {P}_{2n}} (or P 2 n + 1 {\mathcal {P}_{2n + 1}} ), with respect to the norm ‖ f ‖ = ( ( f , f ) ) 1 / 2 \left \| f \right \| = {((f,f))^{1/2}} , where the inner product is defined by ( f , g ) = ∫ − 1 1 f ( x ) g ( x ) w ( x ) d x (f,g) = \smallint _{ - 1}^1f(x)g(x)w(x)dx , with f , g ∈ L 2 [ − 1 , 1 ] = L w 2 [ − 1 , 1 ] f,g \in {L^2}[ - 1,1] = L_w^2[ - 1,1] and w ( x ) = ( 1 − x 2 ) λ − 1 / 2 w(x) = {(1 - {x^2})^{\lambda - 1/2}} . We also consider the general case when f is neither an even nor an odd function but f ∈ L 2 [ − 1 , 1 ] f \in {L^2}[ - 1,1] and f ( − 1 ) = f ( 1 ) = 0 f( - 1) = f(1) = 0 . Using the theory of Gegenbauer polynomials we obtain the approximating polynomials in the form \[ ϕ 2 n ( x ) = ∑ k = 1 n d n , k ( 1 − x 2 ) k when f ∈ F e {\phi _{2n}}(x) = \sum \limits _{k = 1}^n {{d_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}}\,f \in {F_e}} \] and \[ ϕ 2 n + 1 ( x ) = x ∑ k = 1 n e n , k ( 1 − x 2 ) k when f ∈ F o . {\phi _{2n + 1}}(x) = x\sum \limits _{k = 1}^n {{e_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}}\,f \in {F_o}.} \] We apply the general theory to the functions f ( x ) = cos ⁡ ( π x / 2 ) f(x) = \cos (\pi x/2) and f ( x ) = J 0 ( a 0 x ) f(x) = {J_0}({a_0}x) , where a 0 = { min x > 0 : J 0 ( x ) = 0 } {a_0} = \{ \min x > 0:{J_0}(x) = 0\} .