For the extremal problem:
\[
E
n
,
r
(
α
)
:=
min
∥
exp
(
−
|
x
|
α
)
(
x
n
+
⋯
)
∥
L
r
,
α
>
0
,
{E_{n,r}}(\alpha ): = \min \parallel \exp ( - |x{|^\alpha })\,({x^n} + \cdots ){\parallel _{{L^r}}}, \qquad \alpha > 0,
\]
where
L
r
(
0
>
r
⩽
∞
)
{L^r}\,(0 > r \leqslant \infty )
denotes the usual integral norm over
R
{\mathbf {R}}
, and the minimum is taken over all monic polynomials of degree
n
n
, we describe the asymptotic form of the error
E
n
,
r
(
α
)
(
as
n
→
∞
)
{E_{n,r}}(\alpha )\;({\text {as}}\;n \to \infty )
as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case
r
=
2
r = 2
yields new information regarding the polynomials
{
p
n
(
α
;
x
)
=
γ
n
(
α
)
x
n
+
⋯
}
\{ {p_n}(\alpha ;x) = {\gamma _n}(\alpha )\,{x^n} + \cdots \}
which are orthonormal on
R
{\mathbf {R}}
with respect to
exp
(
−
2
|
x
|
α
)
\exp ( - 2|x{|^\alpha })
. In particular, it is shown that a conjecture of Freud concerning the leading coefficients
γ
n
(
α
)
{\gamma _n}(\alpha )
is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud’s conjecture. For
r
=
∞
,
α
>
0
r = \infty ,\alpha > 0
we also prove that, if
deg
P
n
(
x
)
⩽
n
\deg {P_n}(x) \leqslant n
, the norm
∥
exp
(
−
|
x
|
α
)
P
n
(
x
)
∥
L
∞
\parallel \exp ( - |x|^{\alpha })\,{P_n}(x)\parallel _{{L^\infty }}
is attained on the finite interval
\[
[
−
(
n
/
λ
α
)
1
/
α
,
(
n
/
λ
α
)
1
/
α
]
,
where
λ
α
=
Γ
(
α
)
/
2
α
−
2
{
Γ
(
α
/
2
)
}
2
.
\left [ { - {{(n/{\lambda _\alpha })}^{1/\alpha }},{{(n/{\lambda _\alpha })}^{1/\alpha }}} \right ],\quad {\text {where}}\;{\lambda _\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.
\]
Extensions of Nikolskii-type inequalities are also given.