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Fix a positive integer      n n       and      1 &gt; p &gt; ∞  1&gt;p&gt;\infty      . We provide expressions for the weighted      L p L^{p}       distance  \[    inf f ∫  0 2 π  | 1 −  f | p w d λ  , \inf _{f} \int ^{2 \pi }_{0} | 1 - f |^p w\,d\lambda ,     \]  where      d λ  d\lambda       is normalized Lebesgue measure on the unit circle,      w w       is a nonnegative integrable function, and      f f       ranges over the trigonometric polynomials with frequencies in  \[    S 1 = { …  , −  3 , −  2 , −  1 } ∪  { 1 , 2 , 3 , …  , n } , S_1 = \{ \ldots , -3, -2, -1\}\cup \{ 1, 2, 3,\ldots , n\},     \]   \[    S 2 = { …  , −  3 , −  2 , −  1 } ∖  { −  n } , S_2=\{ \ldots , -3, -2,-1\}\setminus \{-n\},     \]  or  \[    S 3 = { …  , −  3 , −  2 , −  1 } ∪  { n } . S_3 =\{ \ldots , -3, -2, - 1\}\cup \{n\}.     \]  These distances are related to other extremal problems, and are shown to be positive if and only if      log ⁡  w \log w       is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with      w w      .

Some extremal problems in 𝐿^{𝑝}(𝑤)