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Let      D D       be a bounded Jordan domain. Define      A q ( D ) {A_q}(D)      , the Bers space, to be the Banach space of holomorphic functions on      D D      , such that      ∬  D | f | λ  D 2 −  q d x d y \iint _D {|f|\lambda _D^{2 - q}dxdy}       is finite, where      λ  D ( z ) {\lambda _D}(z)       is the Poincaré metric for      D D      . It is well known that the polynomials are dense in      A q ( D ) {A_q}(D)       for      2 ≦  q &gt; ∞  2 \leqq q &gt; \infty       and we shall prove they are dense in      A q ( D ) {A_q}(D)       for      1 &gt; q &gt; 2 1 &gt; q &gt; 2       if the boundary of      D D       is rectifiable. Also some remarks are made in case the boundary of      D D       is not rectifiable.

On polynomial density in 𝐴_{𝑞}(𝐷)