Certain Banach algebras in connection with Besov spaces (Q1577942)

Denote by \(C(K)\) the space of all complex-valued continuous functions on the compact set \(K\subset\mathbb{C}\) and by \(\Pi_{n}\) the subspace of all polynomials of degree at most \(n\), and let \(\Pi=\bigcup _{n}\Pi_{n}.\) For \(f\in C( K) \) put \(E_{k}( f) =\inf\{ \|f-p\|_{\infty}:p\in\Pi_{k}\} ,\) \(k=0,1,...,\) and for \(q>0\) let \(S_{q}( f) =( \sum_{k=0}^{\infty}E_{k}^{q}( f)) ^{1/q}\) the \(q\) -minimax series of \(f,\) and for \(\gamma\geq 0\) let \(S_{q,\gamma}( f) =( \sum_{k=0}^{\infty}( k+2) ^{\gamma q-1}E_{k}^{q}( f)) ^{1/q}\) the \(\gamma-q\)-minimax series of \(f.\) One proves the following results: 1\(^{\text{0}}\) The space \(C_{0,\gamma,q}( K) =\{ f\in C( K) :S_{q,\gamma}( f) <\infty,f( z_{0}) =0\} ,\) where \(z_{0}\in K\) is fixed and \(q\geq 1,\) is a Banach algebra with respect to the norm \(S_{q,\gamma}.\) 2\(^{\text{0}}\) The Besov space \[ B_{\gamma,q}^{\infty}( K) =\{ f\in C( K) :\|f\|_{B_{\gamma,q}^{\infty }}:=( \|f\|_{\infty}^{q}+S_{q,\gamma}^{q}( f)) ^{1/2}<\infty\} \] is a Banach algebra. In particular, one obtains that the Besov space \(B_{\gamma ,q}^{\infty}( [ a,b]) \) is a Banach algebra and, if \(\gamma<0,\) then \(B_{\gamma,q}^{\infty}( K) =C( K) .\)