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- W1347251 abstract "AbstractFor a ( mathbb{D} ) function ϑ, form the subspace $$ K_{zvartheta } : = H^2 left( mathbb{D} right) cap (zvartheta H^2 (mathbb{D}))^ bot $$ Since z ϑ H2 ( left( mathbb{D} right) ) is an S-invariant subspace of H2( left( mathbb{D} right) ), then Kzϑ will be an S*-invariant subspace of H2( left( mathbb{D} right) ), where $$ S^* f = frac{{f - f(0)}} {z} $$ is the backward shift operator. It is also easy to see that Kzϑ contains the constants. In fact, by Beurling’s theorem, every S*-invariant subspace, which also contains the constants. takes the form Kzϑ for some ( mathbb{D} )-inner function ϑ. It is well known [16, 26] that functions in Kzϑ have special ‘continuation’ properties. Indeed, recall from (3.3.2) that for h∈L1(m) $$ (Ch)(lambda ) : = int_mathbb{T} {frac{{h(zeta )}} {{zeta - lambda }}dm} (zeta ) $$ denotes the Cauchy transform of h. It is known [16, p. 87] that for any f∈Kzϑ the meromorphic function $$ tilde f(lambda ) : = frac{{C(foverline {zeta vartheta } )(lambda )}} {{C(overline {zeta vartheta } )(lambda )}} $$ (4.1.1) on ( mathbb{D}_e ) is a pseudocontinuation of f in that the non-tangential limits of f (from ( mathbb{D} ) ) and ( tilde f ) (from ( mathbb{D}_e )) are equal almost everywhere on ( mathbb{T} ) . Using the Cauchy integral formula and power series, one can prove the identity $$ tilde f(lambda ) = frac{1} {{vartheta ^* (lambda )}}sumlimits_{n = 1}^infty {frac{1} {{lambda ^{n - 1} }}} widehat{foverline {zeta vartheta } } ( - n), $$ where ( hat cdot (k) ) denotes the k-th Fourier coefficient and $$ vartheta ^* (lambda ): = overline {vartheta left( {begin{array}{*{20}c} 1 {overline{overline lambda } } end{array} } right), } lambda in mathbb{D}_e . $$ (4.1.2) This says that $$ tilde f in frac{1} {{vartheta ^* }}H^2 (mathbb{D}_e ) forall f in K_{zvartheta } . $$ (4.1.3) ." @default.
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- W1347251 date "2009-01-01" @default.
- W1347251 modified "2023-10-17" @default.
- W1347251 title "Nearly invariant and the backward shift" @default.
- W1347251 doi "https://doi.org/10.1007/978-3-0346-0098-9_4" @default.
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