English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In 1996, H. Volkmer observed that the inequality∫   −  1   1    1   |  r  ||   f  ′   |   2   d  x   2   ≤   K  2    ∫   −  1   1|  f  |   2   d  x   ∫   −  1   11  r    f  ′    ′    2   d  x    is satisfied with some positive constant    K  >  0    for a certain class of functions  f  on [ − 1, 1] if the eigenfunctions of the problem    −   y   ′  ′    =  λ    r   (  x  )   y  ,    y   (  −  1  )   =  y   (  1  )   =  0    form a Riesz basis of the Hilbert spaceL   |  r  |   2    (  −  1  ,  1  )    . Here the weight    r  ∈   L  1    (  −  1  ,  1  )is assumed to satisfy    x  r  (  x  )  >  0    a.e. on ( − 1, 1).  We present two criteria in terms of Weyl–Titchmarsh  m ‐functions for the Volkmer inequality to be valid. Note that one of these criteria is new even for the classical HELP inequality. Using these results we improve the result of Volkmer by showing that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if  r  is odd.

On a necessary aspect for the Riesz basis property for indefinite Sturm‐Liouville problems