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

We identify the torus with the unit interval $[0,1)$ and let$n,\nu\in\mathbb{N}$, $1\leq \nu\leq n-1$ and $N:=n+\nu$. Then we define the(partially equally spaced) knots \[ t_{j}=\{[c]{ll}% \frac{j}{2n}, &\text{for}j=0,...,2\nu, \frac{j-\nu}{n}, & \text{for}j=2\nu+1,...,N-1.]Furthermore, given $n,\nu$ we let $V_{n,\nu}$ be the space of piecewise linearcontinuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$.Finally, let $P_{n,\nu}$ be the orthogonal projection operator of$L^{2}([0,1))$ onto $V_{n,\nu}.$ The main result is\[\lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrowL^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrowL^\infty\|=2+\frac{33-18\sqrt{3}}{13}.\] This shows in particular that theLebesgue constant of the classical Franklin orthonormal system on the torus is$2+\frac{33-18\sqrt{3}}{13}$.

The Lebesgue Constant for the Periodic Franklin System