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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A finitely-additive measure $\lambda $ on an infinite-dimensional realHilbert space $E$ which is invariant with respect to shifts and orthogonalmappings has been defined. This measure can be considered as the analog of theLebesgue measure in the sense of its invariance with respect to the abovetransformations. The constructed measure is defined on the ring $\cal R$ ofsubsets of the Hilbert space generated by measurable rectangles. A measurablerectangle is an infinite-dimensional parallelepiped such that the product ofthe lengths of its edges converges unconditionally. The shift androtation-invariant measure is obtained as a continuation of a family ofshift-invariant measures $\lambda _{\cal E}$, where each measure $\lambda_{\cal E}$ is defined on the ring ${\cal R}_{\cal E}$ of measurable rectangleswith edges collinear to the vectors of some orthonormal basis $\cal E$ in thespace $E$. An equivalence relation is introduced on the set of orthonormalbases in terms of the transition matrix from one orthonormal basis to another.The equivalence relation allows to glue measures defined on the subset ringscorresponding to different bases into the one measure $\lambda $ defined on theunique ring $\cal R$. The obtained measure $\lambda $ is invariant with respectto shifts and rotations. The decomposition of the measure $\lambda$ into thesum of mutually singular shift-invariant measures is obtained. The paperdescribes the structure of the space $\cal H$ of numerical functions squareintegrable with respect to the constructed shift and rotation-invariant measure$\lambda $. The decomposition of the space $\cal H$ into the orthogonal sum ofsubspaces corresponding to all possible equivalence classes of bases isobtained. Unitary groups acting by means of orthogonal transformations of theargument in the space $\cal H$ of square integrable functions are investigated.

Measures on a Hilbert space that are invariant with respect to shifts and orthogonal transformations