Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ R n and Solutions of the Helmholtz Equation

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ R n and Solutions of... The purpose of this article is to extend to $$\mathbb {R}^{n}$$ R n known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in $$\mathbb {R} ^{n}$$ R n that are the Fourier transform of measures supported in the unit sphere with density in $$L^{2}(\mathbb {S}^{n-1})$$ L 2 ( S n - 1 ) . As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in $$\mathbb {R}^{n}$$ R n belonging to weighted Sobolev type spaces $$\mathcal {H}^{p}$$ H p having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in $$\mathcal {H}^{p}$$ H p and in mixed norm spaces, the continuity of the orthogonal projection $$\mathcal {P}$$ P of $$\mathcal {H}^{2}$$ H 2 onto the Herglotz wave functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ R n and Solutions of the Helmholtz Equation

Loading next page...
 
/lp/springer_journal/reproducing-kernel-for-the-herglotz-functions-in-mathbb-r-n-r-n-and-37ucD05F0I
Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9492-8
Publisher site
See Article on Publisher Site

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial