Skip to main content Accessibility help
×
Home

On the Zeros of Functions with Derivatives in H 1 and H

  • James Wells (a1) (a2)

Extract

Let {zk },0 < |zk | < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk } which will ensure the existence of a function f analytic in D satisfying

(A)

and whose derivative f′ belongs to the Hardy class H 1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.

THEOREM 1. If

(1)

(2)

and

(3)

then there is a function f analytic in D which satisfies (A) and its derivative fbelongs to H 1.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the Zeros of Functions with Derivatives in H 1 and H
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the Zeros of Functions with Derivatives in H 1 and H
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the Zeros of Functions with Derivatives in H 1 and H
      Available formats
      ×

Copyright

References

Hide All
1. Beurling, A., Ensembles exceptionnels, Acta Math. 72 (1940), 113.
2. Carleson, L., Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325345.
3. Carleson, L., On the zeros of functions with bounded Dirichlet integrals, Math. Z. 56 (1952), 289295.
4. Caughran, James G., Analytic functions with Hv derivative, Thesis, University of Michigan, Ann Arbor, 1967.
5. Caughran, James G., Two results concerning the zeros of functions with finite Dirichlet integral, Can. J. Math. 21 (1969), 312316.
6. Hardy, G. H. and Littlewood, J. E., A convergence theorem for Fourier series, Math. Z. 28 (1928), 565634.
7. Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals. II, Math. Z. 34 (1931), 403439.
8. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).
9. Phillip, Novinger, Holomorphic functions with infinitely differentiate boundary values (to appear in Illinois J. Math.).
10. Walter, Rudin, Real and complex analysis (McGraw-Hill, New York, 1966).
11. Shapiro, H. S. and Shields, A. L., On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217229.
12. Taylor, B. A. and Williams, D. L., On closed ideals in Ax, Notices Amer. Math. Soc. 16 (1969), 144.
13. Zygmund, A., Trigonometric series, Vol. II, 2nd ed. (Cambridge Univ. Press, New York, 1959).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

On the Zeros of Functions with Derivatives in H 1 and H

  • James Wells (a1) (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed