Some remarks on convolution equations

C. A. Berenstein; M. A. Dostal

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 1, page 55-73
  • ISSN: 0373-0956

Abstract

top
Using a description of the topology of the spaces E ' ( Ω ) ( Ω open convex subset of R n ) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution T , T ∈ E ' . We give applications to a class of distributions T satisfying cv. sing. supp. S * T = cv. sing. supp. S + cv. sing. supp. T for all S ∈ E ' .

How to cite

top

Berenstein, C. A., and Dostal, M. A.. "Some remarks on convolution equations." Annales de l'institut Fourier 23.1 (1973): 55-73. <http://eudml.org/doc/74113>.

@article{Berenstein1973,
abstract = {Using a description of the topology of the spaces $\{\bf E\}^\{\prime \}(\Omega )$ ($\Omega $ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in \{\bf E\}^\{\prime \}$. We give applications to a class of distributions $T$ satisfying\begin\{\}\text\{cv.\} \text\{sing.\} \text\{supp.\}\, S*T=\, \text\{cv.\} \text\{sing.\} \text\{supp.\}\, S+ \,\text\{cv.\} \text\{sing.\} \text\{supp.\}\, T\end\{\}for all $S\in \{\bf E\}^\{\prime \}$.},
author = {Berenstein, C. A., Dostal, M. A.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {55-73},
publisher = {Association des Annales de l'Institut Fourier},
title = {Some remarks on convolution equations},
url = {http://eudml.org/doc/74113},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Berenstein, C. A.
AU - Dostal, M. A.
TI - Some remarks on convolution equations
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 1
SP - 55
EP - 73
AB - Using a description of the topology of the spaces ${\bf E}^{\prime }(\Omega )$ ($\Omega $ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in {\bf E}^{\prime }$. We give applications to a class of distributions $T$ satisfying\begin{}\text{cv.} \text{sing.} \text{supp.}\, S*T=\, \text{cv.} \text{sing.} \text{supp.}\, S+ \,\text{cv.} \text{sing.} \text{supp.}\, T\end{}for all $S\in {\bf E}^{\prime }$.
LA - eng
UR - http://eudml.org/doc/74113
ER -

References

top
  1. [1] A.D. ALEKSANDROV, “Die innere Geometrie der konvexen Flächen”, Berlin, 1955. Zbl0065.15102
  2. [2] I.Ja. BAKEL'MAN, “Geometric methods of solutions of elliptic equations” (in Russian), Moscow, 1964. 
  3. [3] C.A. BERENSTEIN and M.A. DOSTAL, Topological properties of the analytically uniform spaces, Trans. Amer. Math. Soc., 154 (1971), 493-513. Zbl0208.37403MR53 #1252
  4. [4] C.A. BERENSTEIN and M.A. DOSTAL, Fourier transforms of the Beurling classes Dw, E'w', Bull. Amer. Math. Soc., 77 (1971), 963-967. Zbl0229.46038MR44 #5769
  5. [5] C.A. BERENSTEIN and M.A. DOSTAL, “Analytically uniform spaces and their applications to convolution equations”, Lecture Notes in Math., vol. 256, Springer-Verlag, 1972. Zbl0237.47025MR58 #12344
  6. [6] N. BOURBAKI, “Espaces vectoriels topologiques”, Eléments de mathématique, Livre V, Hermann et Cie, Paris, 1953, 1955. 
  7. [7] M.A. DOSTAL, On Fourier image of the singular support of a distribution, Czech. Math. J., 16 (1966), 231-237. Zbl0151.18203MR33 #4593
  8. [8] M.A. DOSTAL, An analogue of a theorem of Vladimir Bernstein and its applications to singular supports of distributions, Proc. London Math. Soc., 19 (1969), p. 553-576. Zbl0176.10302MR40 #3302
  9. [9] M.A. DOSTAL, A complex characterization of the Schwartz space D ( A T T ), Math. Ann., 195 (1972), p. 175-191. Zbl0213.40002MR45 #888
  10. [10] L. EHRENPREIS, Solutions of some problems of division I, Amer. J. Math. 76 (1954), 883-893. Zbl0056.10601MR16,834a
  11. [11] L. EHRENPREIS, Solutions of some problems of division II, Amer. J. Math., 77 (1955), 286-292. Zbl0064.11504MR16,1123a
  12. [12] L. HÖRMANDER, On the range of convolution operators, Ann. Math., 76 (1962), 148-170. Zbl0109.08501MR25 #5379
  13. [13] L. HÖRMANDER, Supports and singular supports of convolutions, Acta Math., 110 (1965), 279-302. Zbl0188.19406MR27 #4070
  14. [14] B. MALGRANGE, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955-1956), 271-355. Zbl0071.09002MR19,280a
  15. [15] B. MALGRANGE, Sur la propagation de la régularité des solutions des équations à coefficients constants, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine, 3 (1959), 432-440. Zbl0109.32002
  16. [16] K. REIDEMEISTER, Uber die singulären Randpunkte eines konvexen Körpers, Math. Ann., 83 (1921), 116-118. Zbl48.0835.03JFM48.0835.03
  17. [17] W. RUDIN, “Function theory in polydisks”, W.A. Benjamin, Inc., New York, 1969. Zbl0177.34101
  18. [18] L. SCHWARTZ, “Théorie des distributions”, I, II, Hermann et Cie, Paris, 1950, 1953. Zbl0037.07301

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.