Exact asymptotic behavior of singular values of a class of integral operators

Milutin R. Dostanić

Czechoslovak Mathematical Journal (1999)

  • Volume: 49, Issue: 4, page 707-732
  • ISSN: 0011-4642

Abstract

top
We find an exact asymptotic formula for the singular values of the integral operator of the form ∫ Ω T ( x , y ) k ( x - y ) · d y L 2 ( Ω ) → L 2 ( Ω ) ( Ω ⊂ ℝ m , a Jordan measurable set) where k ( t ) = k 0 ( ( t 1 2 + t 2 2 + ... t m 2 ) m 2 ) , k 0 ( x ) = x α - 1 L ( 1 x ) , 1 2 - 1 2 m < α < 1 2 and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T .

How to cite

top

Dostanić, Milutin R.. "Exact asymptotic behavior of singular values of a class of integral operators." Czechoslovak Mathematical Journal 49.4 (1999): 707-732. <http://eudml.org/doc/30518>.

@article{Dostanić1999,
abstract = {We find an exact asymptotic formula for the singular values of the integral operator of the form $\int _\{\Omega \} T(x,y)k(x-y) \cdot \mathrm \{d\}y \: L^2 (\Omega )\rightarrow L^2(\Omega )$ ($\Omega \subset \mathbb \{R\}^m$, a Jordan measurable set) where $k(t) = k_0((t^2_1 + t^2_2 + \ldots t^2_m)^\{\frac\{m\}\{2\}\})$, $k_0 (x) = x^\{\alpha -1\} L(\tfrac\{1\}\{x\})$, $\tfrac\{1\}\{2\} - \tfrac\{1\}\{2m\}< \alpha < \tfrac\{1\}\{2\}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$.},
author = {Dostanić, Milutin R.},
journal = {Czechoslovak Mathematical Journal},
keywords = {spectrum of integral operator; singular value; asymptotic behavior},
language = {eng},
number = {4},
pages = {707-732},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exact asymptotic behavior of singular values of a class of integral operators},
url = {http://eudml.org/doc/30518},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Dostanić, Milutin R.
TI - Exact asymptotic behavior of singular values of a class of integral operators
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 4
SP - 707
EP - 732
AB - We find an exact asymptotic formula for the singular values of the integral operator of the form $\int _{\Omega } T(x,y)k(x-y) \cdot \mathrm {d}y \: L^2 (\Omega )\rightarrow L^2(\Omega )$ ($\Omega \subset \mathbb {R}^m$, a Jordan measurable set) where $k(t) = k_0((t^2_1 + t^2_2 + \ldots t^2_m)^{\frac{m}{2}})$, $k_0 (x) = x^{\alpha -1} L(\tfrac{1}{x})$, $\tfrac{1}{2} - \tfrac{1}{2m}< \alpha < \tfrac{1}{2}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$.
LA - eng
KW - spectrum of integral operator; singular value; asymptotic behavior
UR - http://eudml.org/doc/30518
ER -

References

top
  1. Asymptotic behavior of the spectrum of weakly polar integral operators, Izv. Akad. Nauk. SSSR, Ser. Mat. Tom 34 (1970), N5, 1151–1168. (1970) 
  2. Eigenvalues of weakly singular integral operators, J. London Math. Soc. (2) 41 (1990), 323–335. (1990) MR1067272
  3. An estimation of singular of convolution operators, Proc. Amer. Math. Soc. 123 (1995), N5, 1399–1409. (1995) MR1246522
  4. Introduction to the Theory of Linear Nonselfadjoint Operators, in “Translation of Math. monographs” Vol. 18, Amer. Math. Soc., Providence, R.I., 1969. (1969) MR0246142
  5. 10.1307/mmj/1028990026, Mich. Math. J. 3 (1955/56), 141–148. (1955/56) MR0085650DOI10.1307/mmj/1028990026
  6. Asymptotic behavior of the spectrum of integral operators with a singularity on the diagonal, Math. USSR Sb. T 94 (136) N3 (7), 1974, pp. 445–451. (1974) MR0361935
  7. Lectures on Mathematics Physics, Moscow, 1968. (1968) 
  8. 10.1016/0022-247X(90)90423-D, J. of Math. Analysis and Applications 145 (1990), 573–605. (1990) Zbl0699.42001MR1038180DOI10.1016/0022-247X(90)90423-D
  9. 10.1017/S0013091500016254, Proceeding of the Edinburgh Math. Soc. 22 (1979), 137–144. (1979) MR0549459DOI10.1017/S0013091500016254
  10. Some results on the asymptotic behavior of eigenvalues for a class of integral equations with translations kernels, J. Math. Mech. 12 (1963), 619–628. (1963) MR0150551
  11. Asymptotic of the spectrum of abstract integral operators, Trudy. Moscow Mat. Obšč. T. 34 (1977), 105–128. (1977) MR0461221
  12. Fractional Integrals and Derivative and Some Applications, Minsk, 1987. (1987) 
  13. Regularly Varying Functions, Springer Verlag, 1976. (1976) Zbl0324.26002MR0453936
  14. 10.1007/BF00282438, Arch. Rational Mech. Analys. 17 (1964), 215–229. (1964) Zbl0183.11701MR0169015DOI10.1007/BF00282438
  15. 10.1090/S0002-9947-1963-0155161-0, Trans. Amer. Math. Soc. 109 (1963), 278–295. (1963) Zbl0178.14501MR0155161DOI10.1090/S0002-9947-1963-0155161-0

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.