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Two-sided estimates of the approximation numbers of certain Volterra integral operators

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E. Edmunds D., Evans W. D., Harris D. J.

Studia Mathematica

1997

tom 124

nr 1

59-80

We consider the Volterra integral operator $T:L^{p}(ℝ^{+}) → L^{p}(ℝ^{+})$ defined by $(Tf)(x) = v(x)ʃ_{0}^{x} u(t)f(t)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n(T)$ of T are established when 1 < p < ∞. When p = 2 these yield $lim_{n→∞} na_{n}(T) = π^{-1} ʃ_{0}^{∞} |u(t)v(t)|dt$. We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (a_{n}(T)) when 1 < α < ∞.

Ograniczanie wyników

1 Studia Mathematica

1 E. Edmunds D.

1 Evans W. D.

1 Harris D. J.

1 1997

Wyniki wyszukiwania

Two-sided estimates of the approximation numbers of certain Volterra integral operators