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

ArticleOriginal scientific text

Title

Authors ¹, ², ²

Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, Brighton BN1 9QH, U.K.
School of Mathematics, University of Wales, Cardiff, Senghennydd Road, Cardiff CF2 4YH, U.K.

We consider the Volterra integral operator

T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})

defined by

(T f) (x) = v (x) \int_{0}^{x} u (t) f (t) d t

. 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 \to \infty} n a_{n} (T) = π^{- 1} \int_{0}^{\infty} | u (t) v (t) | d t

. We also provide upper and lower estimates for the

ℓ^{α}

and weak

ℓ^{α}

norms of

(a_{n} (T))

when 1 < α < ∞.

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