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Volumes of unit balls of Lorentz spaces
Doležalová, Anna ; Vybíral, Jan (advisor)
This thesis studies the volume of the unit ball of finite-dimensional Lorentz sequence spaces p,q n . Lorentz spaces are a generalisation of Lebesgue spaces with a quasinorm described by two parameters 0 < p, q ≤ ∞. The volume of the unit ball Bp,q n of a general finite-dimensional Lorentz space was so far an unknown quantity, even though for the Lebesgue spaces it has been well-known for many years. We present the explicit formula for Vol(Bp,∞ n ) and Vol(Bp,1 n ). We also describe the asymptotic behaviour of the n-th root of Vol(Bp,q n ) with respect to the dimension n and show that [Vol(Bp,q n )]1/n ≈ n−1/p for all 0 < p < ∞, 0 < q ≤ ∞. Furthermore, we study the ratio of Vol(Bp,∞ n ) and Vol(Bp n). We conclude by examining the decay of entropy numbers of embeddings of the Lorentz spaces.
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Entropy numbers
Kossaczká, Marta ; Vybíral, Jan (advisor) ; Hencl, Stanislav (referee)
In this work we study entropy numbers of linear operators. We focus on entropy numbers of identities between real finite-dimensional sequence spaces and present detailed proofs of their estimates. Then we describe relation between entropy numbers of identities between real spaces and between complex spaces, which allows us to establish similar estimates for complex spaces. Powered by TCPDF (www.tcpdf.org)
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Volumes of unit balls of Lorentz spaces
Doležalová, Anna ; Vybíral, Jan (advisor) ; Lang, Jan (referee)
This thesis studies the volume of the unit ball of finite-dimensional Lorentz sequence spaces p,q n . Lorentz spaces are a generalisation of Lebesgue spaces with a quasinorm described by two parameters 0 < p, q ≤ ∞. The volume of the unit ball Bp,q n of a general finite-dimensional Lorentz space was so far an unknown quantity, even though for the Lebesgue spaces it has been well-known for many years. We present the explicit formula for Vol(Bp,∞ n ) and Vol(Bp,1 n ). We also describe the asymptotic behaviour of the n-th root of Vol(Bp,q n ) with respect to the dimension n and show that [Vol(Bp,q n )]1/n ≈ n−1/p for all 0 < p < ∞, 0 < q ≤ ∞. Furthermore, we study the ratio of Vol(Bp,∞ n ) and Vol(Bp n). We conclude by examining the decay of entropy numbers of embeddings of the Lorentz spaces.
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Entropy numbers
Kossaczká, Marta ; Vybíral, Jan (advisor) ; Hencl, Stanislav (referee)
In this work we study entropy numbers of linear operators. We focus on entropy numbers of identities between real finite-dimensional sequence spaces and present detailed proofs of their estimates. Then we describe relation between entropy numbers of identities between real spaces and between complex spaces, which allows us to establish similar estimates for complex spaces. Powered by TCPDF (www.tcpdf.org)
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