Compressive Sensing techniques are used in a short proof of Kashin's decomposition theorem generalized to p-spaces for p ≤ 1. The proof is based on the observation that the null-space of a properly-sized matrix with restricted isometry property is almost Euclidean when endowed with the p-quas...
It is the purpose of this paper to present an alternative–short and simple–proof.doi:10.1016/j.jat.2014.02.001GolitschekManfred v.Elsevier Inc.Journal of Approximation TheoryM.V. Golitschek. On the L∞-norm of the orthogonal projector onto splines: A short proof of A. Shardin's ...
§2. PREPARATION OF THE BOUNDARY AND UNIQUE EXTENSION 2.1. Preparation of a good C∞ boundary. Denote by ||z|| := |z1|2 + ··· + |zn|2 1/2 the Euclidean norm of z = (z1, . . . , zn) ∈ Cn and by Bn(p, δ) := ||z − p|| < δ the open ball of radius ...
bythereproducingpropertytheeigenequationsof LGisS hshTsPJn kc λRJn kc cTRJn kc ImRJn hshTsR 2Jnku 2Jnku ForasquarematrixAcontainingcomplexelements let HStr AA whereA HSisknownastheHilbertSchmidt HS normorFrobeniusnormofA SeeDunfordandSchwarz 1963 1010orHornandJohnson 1990 Chapter5 Notethat HSL...
such that the corresponding one-particle density matrix γψN satisfies ⟨ϕ0,γψNϕ0⟩=N+O(1),with optimal rate in the parameter N. As Corollary of Theorem 1 and Remark 4, we immediately obtain the energy asymptotic of the ground state energy of an N-particle system. Recall, ...
{m \times n}\)is a public matrix, while\(\varvec{s} \in \mathbb {Z}_q^n\)and\(\varvec{e} \in \mathbb {Z}_q^m\)correspond to the randomness of the ciphertext. The vector\(\varvec{e}\)is supposed to be small, i.e.,\(\varvec{c}\)is close to theq-ary lattice\(...
Let T i , i=1,2,,d, be locally compact Hausdorff spaces and let C 0 (T i ) be the Banach space of all scalar valued continuous functions on T i vanishing at infinity (with the supremum norm). Suppose U:∏ 1 d C 0 (T i )→Y is a bounded d- linear operator, where either...
Simple proof of existence of a complex eigenvalue of a complex square matrix... and yet another proof of the fundamental theorem of algebra with linear alg... We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to...
{m \times n}\)is a public matrix, while\(\varvec{s} \in \mathbb {Z}_q^n\)and\(\varvec{e} \in \mathbb {Z}_q^m\)correspond to the randomness of the ciphertext. The vector\(\varvec{e}\)is supposed to be small, i.e.,\(\varvec{c}\)is close to theq-ary lattice\(...
1. Introduction Many real-world signals including physiological signals are irregular in some aspects. They are neither purely periodic nor can they be expressed by an analytic formula. The inherent irregularities imply the uncertainty during the evolution of the underlying process from which the ...