Signals obtained during tonic-clonic epileptic seizures are usually neglected for analysis by the physicians due to the presence of noise caused by muscle contractions. Although noise obscures completely the recording, some information about the underlying brain activity can be obtained with wavelet transform by filtering those frequencies associated with muscle activity. One great advantage of this method over traditional filtering is that the filtered frequencies do not modify the pattern of the remanent ones. An accurate analysis of the different seizure stages was achieved using the wavelet packet method, and through the information cost function the brain dynamical behavior can be accessed. ͓S1063-651X͑97͒09712-2͔

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xn) in X with K ⊆ { P n αnxn : (αn) ∈ B p }. We prove that an operator T : X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T * is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

Compact operators whose adjoints factor through subspaces of p , Studia Math. 150 (2002) 17-33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasip-nuclear operators. This relates the p-AP to Saphar's approximation property AP p . As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.

A Banach space X is said to have the k p -approximation property (k p -AP) if for every Banach space Y , the space F (Y , X) of finite rank operators is dense in the space K p (Y , X) of p-compact operators endowed with its natural ideal norm k p . In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the k p -AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the k p -AP and Saphar's approximation property. As another application, the k p -AP is characterized in terms of a trace condition. Finally, we relate the k p -AP to the (p, p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of L p (μ)-spaces.