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admissible approximation admissible for Xx angle of amplitude arcsin arga averaging schemes averaging theory ball ft ball of radius bounded bounded set closed ball CLOSURE complex numbers satisfying complex plane condition of exponent CONTAINS THE SPECTRUM continued indefinitely continuous mapping satisfying convex hull cross-Lipschitzian mapping defined defined everywhere demicontinuous dense in 3C determined by 36 discs 48 domain of definition follows ft(T Hence Hilbert space HOlder condition homeomorphism indefinitely and converges inequality interior point interval inverse Lemma linear mappings linear operators linear subspace locally cross-bounded monotone operators non-linear non-vanishing complex number NUMERICAL RANGE CONTAINS obtains one-one one-to-one operator origin orthogonal orthogonal projection point in fi(T positive distance pre-image Proof proved range boundary replace right hand member satisfying a cross sequence of succesive succesive averages T-XI Theorem TL(T Tx-y Txxo-yii unit circle unit disc y a vector ye3C