| Based on operator theory and matrix theory and some research on quantum channels, wavelet-preservers, dilations of frames of a Hilbert space, wavelet neural networks, a quantum wavelet neural networks are es-tablished and discussed. The main contents of this article contain the follow-ing four parts.The first part is about the characterizations of quantum channels.A quantum channel on a quantum system with state space H is defined as a trace-preserving completely positive map on the C*-algebra B(H). By using operator theory and matrix theory, two important properties of a completely positive map are proved and the Stinespring expansion theorem is obtained as an application, which profoundly reveals the essence for completely positive maps. A representation theorem of a completely positive map between any two C*-algebras is proved and representations of a family of C*-algebras with the same Hilbert space are obtained.The second part is devoted to research on wavelet-preservers and dila-tions of frames of a Hilbert space.Some relationships among the frames, w-dependent frames and Riesz base for a separable infinite dimensional complex Hilbert space are estab-lished, and some sufficient conditions for dilation of a frame and a necessary and sufficient condition for dilation of a Riesz basis are proved, respectively. The following results are proved.(1) Algebraic properties of the set W(L2(K)) of all wavelets in L1(R)∩ L2(R) are discussed. It is proved that the set GW(L2(R)):=W(L2(R)) U {0} is closed under multiplication, addition and convolution, and becomes a commutative normed algebra.(2) Bounded linear operators which map a wavelet to a wavelet in L2(R), called wavelet-preservers, are discussed and it is proved that set WP(L2(R)) all of these operators forms a multiplicative semigroup with identity.(3) Bounded linear operators which map a wavelet to a wavelet or0in L2(R) are discussed and it is proved that set GWP(L2(R)) all of these opera-tors consists a unital normed sub-algebra of the Banach algebra B(L2(R)).(4) A sufficient condition for a bounded linear operator to be a wavelet preserver is established.The third part is on the interpolation of wavelet neural networks.Given some one-dimensional interpolation samples, a single hidden layer feed forward neural network (called the wavelet neural network) is introduced by using a continuous wavelet function as activation function. Under certain conditions, the existence of a exact interpolation for the given samples by a wavelet neural network is obtained, and the related approximate interpolation is constructed. Moreover, the error between the exact interpolation and the approximate interpolation of the wavelet neural network is estimated. For the multi-variate case, by using inner products of the interpolation nodes and the method for the one-dimensional case, the exact interpolation and approx-imate interpolation by wavelet neural network for multivariate condition are established, respectively, and an error between the two interpolations is also estimated.The last part is devoted to propose and discuss quantum wavelet neural networks and their interpolation problems.Quantum neuron models based on the universal quantum gates and quan-tum preceptor neural network models are discussed first. A characterization of the continuity of a quantum BP neural network model is then proved. A quantum wavelet neural network is constructed by using a one-dimensional continuous wavelet as activation function. Furthermore, an interpolation prob-lem by a quantum wavelet neural network is discussed in light of changing the real-valued description into quantum state description for the samples. |
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