By Anne-Marie Sapse

This publication makes a speciality of what forms of organic difficulties may be solved via quantum mechanical calculations and the way to choose the best equipment.

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When the frame vectors are linearly depen- dent, this uniqueness is indeed lost since we may add an arbitrary function e(m) E %+ to f(m) in the reconstruction formula without changing f (%+ # (0) since the frame vectors are dependent). e. one satisfying the consistency condition. Moreover, as we shall see, it usually happens in practice that the set M , in addition to being a measure space, has some further structure, and the reproducing kernel K(m,m') preserves this structure. For example, M could be a topological space and I< be continuous on M x M , or M could be a differentiable manifold and I< be differentiable, or (as will happen in our treatment of relativistic quantum mechanics) M could be a complex manifold and K be holomorphic.

The smaller 6, the faster the convergence. For a tight frame, (35) S=0 and cG = I , so the series collapses to a single term G-' = c. 3. Generalized fiames and Resolutions of Unity 27 If 0 < S << 1, the frame is called snug (Daubechies [1988a]) and the above formulae merely represent the first terms in the expansions. However, it is found in practice that under certain conditions, this first term gives a good approximation to the reconstruction for snug frames. ” For such frames, the oversampling seems to compensate for the truncation of the series.

The reader interested in learning more about this fascinating subject may consult Aronszajn [1950], Bergman [1950], Meschkowski [1962] or Hille [1972]. Suppose we begin with an arbitrary set M and a set of functions g(m)on M which forms a Hilbert space F under some inner product ( I - ). 3, M happened to a measure space, ,F was %T and the inner product was that of L2(dp). But it is important to + keep in mind that the exact form of the inner product in F does not need to be specified, as far as the general theory of reproducing kernels is concerned.

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