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By Leopoldo Nachbin (Eds.)
A brand new synthesis of the foundations of quantum mechanics and Relativity is proposed within the context of complicated differential geometry. The positivity of the power signifies that wave features and fields could be prolonged to advanced spacetime, and it truly is proven that this complexification has a superior actual interpretation as a longer part area. The prolonged fields may be stated to be reasonable wavelet transforms of the unique fields. a brand new, algebraic conception of wavelets is built.
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Then f"(mF',s) e 0 whenever m # 0. To apply the reconstruction formula, we must still choose a time interval T < T . By the uncertainty principle, T A V> 1/2. The above condition on Av therefore implies that T > 1/2W. It would thus seem that we could get away with a slightly larger sampling interval T than the Nyquist interval TN = 1/2W. Our reconstruction formula reduces to + f(t) X g(t)-' c h(t - nT)f"(0, nT). 6. Eansforms 43 to be. But a small Av means a large r , hence the samples f(0, nT) are smeared over a large time interval.
In this case, %T = L 2 ( d p ) 1 2 ( M )and the consistency condition reduces to an identity. The reconstruction formula becomes the usual expression for f as a linear combination of the (reciprocal) basis vectors. If we further specialize to the case of a tight frame, then G = AI implies that (h,(hk)=A6: and (hmJhk)=A-'6P, (30) so 'HM and 'HM become orthogonal bases. Requiring A = B = 1 means that ' H M = 'KM reduce to a single orthonormal basis. Returning to the general case, we may summarize our findings as follows: E M ,N M ,K(m,m') and g(m,m') (h" I Ghml) are generalizations of the concepts of basis, reciprocal basis, Kronecker delta and metric tensor to the infinite-dimensional case where, in addition, the requirement of linear independence is dropped.
10) Thus if the Fourier transform R(v) of the basic window is concentrated near u = 0 (which, among other things, means that h(t) must be fairly smooth, since any discontinuities or sharp edges introduce high-frequency components), then that of h,,,,, is concentrated near v' = v. In other words, hu,d is actually a window in phase space or time-frequency. The fact that these windows form a tight frame means that signals may be equally well represented in timefrequency as in time. The resolution of unity is 38 1.