Fourier inner product

As for the Fourier Transform (FT), the Continuous Wavelet Transform (CWT) is expressed by the mean of an inner product between the signal to analyze s(t) and a set of analyzing function ... 

For any signal f(t), the Fourier transform f(o)i is obtained by inner product of f t) with a sinusoidal wave... 

Covariance NMR spectroscopy allows acquisition of spin-spin correlation in a more efficient way compared to the traditional two-dimensional Fourier-transformation NMR spectroscopy, leading to reduction in the experimental time or increase in the sensitivity of the spectrum obtainable within a given experimental time. This chapter summarizes recent works on covariance NMR, focusing on its applications to solid-state NMR spectroscopy. In addition to a brief survey of the covariance spectroscopy, an open question of whether "inner-product" spectroscopy is more natural is posted. The usefulness of covariance NMR spectroscopy is presented by exploring its applications to solid-state systems of chemical/biological interest. A number of recent reports to further improve its efficiency or to extend the scope of its applicability are reviewed. 

This reciprocal relationship is a general property of Fourier transforms. The -function of the previous paragraph and its transform demonstrate the same reciprocity. To characterise this property more precisely a bracket function, called a scalar or inner (dot) product, is used to define an overlap integral... 

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