By Vladimir V. Kisil (auth.), Sergei V. Rogosin, Anna A. Koroleva (eds.)
This publication includes survey papers in accordance with the lectures offered on the third foreign iciness university “Modern difficulties of arithmetic and Mechanics” held in January 2010 on the Belarusian nation collage, Minsk. those lectures are dedicated to diversified difficulties of recent research and its purposes. a longer presentation of contemporary difficulties of utilized research will let the reader to get acquainted with new ways of more often than not interdisciplinary personality. the consequences mentioned are program orientated and current new perception into utilized difficulties of becoming value reminiscent of purposes to composite fabrics, anomalous diffusion, and fluid dynamics.
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Examples of covariant transform In this subsection we will provide several examples of covariant transforms. Some of them will be expanded in subsequent sections, however a detailed study of all aspects will not ﬁt into the present work. 7. Let ???? be a Hilbert space with an inner product ⟨⋅, ⋅⟩ and ???? be a unitary representation of a group ???? in the space ???? . Let ???? : ???? → ℂ be a functional ???? → ⟨????, ????0 ⟩ deﬁned by a vector ????0 ∈ ???? . The vector ????0 is often called the mother wavelet in areas related to signal processing or the vacuum state in a quantum framework.
Thus the associated coherent states ????(????, ????) = ????1 (????)????0 (????) = (???? − ????)−1 are completely determined by the point on the unit disk ???? = ????¯???? ¯ −1 . The family of coherent states considered as a function of both ???? and ???? is obviously the Cauchy kernel . The wavelet transform [64, 66] ???? : ????2 (????) → ????2 (????) : ???? (????) → ???????? (????) = ⟨????, ???????? ⟩ is the Cauchy integral ∫ 1 1 ????????. 23) ???? (????) ???????? (????) = 2???????? ???? ????−???? This approach can be extended to an arbitrary connected simply-connected domain.
10 is shown at (a). Contravariant spectrum of the same matrix in the jet space is drawn at (b). 12 is presented at (c). where ????1 = 3 ????????/4 ???? , 4 ????2 = 2 ????5????/6 ???? , 3 ????3 = 2 −????3????/4 ???? , 5 ????4 = 3 −????????/3 ???? . , eigenvalues of ????, and (b) describes spectrum sp ???? arising from the wavelet construction. , the diagonal matrix diag (????1 , ????2 , ????3 , ????4 ) , which even have a diﬀerent dimensionality. At the same time Figure 11(b) completely characterises ???? up to a similarity. Note that each point of sp ???? in Figure 11(b) corresponds to a particular root vector, which spans a primary subrepresentation.