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By Vladimir V. Kisil (auth.), Sergei V. Rogosin, Anna A. Koroleva (eds.)

ISBN-10: 3034804164

ISBN-13: 9783034804165

This publication comprises survey papers in line with the lectures offered on the third overseas iciness tuition “Modern difficulties of arithmetic and Mechanics” held in January 2010 on the Belarusian country collage, Minsk. those lectures are dedicated to various difficulties of contemporary research and its functions. a longer presentation of recent difficulties of utilized research will let the reader to get accustomed to new techniques of quite often interdisciplinary personality. the implications mentioned are software orientated and current new perception into utilized difficulties of transforming into value resembling functions to composite fabrics, anomalous diffusion, and fluid dynamics.

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Paper [91] gives an important example of a covariant transform which provides a simplification even in the case of dim ???? = dim ???? . Erlangen Program at Large: An Overview 23 As we will see below the covariant transform is a close relative of the wavelet transform. The name is chosen due to the following common property of both transformations. 4. 1) intertwines ???? and the left regular representation Λ on ????(????, ???? ): ????????(????) = Λ(????)????. Here Λ is defined as usual by Λ(????) : ???? (ℎ) → ???? (???? −1 ℎ).

In this case the wavelet (covariant) transform is a map into the square integrable functions [26] with respect to the left Haar measure. The map becomes an isometry if ????0 is properly scaled. However square integrable representations and admissible vectors do not cover all interesting cases. 8. 4) ???????? ????????. Its isometric representation on ???? = ???????? (ℝ) is 1 [???????? (????) ???? ](????) = ???? ???? ???? (???????? + ????) , where ???? −1 = (????, ????). We consider the operators ????± : ????2 (ℝ) → ℂ defined by: ∫ 1 ???? (????) ???????? .

7) on the real line generalises to the representation of a square integrable function ???? on the upper half-plane as a sum ∑ ???? (????) = ???????? ???????? (????), ???????? ∈ ???????? (ℝ2± ) ???? for prescribed coefficients ???????? and analytic functions ???????? in question from different irreducible subspaces. Covariant transforms are also meaningful for principal and complementary series of representations of the group SL2 (ℝ), which are not square integrable [64]. 10. Let ???? = SU(2) × Aff be the Cartesian product of the groups SU(2) of unitary rotations of ℂ2 and the ???????? + ???? group Aff.

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Advances in Applied Analysis by Vladimir V. Kisil (auth.), Sergei V. Rogosin, Anna A. Koroleva (eds.)

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