Howe, R., On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) (3) (1980), 821--843.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Check that the following are noncommutative Lie (and thus continuous) groups: i) 76, Chap. 7] The ax b group: set of elements (a, b) a with the group law: a, b) a # , b # ) aa # , ab # b) The identity is (1, 0) and (a, b) 1 = a 1 , b a) ii) The Heisenberg group [25], 76, Chap. 1] a set of triples of real numbers (s, x, y) with the group multiplication: s, x, y) s # , x # , y # ) s s # 1 2 (x # y xy # ) x x # , y y # ) A.1.1) The identity is (0, 0, 0) and (s, x, y) 1 = s, x, y) iii) The SL 2 (R) group [26, 53] a set of 2 2 ....

....det = ad bc equal to 1 and the group law coinciding with matrix multiplication: a # b # c # d # aa # bc # ab # bd # ca # dc # cb # dd # . The identity is the unit matrix and c . The above three groups are behind many important results of real and complex analysis [25, 26, 53] and we meet them many times in these notes. A.2 Homogeneous Spaces and Invariant Measures While abstract group are a suitable language for investigation of their general properties we meet groups in applications as transformation groups acting on a set X. Let X be a set and let be defined an ....

Roger Howe. On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.), 3(2):821--843, 1980.

....found in [15, 37, 39] 4. 1 The Heisenberg Group and Schrodinger Representation We will consider a realization of the previous results in a particular cases of the Fourier transform and Segal Bargmann [5, 59] type spaces F p (C ) They arise from representations of the Heisenberg group H [25, 28, 69] on L p (R ) The Lie algebra h n of H spanned by fT; P j ; Q j g, n = 1; n is defined by the commutation relations: P i ; Q j ] T ffi ij : 4.1) They are known from quantum mechanics as the canonical commutation relations of coordinates and momentum operators. An element g 2 ....

....dx = 2) b f(y)e dx: The condition 2.19.4 MW : B B follows from the composition of two facts W : B B and almost identical to it M : B B, which are proved in standard analysis textbooks (see for example [33, x IV.2.3] To check scaling (2. 14) according to the tradition in analysis [28] we take a probe vector p 0 = e 2 B. Due to well known formula R 1 dy = 2) 1=2 of real analysis we have hbp 0 (x) l 0 i b x dx; l 0 Gamman Z Z Z dye 2xw dx dw n=2 dy = hp 0 ; l 0 i : Thus our scaling is correct. W and M intertwine the left regular ....

Roger Howe. On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.), 3(2):821--843, 1980.

....F (k; l; d . The group convolution and L (H ) are de ned with respect to this measure, the involution is de ned as (17) F (k; l; F ( k; l; F ( k; l; e Thus both and obviously depend on . The two convolutions and are closely related. Following [9, 13] we de ne an embedding j of (H ) as follows: if a 2 (18) j(a) k; l; a(k; l) To describe the range of j, we expand a function F on H into a Fourier series with respect to the third coordinate . Note that if F 2 L (H ) then for xed k; l 2 the function F (k; l; ....

R. Howe. On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.), 3(2):821-843, 1980.

....of optical wave guides, which includes the technologies of lasers and fiber optics. This is explained in [S] The Heisenberg group is the simplest nonabelian Lie group and as such, its ordinary representation theory has been studied extensively. For further bibliography see the survey articles [H] and [R] In a forthcoming paper, I will show how the results of the present paper can be applied to the projective representation theory of the Heisenberg group, which is what [R] and [S] are really studying. In order to address this question for the Heisenberg groups, it is convenient to have ....

R.Howe, On the Role of the Heisenberg Group in Harmonic Analysis, Bull. Amer. Math. Soc 3 (1980), 821---843.

....structures of coherent states gives us the uniform theory for the Bargmann, Bergman and Szego projectors at the Segal Bargmann (Fock) Bergman and Hardy spaces respectively. Applications to wavelets (and other) theory are also possible. The author gratefully acknowledge his inspiration by papers [17, 18, 28, 29, 51]. Besides, there is some interference with the coherent recent paper of Folland [22] The author is also grateful to Dr. V. V. Kravchenko and Prof. N. L. Vasilevski for helpful discussions. 2 Relative Convolutions 2.1 Definitions and Notations Let G be a connected simply connected Lie group, let ....

....of ordered operators in a very similar way. Anderson [1] introduced a generalization of the Weyl calculus for arbitrary set of self adjoint operators in a Banach space exactly by formula (5) A description of different operator calculuses may be found in [43] But it was shown by R. Howe [28, 29], that success of the original Weyl calculus is intimately connected with the structure of the Heisenberg group and its different representations. Thus one can obtain a new fruitful branch in this direction making an assumption, that the operators X j in (5) are not arbitrary but are connected ....

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Roger Howe. On the role of the Heisenberg group in harmonic analysis. Bull. of the AMS (New Series), 3(2):821--843, 1980.

.... context there are many different approaches to quantization with a large number of nice results (for example, the PDO calculus [13] 14] 27] 31] and the Toeplitz operators in the Segal Bargman space [5] 6] The richness of the calculus is rooted in the structure of the Heisenberg group [15], 16] At a second level one can construct a functional calculus from an arbitrary finite set of self adjoint operators fT j g; 1 j m. In this case the number of possible quantizations and the precise results obtained is much smaller [22] and sometimes only a more or less formal power series ....

....of the Weyl calculus for an arbitrary set fT j g of self adjoint operators in a Banach space by the formula K = 2 ) GammaN=2 Z R N b k(x 1 ; x 2 ; xN ) e i P N 1 x j T j dx; 6) A description of several different operator calculus may be found in [22] It was shown by R. Howe [15], 16] that the success of the original Weyl calculus is intimately connected with the structure of the Heisenberg group and its different representations. Remark 2.5 It should be noted, that besides the definition of the Weyl functional calculus in accordance with the Procedure 2.3 it can be ....

R. Howe. On the role of the Heisenberg group in harmonic analysis. Bull. of the AMS (New Series), 3(2):821--843, 1980.

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Howe, R., On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) (3) (1980), 821--843.

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