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G/ and gkp Ä kf k1 kgkp : kf Furthermore, if p D 1, then f g is also continuous. G/. 3 Representation Theory Let H1 and H2 be two Hilbert spaces (the corresponding norms and scalar product are simply denoted by k k and h ; i). Suppose that AW H1 ! H1 ; H2 /. Recall that A is an isometry if kAuk D kuk for every u 2 H1 . Since kAuk2 D hAu; Aui D hA Au; ui and kuk2 D hu; ui, the polarization identity implies that A is an isometry if and only if A A D idH1 . Hence, isometries are injective, but they are not necessarily surjective.
A; b/f ; gi ¤ 0 as a continuous function, because neither fO nor gO can identically vanish. R/ is irreducible. R/. R/ and that D C ˚ . R/. R/ that satisfies it is called a wavelet. 2 The Use of Representations 35 Let us now consider the full affine group. a /j2 Ã da jOg. /j2 d jaj This time, for any nonzero , as a ranges in R the numbers a cover R and the change of variable a 7! a/j2 da jaj Ã ÂZ R jOg. 30) which cannot be zero if both f and g are not zero. This proves that full is an irreducible unitary representation of Gfull .
61. G; ; H ; / is a reproducing system. G/, so that, Proof. G/. x/ i D hF; V . x/: The second statement is obvious, because K is of course in the range of V and hence coincides with its projection onto the range. t u We end this section with a classical result due to Duflo and Moore , later reviewed with a slightly different argument in . The proof would require some results on unbounded operators that defy the scope of this chapter. Here we content ourselves with its statement2 and some comments.
Scientific papers, vol.2 by Rayleigh.