## Alternative Pseudodifferential Analysis: With an Application to Modular Forms, Issue 1935This volume introduces an entirely new pseudodifferential analysis on the line, the opposition of which to the usual (Weyl-type) analysis can be said to reflect that, in representation theory, between the representations from the discrete and from the (full, non-unitary) series, or that between modular forms of the holomorphic and substitute for the usual Moyal-type brackets. This pseudodifferential analysis relies on the one-dimensional case of the recently introduced anaplectic representation and analysis, a competitor of the metaplectic representation and usual analysis. Besides researchers and graduate students interested in pseudodifferential analysis and in modular forms, the book may also appeal to analysts and physicists, for its concepts making possible the transformation of creation-annihilation operators into automorphisms, simultaneously changing the usual scalar product into an indefinite but still non-degenerate one. |

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### Contents

Introduction | 1 |

The Metaplectic and Anaplectic Representations | 11 |

22 Anaplectic Analysis | 16 |

The OneDimensional Alternative Pseudodifferential Analysis | 27 |

31 Ascending Pseudodifferential Analysis | 28 |

32 Classes of Operators | 38 |

33 The Resolvent of the Lowering Operator | 57 |

34 The Composition Formula | 64 |

From Anaplectic Analysis to Usual Analysis | 75 |

42 Ascending Pseudodifferential Calculus in vAnaplectic Analysis | 83 |

Pseudodifferential Analysis and Modular Forms | 93 |

52 Moyal Brackets and RankinCohen Brackets | 106 |

115 | |

117 | |

118 | |