Contents
Preface vii
Chapter 1. The Gamma function extended to
nonpositive integer points 1
1.1. Homogeneous distributions 1
1.2. A meromorphic extension of the Gamma function 4
1.3. Riesz regularisation 5
1.4. Hadamard’s “finite part” method 7
1.5. Discrepancies 12
Chapter 2. The canonical integral
and noncommutative residue on symbols 15
2.1. Classical and log-polyhomogeneous symbols on
Rd
15
2.2. The noncommutative residue on classical symbols 20
2.3. Closed linear forms on symbol valued forms 22
2.4. The noncommutative residue characterised via Stokes’ property. I 24
2.5. The canonical integral characterised via Stokes’ property. I 24
2.6. Characterisations by means of the translation invariance 28
2.7. The noncommutative residue characterised via its covariance. I 32
Chapter 3. The cut-off regularised integral 35
3.1. Cut-off (or Hadamard finite part) integral 35
3.2. Cut-off integrals and periods 38
3.3. Discrepancies of the cut-off integral 39
3.4. Characterisations by means of Stokes’ property. II 48
3.5. The canonical integral characterised by means of its covariance 49
Chapter 4. The noncommutative residue as a complex residue 53
4.1. Regularised evaluators 53
4.2. Meromorphic extensions of integrals on classical symbols 56
4.3. Extension to log-polyhomogeneous symbols 61
4.4. Invariance properties of the noncommutative residue 68
4.5. Dimensional versus cut-off regularised integrals 70
4.6. Discrepancies of regularised integrals 74
Chapter 5. The canonical sum on noninteger order classical symbols 79
5.1. The Euler-Maclaurin formula 79
5.2. The higher dimensional Euler-Maclaurin formula 83
5.3. Cut-off discrete sums on
Zd
subordinated to convex polytopes 86
5.4.
Zd-translation
invariant linear forms on symbols 91
5.5. The noncommutative residue and
Zd-translation
invariance 93
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