| Preface | p. xiii |
| Notations | p. xix |
| Preliminaries | |
| Polytopes | p. 3 |
| Convex Sets | p. 3 |
| Convex Sets | p. 3 |
| Duality | p. 5 |
| Lines in Convex Sets | p. 6 |
| Faces | p. 8 |
| Polyhedra | p. 10 |
| Convex Polyhedra | p. 10 |
| Simplicial Complexes | p. 13 |
| Polyhedral Cones | p. 13 |
| A Dual Picture | p. 15 |
| Variable Polytopes | p. 16 |
| Two Families of Polytopes | p. 16 |
| Faces | p. 17 |
| Cells and Strongly Regular Points | p. 18 |
| Vertices of ¿X(b) | p. 21 |
| Piecewise Polynomial Functions | p. 23 |
| Hyperplane Arrangements | p. 25 |
| Arrangements | p. 25 |
| Hyperplane Arrangements | p. 25 |
| Real Arrangements | p. 27 |
| Graph Arrangements | p. 29 |
| Graphs Are Unimodular | p. 31 |
| Matroids | p. 33 |
| Cocircuits | p. 34 |
| Unbroken Bases | p. 35 |
| Tutte Polynomial | p. 37 |
| Characteristic Polynomial | p. 41 |
| Identities | p. 43 |
| Zonotopes | p. 44 |
| Zonotopes | p. 44 |
| B(X) in the Case of Lattices | p. 51 |
| Root Systems | p. 55 |
| The Shifted Box | p. 55 |
| The Volume of B(X) | p. 57 |
| The External Activity and Tutte Polynomials | p. 61 |
| Exceptional Types | p. 65 |
| Fourier and Laplace Transforms | p. 69 |
| First Definitions | p. 69 |
| Algebraic Fourier Transform | p. 69 |
| Laplace Transform | p. 70 |
| Tempered Distributions | p. 71 |
| Convolution | p. 72 |
| Laplace Versus Fourier Transform | p. 73 |
| Modules over the Weyl Algebra | p. 77 |
| Basic Modules | p. 77 |
| The Polynomials | p. 77 |
| Automorphisms | p. 78 |
| The Characteristic Variety | p. 80 |
| Differential and Difference Equations | p. 85 |
| Solutions of Differential Equations | p. 85 |
| Differential Equations with Constant Coefficients | p. 85 |
| Families | p. 90 |
| Tori | p. 91 |
| Characters | p. 91 |
| Elementary Divisors | p. 93 |
| Difference Equations | p. 96 |
| Difference Operators | p. 96 |
| Recursion | p. 101 |
| Generalized Euler Recursion | p. 101 |
| Approximation Theory I | p. 103 |
| Approximation Theory | p. 103 |
| A Local Approximation Scheme | p. 103 |
| A Global Approximation Scheme | p. 105 |
| The Strang-Fix Condition | p. 107 |
| The Differentiable Case | |
| Splines | p. 113 |
| Two Splines | p. 113 |
| The Box Spline | p. 113 |
| E-splines | p. 115 |
| Shifted Box Spline | p. 118 |
| Decompositions | p. 119 |
| Recursive Expressions | p. 120 |
| Smoothness | p. 124 |
| A Second Recursion | p. 125 |
| RX as a D-Module | p. 127 |
| The Algebra RX | p. 127 |
| The Complement of Hyperplanes as Affine Variety | p. 127 |
| A Prototype D-module | p. 128 |
| Partial Fractions | p. 129 |
| The Generic Case | p. 131 |
| The Filtration by Polar Order | p. 132 |
| The Polar Part | p. 137 |
| Two Modules in Correspondence | p. 138 |
| The Function TX | p. 141 |
| The Case of Numbers | p. 141 |
| Volume | p. 141 |
| An Expansion | p. 142 |
| Local Expansion | p. 142 |
| The Generic Case | p. 143 |
| The General Case | p. 144 |
| A Formula for TX | p. 145 |
| Jeffrey-Kirwan Residue Formula | p. 145 |
| Geometry of the Cone | p. 150 |
| Big Cells | p. 150 |
| Cohomology | p. 155 |
| De Rham Complex | p. 155 |
| Cohomology | p. 155 |
| Poincaré and Characteristic Polynomial | p. 157 |
| Formality | p. 158 |
| Residues | p. 159 |
| Local Residue | p. 160 |
| Differential Equations | p. 163 |
| The First Theorem | p. 163 |
| The Space D(X) | p. 163 |
| The Dimension of D(X) | p. 165 |
| A Remarkable Family | p. 166 |
| The First Main Theorem | p. 167 |
| A Polygraph | p. 169 |
| Theorem 11.8 | p. 171 |
| A Realization of AX | p. 171 |
| Polar Representation | p. 171 |
| A Dual Approach | p. 174 |
| Parametric Case | p. 177 |
| A Filtration | p. 177 |
| Hilbert Series | p. 178 |
| More Differential Equations | p. 179 |
| A Characterization | p. 179 |
| Regions of Polynomiality | p. 181 |
| A Functional Interpretation | p. 182 |
| General Vectors | p. 183 |
| Polynomials | p. 183 |
| Expansion | p. 183 |
| An Identity | p. 185 |
| The Splines | p. 185 |
| A Hyper-Vandermonde Identity | p. 186 |
| The Discrete Case | |
| Integral Points in Polytopes | p. 191 |
| Decomposition of an Integer | p. 191 |
| Euler Recursion | p. 191 |
| Two Strategies | p. 193 |
| First Method: Development in Partial Fractions | p. 194 |
| Second Method: Computation of Residues | p. 195 |
| The General Discrete Case | p. 196 |
| Pick's Theorem and the Ehrhart Polynomial | p. 196 |
| The Space C[¿] of Bi-infinite Series | p. 197 |
| Euler Maclaurin Sums | p. 200 |
| Brion's Theorem | p. 202 |
| Ehrhart's Theorem | p. 205 |
| The Partition Functions | p. 207 |
| Combinatorial Theory | p. 207 |
| Cut-Locus and Chambers | p. 207 |
| Combinatorial Wall Crossing | p. 208 |
| Combinatorial Wall Crossing II | p. 209 |
| The Difference Theorem | p. 211 |
| Topes and Big Cells | p. 211 |
| A Special System | p. 212 |
| On DM(X) | p. 214 |
| A Categorical Interpretation | p. 217 |
| Special Functions | p. 217 |
| Convolutions and the Partition Function | p. 218 |
| Constructing Elements in DM(X) | p. 219 |
| A Remarkable Space | p. 221 |
| A Basic Formula | p. 221 |
| The Abelian Group F(X) | p. 222 |
| Some Properties of F(X) | p. 223 |
| The Main Theorem | p. 224 |
| Localization Theorem | p. 226 |
| Wall-Crossing Formula | p. 229 |
| The Partition Function | p. 231 |
| The space F(X) | p. 233 |
| Generators of F(X) | p. 234 |
| Continuity | p. 235 |
| Reciprocity | p. 236 |
| The Reciprocity Law | p. 236 |
| Appendix: a Complement | p. 238 |
| A Basis for DM(X) | p. 238 |
| Toric Arrangements | p. 241 |
| Some Basic Formulas | p. 241 |
| Laplace Transform and Partition Functions | p. 241 |
| The Coordinate Algebra | p. 242 |
| Basic Modules and Algebras of Operators | p. 244 |
| Two Algebras as Fourier Transforms | p. 244 |
| Some Basic Modules | p. 246 |
| Induction | p. 248 |
| A Realization | p. 251 |
| The Toric Arrangement | p. 252 |
| The Coordinate Ring as a W(¿) Module | p. 252 |
| Two Isomorphic Modules | p. 259 |
| A Formula for the Partition Function TX | p. 260 |
| The Generic Case | p. 261 |
| Local Reciprocity Law | p. 265 |
| Cohomology of Toric Arrangements | p. 269 |
| de Rham Complex | p. 269 |
| The Decomposition | p. 269 |
| The Unimodular Case | p. 271 |
| A Basic Identity | p. 271 |
| Formality | p. 273 |
| Polar Parts | p. 277 |
| From Volumes to Partition Functions | p. 277 |
| DM(X) as Distributions | p. 277 |
| Polar Parts | p. 278 |
| A Realization of C[¿]/JX | p. 280 |
| A Residue Formula | p. 281 |
| The Operator Formula | p. 282 |
| Local Reciprocity | p. 284 |
| Roots of Unity | p. 285 |
| Dedekind Sums | p. 285 |
| Back to Decomposing Integers | p. 287 |
| Universal Formulas | p. 287 |
| Some Explicit Formulas | p. 289 |
| Computing Dedekind Sums | p. 291 |
| Algorithms | p. 294 |
| Rational Space DMQ(X) | p. 294 |
| Approximation Theory | |
| Convolution by B(X) | p. 301 |
| Some Applications | p. 301 |
| Partition of 1 | p. 301 |
| Semidiscrete Convolution | p. 302 |
| Linear Independence | p. 307 |
| Approximation by Splines | p. 311 |
| Approximation Theory | p. 311 |
| Scaling | p. 311 |
| Mesh Functions and Convolution | p. 313 |
| A Complement on Polynomials | p. 315 |
| Superfunctions | p. 316 |
| Approximation Power | p. 318 |
| An Explicit Superfunction | p. 320 |
| Derivatives in the Algorithm | p. 323 |
| Super and Nil Functions | p. 326 |
| Nil Functions | p. 326 |
| Quasi-Interpolants | p. 329 |
| Projection Algorithms | p. 329 |
| Explicit Projections | p. 330 |
| Stationary Subdivisions | p. 333 |
| Refinable Functions and Box Splines | p. 333 |
| Refinable Functions | p. 333 |
| Cutting Corners | p. 335 |
| The Box Spline | p. 337 |
| The Functional Equation | p. 339 |
| The Role of Box Splines | p. 341 |
| The Wonderful Model | |
| Minimal Models | p. 347 |
| Irreducibles and Nested Sets | p. 347 |
| Irreducibles and Decompositions | p. 347 |
| Nested Sets | p. 348 |
| Non Linear Coordinates | p. 351 |
| Proper Nested Sets | p. 351 |
| Residues and Cycles | p. 353 |
| A Special Case: Graphs and Networks | p. 354 |
| Complete Sets | p. 354 |
| Irreducible Graphs | p. 355 |
| Proper Maximal Nested Sets | p. 356 |
| Two Examples: An and Magic Arrangements | p. 357 |
| Wonderful Models | p. 362 |
| A Minimal Model | p. 362 |
| The Divisors | p. 366 |
| Geometric Side of Residues | p. 368 |
| Building Sets | p. 368 |
| A Projective Model | p. 369 |
| References | p. 373 |
| Index | p. 381 |
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