{"product_id":"distributions-9781786305251","title":"Distributions","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThis book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same “weak” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.\u003cbr\u003e\u003cbr\u003eThis operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eIntroduction ix\u003c\/p\u003e \u003cp\u003eNotations xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1 Semi-Normed Spaces and Function Spaces 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1. Semi-normed spaces 1\u003c\/p\u003e \u003cp\u003e1.2. Comparison of semi-normed spaces 4\u003c\/p\u003e \u003cp\u003e1.3. Continuous mappings 6\u003c\/p\u003e \u003cp\u003e1.4. Differentiable functions 8\u003c\/p\u003e \u003cp\u003e1.5. Spaces C\u003csup\u003em\u003c\/sup\u003e (Ω; E), C\u003csup\u003em\u003c\/sup\u003e\u003csub\u003eb\u003c\/sub\u003e (Ω; E) and C\u003csup\u003em\u003c\/sup\u003e\u003csub\u003eb\u003c\/sub\u003e (Ω; E) 11              \u003c\/p\u003e \u003cp\u003e1.6. Integral of a uniformly continuous function 14\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2 Space of Test Functions 17\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1. Functions with compact support 17\u003c\/p\u003e \u003cp\u003e2.2. Compactness in their whole of support of functions 19\u003c\/p\u003e \u003cp\u003e2.3. The space D(Ω) 21\u003c\/p\u003e \u003cp\u003e2.4. Sequential completeness of D(Ω) 24\u003c\/p\u003e \u003cp\u003e2.5. Comparison of D(Ω) to various spaces 26\u003c\/p\u003e \u003cp\u003e2.6. Convergent sequences in D(Ω) 28\u003c\/p\u003e \u003cp\u003e2.7. Covering by crown-shaped sets and partitions of unity 33\u003c\/p\u003e \u003cp\u003e2.8. Control of the CK m (Ω)-norms by the semi-norms of D(Ω) 35\u003c\/p\u003e \u003cp\u003e2.9. Semi-norms that are continuous on all the CK ∞ (Ω) 38\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3 Space of Distributions 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1. The space D ′ (Ω; E) 41\u003c\/p\u003e \u003cp\u003e3.2. Characterization of distributions 46\u003c\/p\u003e \u003cp\u003e3.3. Inclusion of C(Ω; E) into D ′ (Ω; E) 48\u003c\/p\u003e \u003cp\u003e3.4. The case where E is not a Neumann space 53\u003c\/p\u003e \u003cp\u003e3.5. Measures 57\u003c\/p\u003e \u003cp\u003e3.6. Continuous functions and measures 63\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4 Extraction of Convergent Subsequences 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1. Bounded subsets of D ′ (Ω; E) 65\u003c\/p\u003e \u003cp\u003e4.2. Convergence in D ′ (Ω; E) 67\u003c\/p\u003e \u003cp\u003e4.3. Sequential completeness of D ′ (Ω; E) 69\u003c\/p\u003e \u003cp\u003e4.4. Sequential compactness in D ′ (Ω; E) 71\u003c\/p\u003e \u003cp\u003e4.5. Change of the space E of values 74\u003c\/p\u003e \u003cp\u003e4.6. The space E-weak 76\u003c\/p\u003e \u003cp\u003e4.7. The space D ′ (Ω; E-weak) and extractability 78\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5 Operations on Distributions 81\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1. Distributions fields 81\u003c\/p\u003e \u003cp\u003e5.2. Derivatives of a distribution 84\u003c\/p\u003e \u003cp\u003e5.3. Image under a linear mapping 91\u003c\/p\u003e \u003cp\u003e5.4. Product with a regular function 94\u003c\/p\u003e \u003cp\u003e5.5. Change of variables 100\u003c\/p\u003e \u003cp\u003e5.6. Some particular changes of variables 107\u003c\/p\u003e \u003cp\u003e5.7. Positive distributions 109\u003c\/p\u003e \u003cp\u003e5.8. Distributions with values in a product space 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6 Restriction, Gluing and Support 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1. Restriction 117\u003c\/p\u003e \u003cp\u003e6.2. Additivity with respect to the domain 121\u003c\/p\u003e \u003cp\u003e6.3. Local character 122\u003c\/p\u003e \u003cp\u003e6.4. Localization-extension 125\u003c\/p\u003e \u003cp\u003e6.5. Gluing 128\u003c\/p\u003e \u003cp\u003e6.6. Annihilation domain and support 130\u003c\/p\u003e \u003cp\u003e6.7. Properties of the annihilation domain and support 133\u003c\/p\u003e \u003cp\u003e6.8. The space DK ′ (Ω; E) 137\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7 Weighting 141\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1. Weighting by a regular function 141\u003c\/p\u003e \u003cp\u003e7.2. Regularizing character of the weighting by a regular function 144\u003c\/p\u003e \u003cp\u003e7.3. Derivatives and support of distributions weighted by a regular weight 148\u003c\/p\u003e \u003cp\u003e7.4. Continuity of the weighting by a regular function 150\u003c\/p\u003e \u003cp\u003e7.5. Weighting by a distribution 153\u003c\/p\u003e \u003cp\u003e7.6. Comparison of the definitions of weighting 156\u003c\/p\u003e \u003cp\u003e7.7. Continuity of the weighting by a distribution 159\u003c\/p\u003e \u003cp\u003e7.8. Derivatives and support of a weighted distribution 161\u003c\/p\u003e \u003cp\u003e7.9. Miscellanous properties of weighting 165\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8 Regularization and Applications 169\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1. Local regularization 169\u003c\/p\u003e \u003cp\u003e8.2. Properties of local approximations 174\u003c\/p\u003e \u003cp\u003e8.3. Global regularization 175\u003c\/p\u003e \u003cp\u003e8.4. Convergence of global approximations 178\u003c\/p\u003e \u003cp\u003e8.5. Properties of global approximations 180\u003c\/p\u003e \u003cp\u003e8.6. Commutativity and associativity of weighting 183\u003c\/p\u003e \u003cp\u003e8.7. Uniform convergence of sequences of distributions 188\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9 Potentials and Singular Functions 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1. Surface integral over a sphere 191\u003c\/p\u003e \u003cp\u003e9.2. Distribution associated with a singular function 193\u003c\/p\u003e \u003cp\u003e9.3. Derivatives of a distribution associated with a singular function 196\u003c\/p\u003e \u003cp\u003e9.4. Elementary Newtonian potential 197\u003c\/p\u003e \u003cp\u003e9.5. Newtonian potential of order n 201\u003c\/p\u003e \u003cp\u003e9.6. Localized potential 208\u003c\/p\u003e \u003cp\u003e9.7. Dirac mass as derivatives of continuous functions 210\u003c\/p\u003e \u003cp\u003e9.8. Heaviside potential 214\u003c\/p\u003e \u003cp\u003e9.9. Weighting by a singular weight 217\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 10 Line Integral of a Continuous Field 221\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1. Line integral along a C\u003csup\u003e1\u003c\/sup\u003e path 221\u003c\/p\u003e \u003cp\u003e10.2. Change of variable in a path 225\u003c\/p\u003e \u003cp\u003e10.3. Line integral along a piecewise C\u003csup\u003e1\u003c\/sup\u003e path 228\u003c\/p\u003e \u003cp\u003e10.4. The homotopy invariance theorem 231\u003c\/p\u003e \u003cp\u003e10.5. Connectedness and simply connectedness 235\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 11 Primitives of Functions 237\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1. Primitive of a function field with a zero line integral 237\u003c\/p\u003e \u003cp\u003e11.2. Tubular flows and concentration theorem 239\u003c\/p\u003e \u003cp\u003e11.3. The orthogonality theorem for functions 243\u003c\/p\u003e \u003cp\u003e11.4. Poincaré’s theorem 244\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 12 Properties of Primitives of Distributions 247\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1. Representation by derivatives 247\u003c\/p\u003e \u003cp\u003e12.2. Distribution whose derivatives are zero or continuous 251\u003c\/p\u003e \u003cp\u003e12.3. Uniqueness of a primitive 253\u003c\/p\u003e \u003cp\u003e12.4. Locally explicit primitive 254\u003c\/p\u003e \u003cp\u003e12.5. Continuous primitive mapping 256\u003c\/p\u003e \u003cp\u003e12.6. Harmonic distributions, distributions with a continuous Laplacian 261\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 13 Existence of Primitives 265\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1. Peripheral gluing 266\u003c\/p\u003e \u003cp\u003e13.2. Reduction to the function case 268\u003c\/p\u003e \u003cp\u003e13.3. The orthogonality theorem 270\u003c\/p\u003e \u003cp\u003e13.4. Poincaré’s generalized theorem 274\u003c\/p\u003e \u003cp\u003e13.5. Current of an incompressible two dimensional field 277\u003c\/p\u003e \u003cp\u003e13.6. Global versus local primitives 279\u003c\/p\u003e \u003cp\u003e13.7. Comparison of the existence conditions of a primitive 282\u003c\/p\u003e \u003cp\u003e13.8. Limits of gradients 283\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 14 Distributions of Distributions 285\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1. Characterization 285\u003c\/p\u003e \u003cp\u003e14.2. Bounded sets 288\u003c\/p\u003e \u003cp\u003e14.3. Convergent sequences 289\u003c\/p\u003e \u003cp\u003e14.4. Extraction of convergent subsequences 293\u003c\/p\u003e \u003cp\u003e14.5. Change of the space of values 294\u003c\/p\u003e \u003cp\u003e14.6. Distributions of distributions with values in E-weak 295\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 15 Separation of Variables 297\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1. Tensor products of test functions 297\u003c\/p\u003e \u003cp\u003e15.2. Decomposition of test functions on a product of sets 301\u003c\/p\u003e \u003cp\u003e15.3. The tensorial control theorem 303\u003c\/p\u003e \u003cp\u003e15.4. Separation of variables 309\u003c\/p\u003e \u003cp\u003e15.5. The kernel theorem 311\u003c\/p\u003e \u003cp\u003e15.6. Regrouping of variables 317\u003c\/p\u003e \u003cp\u003e15.7. Permutation of variables 318\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 16 Banach Space Valued Distributions 323\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1. Finite order distributions 323\u003c\/p\u003e \u003cp\u003e16.2. Weighting of a finite order distribution 326\u003c\/p\u003e \u003cp\u003e16.3. Finite order distribution as derivatives of continuous functions 328\u003c\/p\u003e \u003cp\u003e16.4. Finite order distribution as derivative of a single function 333\u003c\/p\u003e \u003cp\u003e16.5. Distributions in a Banach space as derivatives of functions 335\u003c\/p\u003e \u003cp\u003e16.6. Non-representability of distributions with values in a Fréchet space 339\u003c\/p\u003e \u003cp\u003e16.7. Extendability of distributions with values in a Banach space 342\u003c\/p\u003e \u003cp\u003e16.8. Cancellation of distributions with values in a Banach space 347\u003c\/p\u003e \u003cp\u003eAppendix 349\u003c\/p\u003e \u003cp\u003eBibliography 367\u003c\/p\u003e \u003cp\u003eIndex 371\u003c\/p\u003e","brand":"ISTE Ltd and John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":51042426847575,"sku":"9781786305251","price":112.5,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781786305251.jpg?v=1750954106","url":"https:\/\/bookcurl.com\/products\/distributions-9781786305251","provider":"Book Curl","version":"1.0","type":"link"}