By Steven G. Krantz

This publication treats the topic of analytic services of 1 or extra actual variables utilizing, nearly exclusively, the recommendations of actual research. This technique dramatically alters the traditional development of principles and brings formerly overlooked arguments to the fore. the 1st bankruptcy calls for just a heritage in calculus; the remedy is almost self-contained. because the ebook progresses, the reader is brought to extra refined themes requiring extra historical past and perseverance. while actually complex subject matters are reached, the booklet shifts to a extra expository mode, with targets of introducing the reader to the theorems, supplying context and examples, and indicating resources within the literature.

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In what follows, when Q C W N is a cube with center xo and c > 0 we let cQ denote the set {x E W N : x0 (l/c)(x - xO)E Q). 2 The Whitney decomposition of an open set R E W~ an be taken so that no point of R is contained in more than 1 2 of ~ the cubes. 3 (Partition of Unity) Let R E lRN be an open set and {Qj) a Whitney decomposition for R.

Bj for all j. Set f = F G. It is now obvious that f satisfies the conclusions of the theorem. + In the next section we shall give Hassler Whitney's dramatic generalization of these results to N dimensions. 3. 3 Whitney9sExtension and Approximation Theorems When compared with higher dimensions, the analysis of one real variable is relatively simple at least in part because any open set in W is the disjoint union of countably many open intervals. It was Hassler Whitney [WHI] who discovered the correct multi-variable analogue for this fact.

F,+l. Define the function f :R" -+ Rn by setting There is thus a real analytic function g defined near 0 E Rn such that g ( f ( x ) )= for x E R". By the usual Inverse Function Theorem, F-' is defined; let us write F-' in terms of its component functions as ( u l ,. l,O)= g i ( y l ? - * Y Y Y Z ) and where AiYnis the algebraic function of the components of an (n + 1)x ( n+ 1) matrix which gives the entry of the inverse matrix in the ith row and (n 1 ) row. ~ ~Thus we see that the component functions u1, .