Download Analysis IV: Integration and Spectral Theory, Harmonic by Roger Godement PDF

By Roger Godement

Research quantity IV introduces the reader to useful research (integration, Hilbert areas, harmonic research in workforce idea) and to the tools of the speculation of modular services (theta and L sequence, elliptic features, use of the Lie algebra of SL2). As in volumes I to III, the inimitable variety of the writer is recognizable right here too, not just due to his refusal to write down within the compact kind used these days in lots of textbooks. the 1st half (Integration), a smart mixture of arithmetic stated to be 'modern' and 'classical', is universally necessary while the second one half leads the reader in the direction of a truly lively and really expert box of study, with probably extensive generalizations.

Show description

Read Online or Download Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext) PDF

Similar functional analysis books

Uniform algebras

From the Preface: ``The functional-analytic method of uniform algebras is inextricably interwoven with the speculation of analytic services . .. [T]he ideas and methods brought to house those difficulties [of uniform algebras], comparable to ``peak points'' and ``parts,'' offer new insights into the classical conception of approximation through analytic services.

Lectures on the Edge of the Wedge theorem

Advent The one-variable case Tubes with a standard side evidence of the continual model The Banach-Steinhaus theorem try out capabilities A lemma concerning the radius of convergence facts of the distribution model a mirrored image theorem functions to operate idea in polydiscs Epstein's generalization References

A Course in Abstract Harmonic Analysis

Summary idea is still an essential origin for the examine of concrete situations. It exhibits what the final photo may still appear like and gives effects which are important repeatedly. regardless of this, in spite of the fact that, there are few, if any introductory texts that current a unified photograph of the overall summary idea.

Functional and Shape Data Analysis

This textbook for classes on functionality facts research and form facts research describes how to find, evaluate, and mathematically symbolize shapes, with a spotlight on statistical modeling and inference. it really is geared toward graduate scholars in research in records, engineering, utilized arithmetic, neuroscience, biology, bioinformatics, and different comparable parts.

Extra info for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)

Example text

The same argument applies to functions |gn |, which converge ae. to |f |p , whence µ (|f |p ) = lim µ (|gn |) = lim µ (|sn |p ) = lim Np (sn )p since the sn ∈ L(X) are integrable. Since the series Lp , it finally follows that fn converges to f in µ (|f |p ) = Np (f )p , whence (2). Conversely, suppose that g = |f |p−1 f ∈ Lp . The theorem being trivial for p = 1, one may suppose that p > 1. The map z → |z|p−1 z from C to C is a homeomorphism. Its inverse is z → |z|−1/q z, with 1/p + 1/q = 1 and so 0 < 1/q < 1 .

Let us suppose that f ∈ Lp , choose (corollary 1 of theorem 7) a series of functions fn ∈ L(X) such that Np (fn ) < +∞ , f (x) = fn (x) ae. and set sn (x) = f1 (x) + . . + fn (x) , S(x) = |fn (x)| . § 2. Lp Spaces 31 The functions gn = |sn |p−1 sn are still in L(X) since p > 1. On the other hand, |gn | = |sn |p ≤ S p = G with a function G having values in [0, +∞] satisfying p N1 (G) = N1 (S p ) = Np |fk | p ≤ Np (fk ) < +∞ . The dominated convergence theorem can therefore be applied in L1 to the sequence (gn ), and as it converges ae.

This result explains its importance in applications since the theory of Hilbert spaces is far simpler and more complete than that of other classes of topological vector spaces. We will again come across the L2 spaces in relation to Fourier transforms, but they have many other applications. 14 One can formulate the same definition on F 2 , but the result would not be an inner product: the function µ∗ is not linear on F 1 . § 2. 9) |µ(f g)| ≤ N1 (f g) ≤ Np (f )Nq (g) . As a result, for given g ∈ Lq , the map f −→ µ(f g) is a continuous linear functional on Lp .

Download PDF sample

Rated 4.92 of 5 – based on 20 votes