Analytic Functionals on the Sphere - download pdf or read online

By Mitsuo Morimoto

ISBN-10: 0821805851

ISBN-13: 9780821805855

This ebook treats round harmonic enlargement of actual analytic features and hyperfunctions at the sphere. simply because a one-dimensional sphere is a circle, the easiest instance of the speculation is that of Fourier sequence of periodic services. the writer first introduces a method of complicated neighborhoods of the field via the Lie norm. He then reviews holomorphic services and analytic functionals at the advanced sphere. within the one-dimensional case, this corresponds to the learn of holomorphic capabilities and analytic functionals at the annular set within the advanced airplane, counting on the Laurent sequence growth. during this quantity, it really is proven that an identical inspiration nonetheless works in a higher-dimensional sphere. The Fourier-Borel transformation of analytic functionals at the sphere can be tested; the eigenfunction of the Laplacian should be studied during this method.

Show description

Read or Download Analytic Functionals on the Sphere PDF

Similar mathematical analysis books

Download e-book for iPad: The Immersed Interface Method: Numerical Solutions of PDEs by Zhilin Li

Interface difficulties come up whilst there are diverse fabrics, similar to water and oil, or a similar fabric at various states, reminiscent of water and ice. If partial or traditional differential equations are used to version those functions, the parameters within the governing equations tend to be discontinuous around the interface isolating the 2 fabrics or states, and the resource phrases are frequently singular to reflect source/sink distributions alongside codimensional interfaces.

Download e-book for iPad: Statistics of Random Processes: I. General Theory by Robert S. Liptser, Albert N. Shiryaev, B. Aries

The topic of those volumes is non-linear filtering (prediction and smoothing) concept and its program to the matter of optimum estimation, keep watch over with incomplete facts, info idea, and sequential checking out of speculation. the mandatory mathematical history is gifted within the first quantity: the idea of martingales, stochastic differential equations, absolutely the continuity of likelihood measures for diffusion and Ito methods, parts of stochastic calculus for counting strategies.

Download PDF by M. J. Lighthill: An Introduction to Fourier Analysis and Generalised

This monograph on generalised services, Fourier integrals and Fourier sequence is meant for readers who, whereas accepting idea the place each one element is proved is healthier than one in accordance with conjecture, however search a remedy as straight forward and loose from problems as attainable. Little precise wisdom of specific mathematical suggestions is needed; the booklet is acceptable for complicated college scholars, and will be used because the foundation of a quick undergraduate lecture path.

Extra info for Analytic Functionals on the Sphere

Sample text

2. Let f, g ∈ S, and let μ be a (say) compactly supported measure. Then (75) fˆgˆdμ = (ˆ μ ∗ g) · f dx. Proof. Recall that gˆ˜ = gˆ, so that ˆ gˆˆ = gˆ = g 7. THE RESTRICTION PROBLEM 49 by the inversion theorem. Now apply the duality relation and (74), obtaining fˆgˆdμ = f · (ˆ g μ)ˆdx f · (g ∗ μ ˆ)dx = as claimed. 3. Let μ be a finite positive measure. The following are equivalent for any q and any C. (1) f dμ q ≤ C f 2 , f ∈ L2 (dμ). (2) gˆ L2 (dμ) ≤ C g q , g ∈ S. (3) μ ˆ ∗ f q ≤ C 2 f q , f ∈ S.

Morse Lemma Suppose Ω ⊂ Rn is open, f : Ω → R is C ∞ , p ∈ 2f Ω, ∇f (p) = 0, and suppose that the Hessian matrix Hf (p) = ∂x∂i ∂x (p) is j invertible. Then, for a unique k (= number of positive eigenvalues of Hf ; see 37 38 6. 3 below) there are neighborhoods U and V of 0 and p respectively and a C ∞ diffeomorphism G : U → V with G(0) = p and k n f ◦ G(x) = f (p) + x2j j=1 − x2j . j=k+1 We consider now I(λ) first when a is supported near a regular point, and then when a is supported near a nondegenerate critical point.

This gives the result, since |ˆ a(ξ)|ξ|2N +2 dξ is bounded in terms of Schwartz space seminorms of a ˆ, and therefore in terms of derivatives of a. Now we consider the case of a general phase function with a nondegenerate critical point. It is clear that this should be reducible to the Gaussian case using the Morse lemma and remark 3. above. 1, since we need to obtain the correct form for the asymptotic expansion. 3. Suppose that φ is smooth, ∇φ(p) = 0 and G is a smooth diffeomorphism, G(0) = p.

Download PDF sample

Analytic Functionals on the Sphere by Mitsuo Morimoto

by Jeff

Rated 4.17 of 5 – based on 7 votes

About admin