By Feng Dai, Yuan Xu
1 round Harmonics.- 2 Convolution and round Harmonic Expansion.- three Littlewood-Paley concept and Multiplier Theorem.- four Approximation at the Sphere.- five Weighted Polynomial Inequalities.- 6 Cubature formulation on Spheres.- 7 Harmonic research linked to mirrored image Groups.- eight Boundedness of Projection Operator and Cesaro Means.- nine Projection Operators and Cesaro potential in L^p Spaces.- 10 Weighted top Approximation via Polynomials.- eleven Harmonic research at the Unit Ball.- 12 Polynomial Approximation at the Unit Ball.- thirteen Harmonic research at the Simplex.- 14 Applications.- A Distance, distinction and critical Formulas.- B Jacobi and comparable Orthogonal Polynomials.- References.- Index.- image Index
Read Online or Download Approximation Theory and Harmonic Analysis on Spheres and Balls PDF
Similar mathematical analysis books
For the second one variation of this very winning textual content, Professor Binmore has written chapters on research in vector areas. The dialogue extends to the thought of the by-product of a vector functionality as a matrix and using moment derivatives in classifying desk bound issues. a few valuable thoughts from linear algebra are incorporated the place applicable.
Книга getting to know arithmetic: The artwork of research learning arithmetic: The paintings of research Книги Математика Автор: Anthony Gardiner Год издания: 1987 Формат: djvu Издат. :Oxford college Press, united states Страниц: 220 Размер: 1,6 Mb ISBN: 0198532652 Язык: Английский0 (голосов: zero) Оценка:One of the main extraordinary features of arithmetic is that considerate and chronic mathematical research frequently provokes absolutely unforeseen insights into what could first and foremost have seemed like an boring or intractable challenge.
I've got used this e-book largely as a reference for my very own study. it truly is an exceptional presentation from leaders within the box. My in basic terms feedback is that the examples offered within the e-book are usually trivial (namely, one-dimensional), a lot extra paintings is needed to really enforce the spectral tools defined within the textual content.
- Lectures on the Theory of Functions of Several Complex Variables
- Harmonic Analysis, the Trace Formula, and Shimura Varieties: Proceedings of the Clay Mathematics Institute, 2003 Summer School, the Fields Institute, (Clay Mathematics Proceedings,)
- Non-smooth Deterministic or Stochastic Discrete Dynamical Systems
- Symmetric Hilbert Spaces and Related Topics
Additional resources for Approximation Theory and Harmonic Analysis on Spheres and Balls
3) Pr ( x, y ) ≥ 0 and ωd−1 Sd−1 Pr ( x, y )dσ (y) = 1. Proof. 8). The second item follows from the first. The infinite series converges uniformly, since r < 1. Integration term by term shows that ωd−1 Sd−1 Pr ( x, y )dσ (y) = 1. 5. Let f be a continuous function on Sd−1 . For 0 ≤ r < 1, u(rξ ) := Pr f (ξ ) is a harmonic function in x = rξ and limr→1− u(rξ ) = f (ξ ), ∀ξ ∈ Sd−1 . Proof. The proof is standard, and we shall be brief. 4, |u(rξ ) − f (ξ )| = ≤ 1 ωd Sd−1 [ f (y) − f (ξ )]Pr ( ξ , y )dσ (y) sup | f (y) − f (ξ )| + 2 f ξ −y ≤δ ∞ ξ −y ≥δ Pr ( ξ , y )dσ (y) for every δ > 0.
In particular, for f ∈ L2 (Sd−1 ), the partial sum operator Sn f converges to f in the · 2 norm, and the operator norm Sn 2 is uniformly bounded. 1. Let d > 2. Then Sn Λn := max x∈Sd−1 1 ωd ∞ Sd−1 = Sn Sn f 1 p. = Λn , where |Kn (x, y)|dσ (y) ∼ n d−2 2 . Proof. That Sn ∞ = Sn 1 = Λn follows from a standard argument for linear integral operators. 8). In the case of d = 2, we have Sn ∞ ∼ log n, as shown in classical Fourier analysis. The constant Λn is often called the Lebesgue constant. Since it is unbounded as n → ∞, the uniform boundedness principle implies that there is a function f ∈ C(Sd−1 ) for which Sn f does not converge to f in the uniform norm.
The spherical part of the Laplace operator, can then be derived from Eq. 1) by the change of variables x → (r, ξ1 , . . , ξd−1 ), where r > 0 and ξ = (ξ1 , . . , ξd ) ∈ Sd−1 . For this approach and a derivation of Eq. 1), see . We shall adopt an approach that is elementary and selfcontained. 1. 2) where Δ0 = d−1 ∂2 d−1 d−1 i=1 i i=1 j=1 ∂2 d−1 ∂ ∑ ∂ ξ 2 − ∑ ∑ ξi ξ j ∂ ξi ∂ ξ j − (d − 1) ∑ ξi ∂ ξi . 3) i=1 Proof. Since ξ ∈ Sd−1 , we have ξ12 + · · · + ξd2 = 1. We evaluate the Laplacian Δ under a change of variables (x1 , .