By Dmitriy Bilyk; et al (eds.)
Fresh Advances in Harmonic research and functions is devoted to the sixty fifth birthday of Konstantin Oskolkov and contours contributions from analysts around the globe. the amount comprises expository articles by way of major specialists of their fields, in addition to chosen top of the range learn papers that discover new effects and traits in classical and computational harmonic research, approximation conception, combinatorics, convex research, differential equations, sensible research, Fourier research, graph concept, orthogonal polynomials, designated capabilities, and trigonometric sequence. quite a few articles within the quantity emphasize striking connections among harmonic research and different likely unrelated components of arithmetic, equivalent to the interplay among summary difficulties in additive quantity conception, Fourier research, and experimentally came across optical phenomena in physics. Survey and examine articles supply an up to date account of assorted important instructions of recent research and should particularly be of curiosity to younger researchers who're simply beginning their profession. This ebook can be valuable to specialists in research, discrete arithmetic, physics, sign processing, and different components of technology. learn more...
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Extra resources for Recent advances in harmonic analysis and applications : in honor of Konstantin Oskolkov
Let RR ∑ cn einx = 0 for every x ∈ T2 . Does this imply that cn = 0 for every n ∈ Z2 ? Question 5 (Square). Let Sq ∑ cn einx = 0 for every x ∈ T2 . Does this imply that cn = 0 for every n ∈ Z2 ? Question 6 (Simplex). Let Si ∑ cn einx = 0 for every x ∈ T2 . Does this imply that cn = 0 for every n ∈ Z2 ? Many additional questions can be asked. For a lot of one-dimensional generalizations of Theorem C, see Chap. 9 of . We will later need to mention one higher dimensional extension of Theorem 3 which involves replacing the assumption of spherical convergence: Sp ∑ cn einx = 0, by the weaker assumption of spherical Abel summability: Sp ∑ cn einx r n exists for all positive r < 1, where n = n21 + · · · + n2d , and limr→1− Sp ∑ cn einx r n = 0.
Theorem 2. There exist positive absolute constants c and c such that for any sequence ε ∈ H 2n c εk ∑ k−n+1 ≤ k=n εk . k − n+1 k=n 2n sup || f − Sn( f )||C ≤ c ∑ f ∈C(ε ) (3) Thus, inequality (2) is sharp for any order of decay of En ( f ). Similar results are also valid in the space L. In the proof of inequality (2) Oskolkov used a partition of the sequence En ( f ) in a geometric progression. That is, he considered the sequence of natural numbers nk such that 1 (4) Enk+1 ( f ) ≤ Enk ( f ). 2 Afterwards Stechkin  extended the results of Kostya’s work  to the approximation of periodic functions by de la Vall´ee Poussin sums.
163, 401–436 (1972), MR 45 # 9057 2. : Uniqueness of rectangularly convergent trigonometric series. Ann. of Math. 137, 145–166 (1993) 3.
Recent advances in harmonic analysis and applications : in honor of Konstantin Oskolkov by Dmitriy Bilyk; et al (eds.)