Growth and Integrability of Fourier Transforms on Euclidean Space

Bray, William O.

doi:10.1007/s00041-014-9354-1

Cited by 23 publications

(16 citation statements)

References 17 publications

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“…Similar problems for the Fourier transform/coefficients in L p (R d ) and L p (T d ) have been recently investigated in [4,5,25]. In this paper we not only extend these results for the Dunkl setting but also obtain completely new Fourier inequalities.…”

Section: Weighted Fourier Inequalities In Dunkl Settingsupporting

confidence: 57%

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

Горбачев

Ivanov

Tikhonov

2020

Journal of Approximation Theory

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In this paper we study direct and inverse approximation inequalities in L p (R d ), 1 < p < ∞, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function f via the fractional powers of the Dunkl Laplacian of approximants of f . Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier-Dunkl inequalities are derived. c

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Section: Weighted Fourier Inequalities In Dunkl Settingsupporting

confidence: 57%

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

Горбачев

Ivanov

Tikhonov

2020

Journal of Approximation Theory

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“…This section begins with the deduction of basic inequalities for sums of Fourier coefficients of integrable functions and kernels, generalizations of both Theorem 6.1 in [6] and the Hausdorff-Young inequalities mentioned in the previous section. If one replaces the sphere with the Euclidean space where it sits and the Fourier sums with the standard Fourier transformation, then the results are comparable to those proved in [3]. A differential in our favor is the fact that we do not need to make use of either K-functionals or moduli of smoothness in the arguments leading to the inequalities.…”

Section: Estimates Of Fourier-like Sumsmentioning

confidence: 65%

Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere

Jordao

Menegatto

2015

Proc. Amer. Math. Soc.

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We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted Fourier sums of integrable kernels on the sphere that satisfy an abstract Hölder condition based on a parameterized family of multiplier operators defining an approximate identity. This general estimation approach includes an important class of multipliers operators, namely, that defined by convolutions with zonal measures. The estimates are used to obtain decay rates for the eigenvalues of positive integral operators on L 2 (S m ) and generated by a kernel satisfying the Hölder condition based on multiplier operators on L 2 (S m ).

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“…a result of the type Titchmarsh Theorem 2, with Lipschitz or Dini-Lipschitz conditions). On the other hand, from the estimate for small λ, an integrability result can be achieved as done by Titchmarsh in Theorem 1 (see also [4,Theorem 3.4]).…”

Section: Theorem 1 (Cf [26 Theorem 84]) If F Belongs To the Lipschitz Class Lip(η P) In The L P Norm On The Real Line That Ismentioning

confidence: 99%