scholarly journals On the support of tempered distributions

2010 ◽  
Vol 53 (1) ◽  
pp. 255-270 ◽  
Author(s):  
Jasson Vindas ◽  
Ricardo Estrada
Keyword(s):  
Inversion Formula ◽  
Cesàro Summability ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Abel Summability ◽  
Open Set ◽  
Cesaro Summability

AbstractWe show that if the summability means in the Fourier inversion formula for a tempered distribution f ∈ S′(ℝn) converge to zero pointwise in an open set Ω, and if those means are locally bounded in L1(Ω), then Ω ⊂ ℝn\supp f. We prove this for several summability procedures, in particular for Abel summability, Cesàro summability and Gauss-Weierstrass summability.

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

scholarly journals Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 235 ◽  
Author(s):  
Jagdish Pandey ◽  
Jay Maurya ◽  
Santosh Upadhyay ◽  
Hari Srivastava
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Weak Topology ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Wavelet Inversion ◽  
Constant Distribution

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

Summability methods which include the Riesz typical means. I

1971 ◽  
Vol 69 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Dennis C. Russell
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternative Form ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Riesz Summability ◽  
Necessary And Sufficient

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

scholarly journals A Quantum Mechanical Proof of the Fourier Inversion Formula

2011 ◽  
Vol 30 (3) ◽  
pp. 441-457
Author(s):  
Nelson N. de O Castro ◽  
Jacqueline Rojas ◽  
Ramon Mendoza
Keyword(s):  
Inversion Formula ◽  
Quantum Mechanical ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
Mechanical Proof

On ‘translated quasi-Cesàro’ summability. II

1969 ◽  
Vol 66 (1) ◽  
pp. 39-41 ◽  
Author(s):  
B. Kuttner
Keyword(s):  
Summability Method ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Cesaro Summability ◽  
Sequence Transformation ◽  
Image Position ◽  
Necessary And Sufficient

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformationWe note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is thatshould converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.

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