On the dual space for the strict topology 𝛽₁ and the space 𝑀(𝑋) in function space

This paper extends the Meijer transformation,Mμ, given by(Mμf)(p)=2pΓ(1+μ)∫0∞f(t)(pt)μ/2Kμ(2pt)dt, wherefbelongs to an appropriate function space,μ ϵ (−1,∞)andKμis the modified Bessel function of third kind of orderμ, to certain generalized functions. A testing space is constructed so as to contain the Kernel,(pt)μ/2Kμ(2pt), of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform,M¯μf, is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.

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Annihilator, Completeness and Convergence of Wavelet System

Nagoya Mathematical Journal ◽

10.1017/s0027763000009454 ◽

2007 ◽

Vol 188 ◽

pp. 59-105 ◽

Cited By ~ 1

Author(s):

Kwok-Pun Ho

Keyword(s):

Function Space ◽

Dual Space ◽

Linear Span ◽

Dual Frame ◽

Wavelet System ◽

One Dimensional ◽

Trivial Intersection ◽

The One ◽

Definition Of ◽

Dyadic Cubes

AbstractWe show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that(0.1) in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

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Periodic Tempered Distributions of Beurling Type and Periodic Ultradifferentiable Functions with Arbitrary Support

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2018/6563823 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8

Author(s):

Byung Keun Sohn

Keyword(s):

Function Space ◽

Dual Space ◽

Ultradifferentiable Functions ◽

Tempered Distributions ◽

Test Function ◽

Test Function Space

Let Sω′(R) be the space of tempered distributions of Beurling type with test function space Sω(R) and let Eω,p be the space of ultradifferentiable functions with arbitrary support having a period p. We show that Eω,p is generated by Sω(R). Also, we show that the mapping Sω(R)→Eω,p is linear, onto, and continuous and the mapping Sω,p′(R)→Eω,p′ is linear and onto where Sω,p′(R) is the subspace of Sω′(R) having a period p and Eω,p′ is the dual space of Eω,p.

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On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010018135x ◽

1990 ◽

Vol 48 (1) ◽

pp. 520-521

Author(s):

Neng-Yu Zhang ◽

Bruce F. McEwen ◽

Joachim Frank

Keyword(s):

Function Space ◽

Convex Sets ◽

Electron Tomography ◽

Spinal Ganglion ◽

Fourier Space ◽

Missing Information ◽

Voltage Electron ◽

High Voltage Electron Microscope ◽

Energy Bound ◽

Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

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Finite-order Integration Weights Can be Dangerous

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2007-0015 ◽

2007 ◽

Vol 7 (3) ◽

pp. 239-254 ◽

Cited By ~ 3

Author(s):

I.H. Sloan

Keyword(s):

Function Space ◽

Finite Order ◽

Small Subset ◽

Worst Case ◽

Integration Rule ◽

Dimensional Lattice ◽

3 Dimensional ◽

Sum Of Functions ◽

Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

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