A CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONDITIONAL CONVOLUTION PRODUCT WITH CHANGE OF SCALES ON A FUNCTION SPACE I

Dong Hyun Cho

doi:10.4134/bkms.b160278

Conditional Fourier-Feynman Transforms with Drift on a Function Space

Journal of Function Spaces ◽

10.1155/2019/9483724 ◽

2019 ◽

Vol 2019 ◽

pp. 1-16

Author(s):

Dong Hyun Cho ◽

Suk Bong Park

Keyword(s):

Banach Algebra ◽

Function Space ◽

Fourier Transforms ◽

Convolution Product ◽

Borel Measures ◽

Conditional Expectations ◽

Change Of Scale ◽

Convolution Products ◽

Complex Borel Measures ◽

Generalized Cylinder

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on L2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish various changes of scale formulas for the conditional transforms and the conditional convolution products.

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Conditional integral transforms with related topics on function space

Filomat ◽

10.2298/fil1206151c ◽

2012 ◽

Vol 26 (6) ◽

pp. 1151-1162 ◽

Cited By ~ 8

Author(s):

Hyun Chung ◽

Jae Choi ◽

Seung Chang

Keyword(s):

Function Space ◽

Integral Transform ◽

Integral Transforms ◽

Convolution Product ◽

First Variation ◽

Convolution Products ◽

The First Variation

In this paper we study the conditional integral transform, the conditional convolution product and the first variation of functionals on function space. For our research, we modify the class S? of functionals introduced in [7]. We then give the existences of the conditional integral transform, the conditional convolution product and the first variation for functionals in S?. Finally, we give various relationships and formulas among conditional integral transforms, conditional convolution products and first variations of functionals in S?.

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Some Relationships for the Generalized Integral Transform on Function Space

Mathematics ◽

10.3390/math8122246 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2246

Author(s):

Hyun Chung

Keyword(s):

Gaussian Process ◽

Function Space ◽

Integral Transform ◽

Linear Operators ◽

Convolution Product ◽

Generalized Convolution ◽

Bounded Linear Operators ◽

Bounded Linear ◽

First Variation

In this paper, we recall a more generalized integral transform, a generalized convolution product and a generalized first variation on function space. The Gaussian process and the bounded linear operators on function space are used to define them. We then establish the existence and various relationships between the generalized integral transform and the generalized convolution product. Furthermore, we obtain some relationships between the generalized integral transform and the generalized first variation with the generalized Cameron–Storvick theorem. Finally, some applications are demonstrated as examples.

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Relationships of convolution products, generalized transforms, and the first variation on function space

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202006361 ◽

2002 ◽

Vol 29 (10) ◽

pp. 591-608 ◽

Cited By ~ 2

Author(s):

Seung Jun Chang ◽

Jae Gil Choi

Keyword(s):

Brownian Motion ◽

Banach Algebra ◽

Function Space ◽

Standard Wiener Process ◽

Convolution Product ◽

Brownian Motion Process ◽

First Variation ◽

Motion Process ◽

Generalized Brownian Motion Process ◽

The First Variation

We use a generalized Brownian motion process to define the generalized Fourier-Feynman transform, the convolution product, and the first variation. We then examine the various relationships that exist among the first variation, the generalized Fourier-Feynman transform, and the convolution product for functionals on function space that belong to a Banach algebraS(Lab[0,T]). These results subsume similar known results obtained by Park, Skoug, and Storvick (1998) for the standard Wiener process.

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A REPRESENTATION FOR THE INVERSE GENERALISED FOURIER–FEYNMAN TRANSFORM VIA CONVOLUTION PRODUCT ON FUNCTION SPACE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716001039 ◽

2017 ◽

Vol 95 (3) ◽

pp. 424-435

Author(s):

SEUNG JUN CHANG ◽

JAE GIL CHOI

Keyword(s):

Brownian Motion ◽

Vector Space ◽

Function Space ◽

Exponential Type ◽

Convolution Product ◽

Brownian Motion Process ◽

Inverse Transform ◽

Motion Process

We study a representation for the inverse transform of the generalised Fourier–Feynman transform on the function space $C_{a,b}[0,T]$ which is induced by a generalised Brownian motion process. To do this, we define a transform via the concept of the convolution product of functionals on $C_{a,b}[0,T]$. We establish that the composition of these transforms acts like an inverse generalised Fourier–Feynman transform and that the transforms are vector space automorphisms of a vector space ${\mathcal{E}}(C_{a,b}[0,T])$. The vector space ${\mathcal{E}}(C_{a,b}[0,T])$ consists of exponential-type functionals on $C_{a,b}[0,T]$.

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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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