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Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c,...
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Related Experiment Video

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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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Application of the generalized shift operator to the Hankel transform.

Natalie Baddour1

  • 1Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario K1N 6N5 Canada.

Springerplus
|May 31, 2014
PubMed
Summary

The Hankel transform now has convolution and shift theorems. Applying Levitan's generalized shift operator enables these properties, unlike the standard Hankel transform.

Area of Science:

  • Integral transforms
  • Harmonic analysis
  • Mathematical physics

Background:

  • The Hankel transform is a key tool in fields like signal processing and physics.
  • Standard Hankel transforms lack shift-modulation and convolution-multiplication rules, limiting their applicability.
  • These missing rules are crucial for many other integral transforms.

Observation:

  • This study investigates the application of Levitan's generalized shift operator to the Hankel transform.
  • The generalized shift operator modifies the standard definition of a shift.
  • A comparison is made between the generalized shift and a simple shift using illustrative examples.

Findings:

  • The generalized shift operator successfully introduces convolution and shift theorems to the Hankel transform.

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  • These newly applicable theorems mirror those found in other integral transforms.
  • The study demonstrates the practical implementation of these theorems.

    • Implications:

    • The findings extend the utility of the Hankel transform in mathematical physics and signal analysis.
    • This work opens new avenues for applying Hankel transform-based methods where convolution and shift properties are essential.
    • The generalized shift provides a powerful new tool for researchers working with Hankel transforms.