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Necessary Conditions for Integrability of the Fourier Transform
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Elijah Liflyand
Published/Copyright:
March 10, 2010
Abstract
We prove the necessary conditions for the integrability of the Fourier transform. The result is a generalization, on one hand, of the well known necessary condition for absolutely convergent Fourier series and, on the other hand, of an earlier multidimensional result of Trigub.
Key words and phrases:: Integrable Fourier transform; Bessel function; radial part; fractional integral/derivative
Received: 2008-12-12
Published Online: 2010-03-10
Published in Print: 2009-September
© Heldermann Verlag
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- On the Integrability and Uniform Convergence of Multiplicative Fourier Transforms
- A Note on the Boundedness of the Hilbert Transform in Weighted Grand Lebesgue Spaces
- Necessary Conditions for Integrability of the Fourier Transform
- The Diagonal Mapping in Bmoa-Type Spaces of Analytic Functions on the Polydisk
- Representation of Quasi-Measure by Henstock–Kurzweil Type Integral on a Compact-Zero Dimensional Metric Space
- T-Direction and Borel Direction of Algebroid Functions of Finite and Positive Order
Keywords for this article
Integrable Fourier transform;
Bessel function;
radial part;
fractional integral/derivative
Articles in the same Issue
- Boundary Value Problems for Fractional Differential Equations
- On the Approximate Properties of Generalized Cesàro Means of Conjugate Trigonometric Fourier Series
- Another Version of Fuglede–Putnam Theorem
- On Divergence of Fourier Series with Respect to Multiplicative Systems on the Sets of Measure Zero
- Uniqueness of Entire or Meromorphic Functions Concerning Differential Polynomials
- An Almost Greedy Uniformly Bounded Orthonormal Basis in 𝐿𝑝(·)([0, 1]) Spaces
- Rate of Approximation for Certain Szasz–Mirakyan–Durrmeyer Operators
- On the (𝐶, α)-Means of Quadratic Partial Sums of Double Walsh–Kaczmarz–Fourier Series
- Convergence in Measure of Partial Sums of Double Vilenkin–Fourier Series
- On the Exponential Uniform Strong Summability of Multiple Trigonometric Fourier Series
- On the Integrability and Uniform Convergence of Multiplicative Fourier Transforms
- A Note on the Boundedness of the Hilbert Transform in Weighted Grand Lebesgue Spaces
- Necessary Conditions for Integrability of the Fourier Transform
- The Diagonal Mapping in Bmoa-Type Spaces of Analytic Functions on the Polydisk
- Representation of Quasi-Measure by Henstock–Kurzweil Type Integral on a Compact-Zero Dimensional Metric Space
- T-Direction and Borel Direction of Algebroid Functions of Finite and Positive Order
