Fourier-Ehrenpreis integral formula for harmonic functions

Yamane, Hideshi

doi:10.2969/jmsj/1191334083

Cited by 1 publication

(4 citation statements)

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“…Proposition 3 was used in [8]. Here we sketch its main result, which has some similarities with (1).…”

Section: Fourier-ehrenpreis Integral Formulamentioning

confidence: 80%

“…We give an alternative proof of Proposition 4. It appeared in [8] as a proof of Proposition 3. First, Proposition 4 holds for n = 0, 1.…”

Section: Integrating a Rational Function Of Trigonometric Functionsmentioning

confidence: 98%

“…Now we prove Proposition 5. Let f (a, b) be the left hand side of (8). It is positively homogeneous in the sense that f (λa, λb) = λ −(n+1)/2 f (a, b), λ > 0.…”

Section: Integrating a Rational Function Of Trigonometric Functionsmentioning

confidence: 99%

“…We consider V = z ∈ C n ; z 2 = n j=1 z 2 j = 0 . The exponential function f z (t) = exp(−iz • t) is harmonic in t. The main result of [8] states that any harmonic function is a superposition of such harmonic exponentials. Let B n be the open unit ball of R n (n ≥ 3).…”

Section: Fourier-ehrenpreis Integral Formulamentioning

confidence: 99%

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Some trigonometric integrals and the Fourier transform of a spherically symmetric exponential function

Yamane¹

2018

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We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of trigonometric functions. Its evaluation turns out to be much easier than expected because of homogeneity and a hidden symmetry. Relationship with a Fourier integral representation formula for harmonic functions is explained.

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“…Proposition 3 was used in [8]. Here we sketch its main result, which has some similarities with (1).…”

Section: Fourier-ehrenpreis Integral Formulamentioning

confidence: 80%

“…We give an alternative proof of Proposition 4. It appeared in [8] as a proof of Proposition 3. First, Proposition 4 holds for n = 0, 1.…”

Section: Integrating a Rational Function Of Trigonometric Functionsmentioning

confidence: 98%

“…Now we prove Proposition 5. Let f (a, b) be the left hand side of (8). It is positively homogeneous in the sense that f (λa, λb) = λ −(n+1)/2 f (a, b), λ > 0.…”

Section: Integrating a Rational Function Of Trigonometric Functionsmentioning

confidence: 99%

Section: Fourier-ehrenpreis Integral Formulamentioning

confidence: 99%

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