Semyon B . Yakubovich A DISTRIBUTION ASSOCIATED WITH THE KONTOROVICH – LEBEDEV TRANSFORM
نویسنده
چکیده
We show that in a sense of distributions lim ε→0+ 1 π2 τ sinhπτ ∞ Z ε Kiτ (y)Kix(y) dy y = δ(τ − x), where δ is the Dirac distribution, τ , x ∈ R and Kν(x) is the modified Bessel function. The convergence is in E ′(R) for any even φ(x) ∈ E(R) being a restriction to R of a function φ(z) analytic in a horizontal open strip Ga = {z ∈ C : |Im z| < a, a > 0} and continuous in the strip closure. Moreover, it satisfies the condition φ(z) = O ` |z|− Im z−α e−π|Re z|/2 ́ , |Re z| → ∞, α > 1 uniformly in Ga. The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.
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