# summable function

## summable function

[¦səm·ə·bəl ′fəŋk·shən]
(mathematics)
A function whose Lebesgue integral exists.
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The following lemma describes the asymptotic behavior of the Riemann-Liouville fractional integral of a summable function.
(iii) f has linear growth on Q: there is a summable function N such that x [member of] Q, u [member of] [OMEGA] [??]
(H2) h (t) is absolutely continuous and of bounded variation on (0, [infinity]) and h' (t) [less than or equal to] [xi] (t) for some non-negative summable function [xi] (t) (= max {0, h' (t)}) and almost all t > 0.
This fact and the fact that the Fourier coefficients of a summable function with respect to the cosine (with respect to the exponential) system are uniquely determined and imply that f(t)[omega](t) = 0 a.e.
Therefore f(t)[omega](t) sin t [member of] [L.sub.1](0, [pi]) and the fact that the Fourier coefficients of a summable function with respect to the cosine system is unique (along with (11)) imply that f(t)[omega](t) sin t = 0 a.e.
A subset [member of] [subset] [L.sup.1]([0,T];E) is said to be integrably bounded if there exists a summable function [mathematical expression not reproducible] such that for any g [member of] G,
By the symbol [L.sup.1]([0, T]; E) we denote the space of all Bochner summable functions defined on [0, T] with values in E.
Let F be a distribution in D' and let f be a locally summable function. We say that the neutrix composition F(/(x)) exists and is equal to h on the open interval (a, b) if
If [f.sub.d]([??]) is a local summable function then one can define the digital distribution [f.sub.d] by the formula
Grande, "Convolution on spaces of locally summable functions," Journal of Function Spaces and Applications, vol.
On Classes of Summable Functions and their Fourier Series.
Sobolevskii, Integral Operators in Space of Summable Functions, Noordhoff International Publishing, Leyden, Ill, USA, 1976.

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