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Integral of Complex-Valued Measurable Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Integral operators and weighted amalgams

C. Carton-Lebrun, H. Heinig, S. Hofmann (1994)

Studia Mathematica

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ℓ q ̅ ( L v p ̅ ) into ℓ q ( L u p ) . For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted L p -spaces. Amalgams of the form ℓ q ( L w p ) , 1 < p,q < ∞ , q ≠ p, w ∈ A p , are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

Integral Transforms Method to Solve a Time-Space Fractional Diffusion Equation

Nikolova, Yanka, Boyadjiev, Lyubomir (2010)

Fractional Calculus and Applied Analysis

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.

Integrals and Banach spaces for finite order distributions

Erik Talvila (2012)

Czechoslovak Mathematical Journal

Let ℬ c denote the real-valued functions continuous on the extended real line and vanishing at - ∞ . Let ℬ r denote the functions that are left continuous, have a right limit at each point and vanish at - ∞ . Define 𝒜 c n to be the space of tempered distributions that are the n th distributional derivative of a unique function in ℬ c . Similarly with 𝒜 r n from ℬ r . A type of integral is defined on distributions in 𝒜 c n and 𝒜 r n . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ , the spaces...

Integration of some very elementary functions

Jan Mařík (1993)

Mathematica Bohemica

Let m be a natural number. Let f , g and Q be real polynomials such that { d e g f , d e g g } ⊂ { 1 , 2 } , d e g Q < m d e g f , g is not a square and f has imaginary roots, if it is not linear. Effective methods for the integration of Q / ( f m g are exhibited.

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