La théorie de Karamata en bref
In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their...
We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative over I,...
Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for...
Mathematics Subject Classification: 26A33, 47A60, 30C15.In this paper we treat the question of existence and uniqueness of solutions of linear fractional partial differential equations. Along examples we show that, due to the global definition of fractional derivatives, uniqueness is only sure in case of global initial conditions.