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In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

The limit point 𝓧 of an approximating rank-R sequence of a tensor Ƶ can be obtained by fitting a decomposition (S, T, U) ⋅ 𝓖 to Ƶ . The decomposition of the limit point 𝓧 = (S, T, U) ⋅ 𝓖 with 𝓖 = blockdiag(𝓖1, … , 𝓖 m ) can be seen as a three order generalization of the real Jordan canonical form. The main aim of this paper is to study under what conditions we can turn 𝓖 j into canonical form if some of the upper triangular entries of the last three slices of 𝓖 j are zeros. In addition, we show how to turn 𝓖 j into canonical form under these conditions.

The paper is devoted to the development of control procedures with a guide for conflict-controlled dynamical systems described by ordinary fractional differential equations with the Caputo derivative of an order $\alpha \in (0, 1).$ For the case when the guide is in a certain sense a copy of the system, a mutual aiming procedure between the initial system and the guide is elaborated. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an essentially bounded measurable function. This estimate can be considered as a generalization of the known estimates of such type. An example is considered which illustrates the workability of the proposed control procedures., Comment: Submitted to "Fract. Calc. Appl. Anal."

We establish rates of convergences in time series forecasting using the statistical learning approach based on oracle inequalities. A series of papers extends the oracle inequalities obtained for iid observations to time series under weak dependence conditions. Given a family of predictors and $n$ observations, oracle inequalities state that a predictor forecasts the series as well as the best predictor in the family up to a remainder term $\Delta_n$. Using the PAC-Bayesian approach, we establish under weak dependence conditions oracle inequalities with optimal rates of convergence. We extend previous results for the absolute loss function to any Lipschitz loss function with rates $\Delta_n\sim\sqrt{ c(\Theta)/ n}$ where $c(\Theta)$ measures the complexity of the model. We apply the method for quantile loss functions to forecast the french GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and on the time series, we refine the rates of convergence to $\Delta_n \sim c(\Theta)/n$. We achieve for the first time these fast rates for uniformly mixing processes. These rates are known to be optimal in the iid case and for individual sequences. In particular, we generalize the results of Dalalyan and Tsybakov on sparse regression estimation to the case of autoregression.

This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.

In this paper we prove that a subharmonic function in R m of flnite ‚-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of flnite ‚-type which tends to zero uniformly on compacts of R m . The known Brelot-Hadamard representation of subharmonic functions in R m of flnite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of ‚-admissible mass distributions are investigated. c