REMARKS ON SOME CLASSES OF POSITIVE CONTINUOUS FUNCTIONS IN C^n

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

We obtained growth estimates for bivariate functions which are analytic in the unit bidisc and have bounded $\mathbf{L}$-index in joint variables. The positive continuous function $\mathbf{L}(z_1,z_2)=(l_1(z_2,z_2),l_2(z_1,z_2))$ satisfies additional behavior condition: for every point $z=(z_1,z_2)$ belonging the unit bidisc $\mathbb{D}^2$ the appropriate value of the function $l_j$ at this point is greater than the reciprocal to $1-|z_j|$ multiplied by $\beta$, i.e. $l_j(z)>{\beta}/{(1-|z_j|)}$ for each $j\in\{1,2\}$ and some constant $\beta>1.$ Also we prove that for every analytic functions in the unit bidisc with bounded multiplciities of zero points there exists a positive continuous function $\mathbf{L}(z_1,z_2)=(l_1(z_2,z_2),l_2(z_1,z_2))$ providing boundedness of the $\mathbf{L}$-index in joint variables for primary analytic function.

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

A new class of functions, called slightly generalized β-continuous functions is introduced.Basic properties of slightly generalized β-continuous functions are studied.The class of slightly generalized βcontinuous functions properly includes the class of slightly β-continuous functions and generalized β-continuous functions.Also, by using slightly generalized β-continuous functions, some properties of domain/range of functions are characterized.

A new class of closed sets, known as semi-regular weekly closed sets, is introduced in ideal topological space. In addition, we define a new class of continuous functions known as semi-regular weekly -continuous functions is introduced and studied. Some of their characteristics have also been examined and discussed interesting properties of semi-regular weekly -irresolute functions, strongly semi-regular weekly -continuous functions, and semi- regular weekly -continuous functions.

In this note it is proved that entire functions of the form $$ F_a \left( z \right) = H\left( z \right) - a\alpha \left( z \right) $$ , where H and α are entire functions, with α having at least one zero, H′ and α′ having no common zeros and T(r, α) = S(r, H), are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

In this paper we survey some author’s results and developments relating the so-called Gel’fond-Leont’ev (G-L) operators of generalized integration and differentiation, classes of special functions (SF) of generalized hypergeometric type and the operators of generalized fractional calculus (GFC). The G-L operators have been introduced by Gel’fond-Leont’ev [9] in the classes of analytic functions in disks ΔR = {|z| < R}, by means of of multipliers’ sequences composed by the coeffcients of suitable entire (generating) functions. Introducing classes of SF related to Fractional Calculus (FC), as the Mittag-Leffer (ML) function, the multi-index Mittag-Leffer (multi-ML) function and its various particular cases ([16]–[18]), we specify the G-L operators generated by these entire functions. It is shown that in these cases, the G-L operators can be extended to analytic functions in wider complex domains Ω starlike with respect to the origin z = 0 and represented by operators of the Generalized Fractional Calculus (GFC), Kiryakova [14], i.e. operators of generalized integration and differentiation of arbitrary fractional multi-order. Illustrative examples and some open problems are proposed.

We introduced and studied some new classes of functions called (e*, s)-continuous functions, (e, s)-continuous functions and (a, s)-continuous functions. These new notions of functions generalize the class of almost contra-super-continuous functions. Some properties and several characterizations of these types of functions are obtained. We investigate the relationships between these classes of functions and other classes of non-continuous functions. .

Comonotone approximation and interpolation by entire functions

Attitude control of a space structure using a 3-R rigid manipulator

The main task in this article is to give the necessary and sufcient conditions guarantees that the product of two positive definite functions defined on a hypergroup X is also positive definite on X. Also, we prove that a continuous function with compact support ψ is negative definite if and only if exp(‐tψ) is positive definite for each t>0. Moreover, we will give some relations between the class of completely monotonic functions on a hypergroup and the set of τ‐ positive functions.

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form sum _{i=1}^np_i cdot m_i, where m_1 ,ldots ,m_n are exponentials belonging to V, and p_1 ,ldots ,p_n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra {{mathcal {A}}} of complex valued functions such that whenever an exponential m belongs to {{mathcal {A}}}, then m^{-1}in {{mathcal {A}}}. As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary sigma -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

Cauchy-subregular functions vis-à-vis different types of continuity

We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions.