Mediterr. J. Math. (2017) 14:177 DOI 10.1007/s00009-017-0981-z c  Springer International Publishing AG 2017 On Approximation Properties of Phillips Operators Preserving Exponential Functions Vijay Gupta and Gancho Tachev Abstract. In the present paper, we study a modification of the Phillips operators, which reproduces constant and the exponential functions. We obtain the moments using the concept of moment-generating function for the Phillips operators. Here we discuss a uniform convergence esti- mate for this modified forms. Also some direct estimates, which also involve the asymptotic-type result are established. Mathematics Subject Classification. 41A25, 41A36. Keywords. Phillips operators, moment generating function, exponential functions, moments, quantitative results. 1. Introduction The Phillips operator [10] is defined as S n (f ; x)= n ∞  k=1 e −nx (nx) k k! ∞  0 e −nt (nt) k−1 (k − 1)! f (t)dt + e −nx f (0). These operators preserve constant as well linear functions. Some approx- imation results on these operators have been discussed in [4, 11, 12]. After the work of King [9] on the well-known Bernstein polynomials, in the year 2010 the author [5] modified the Phillips operators so as to preserve the test func- tion e 2 . It was observed that the modified form provides better approximation over the usual Phillips operators. By simple computation, we have M x (θ)= S n ( e θt ; x ) = exp  nxθ n − θ  , (1) which is the moment generating-function (abbrev. m.g.f.) of the operators S n and it may be utilized to find the moments of the Phillips operators. Using the software Mathematica, we find first few moments as below: 0123456789().: V,-vol