# Results on Approximation Theory obtained at the Institute

## Approximation and best approximation in function spaces and abstract spaces.

Results regarding the uniqueness and the characterization of the elements of best approximation were obtained for the problem of best approximation in a metric space and in the spaces with asymmetric norms:

- C. Mustăţa,
*On the best approximation in metric spaces*, Rev. Anal. Numer. Theor. Approx., 31 (2002) no. 1, 103-108. - C. Mustăţa,
*On the extremal semi-Lipschitz function*, Mathematica – Rev. Anal. Numer. Theor. Approx 4 (1975) 1, 45-50. - C. Mustăţa, S. Cobzaş,
*Extension of bilinear functionals and best approximation in 2-normed space*, Studia Univ. “Babes-Bolyai”, Seria Mathematica, XLIII (1998) no. 2, 1-13. - C. Mustăţa, S. Cobzaş,
*Extension of bounded linear functionals and best approximation in space with asymmetric norm*, Rev. Anal. Numer. Theor. Approx., 33 (2004) no. 1, 39-50. - S. Cobzaş, C. Mustăţa,
*Best approximation in spaces with asymmetric norm*, Rev. Anal. Numer. Theor. Approx., 35 (2006) no. 1, 17-31. - C. Mustăţa, On the approximation of the global extremum of a semi-Lipschitz function, Mediterr. J. Math. 6 (2009), pp. 169-180.

## Theorems of extension and relationship with problems of best approximation.

Results regarding the extention for Lipschitz and Holder functions, preserving the smallest constants or supplementary properties such as convexity, boundedness, etc.:

- C. Mustăţa,
*Best approximation and unique extension of Lipschitz functions*, J. Approx. Theory, 19 (1977) 3, 222-230 - C. Mustăţa,
*Extension of Holder functions and some related problems of best approximation*, “Babes-Bolyai” Univ., Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7 (1991), 71-86. - C. Mustăţa,
*Extension of semi Lipschitz function on quasi-metric spaces*, Rev. Anal. Numer. Theor. Approx., 30 (2001) nr. 1, 61-67. - C. Mustăţa,
*A Phelps type result for spaces with asymmetric norms*, Bul. St. Univ. Baia Mare, XVIII, Serie B (2002) no. 2, 275-280. - C. Mustăţa, On the Extensions Preserving the Shape of Semi-Holder Function, Results Math. 63 (2013) 425-433.

## Existence and properties of selections associated to the metric projection.

Theorems for selection for metric projection over a cone in a normed space with applicatons in concrete cases are given.

- C. Mustăţa,
*Selections associated to Mc Shane’s extension theorem for Lipschitz functions*, Rev. Anal. Numer. Theor. Approx., 21 (1992) 2, 135-145. - C. Mustăţa, S. Cobzaş,
*Selections associated to the metric projection*, Rev. Anal. Numer. Theor. Approx., 24 (1995) nos. 1-2, 45-52.

## Approximation by linear and positive operators. Umbral calculus.

Results related to the construction and the properties of some approximation operators in the expressions of which appear binomial sequences, Appell sequences and Sheffer sequences:

- M. Crăciun,
*Procedee de aproximare construite cu ajutorul calculului umbral*, Editura Risoprint, Cluj-Napoca, 2008, iv+180 pp, ISBN 978-973-751-908-5. - M. Crăciun,
*Approximation operators constructed by means of Sheffer sequences*, Rev. Anal. Numer. Theor. Approx., 30 (2001) no. 2, 135-150.