A multiparametric approximation theory via certain inequalities of Jackson and Bernstein type is developed. An approximation space is defined and it is shown that it is actually an interpolation space among 2d Banach spaces. As applications, direct and converse approximation theorems in function spaces with a dominant mixed derivative are given. © 1983.383240257Amanov, Representation and imbedding theorems for the function space Sp,0 RB(Rn) (1965) Proc. Steklov Inst. Math., 77, pp. 3-36Benedek, Panzone, The space Lp, with mixed norm (1961) Duke Mathematical Journal, 28, pp. 301-324Berg, Löfström, (1976) Interpolation Spaces, , Springer-Verlag, Berlin/HeidelbergButzer, A survey of work on approximation at Aachen, 1968–1972 (1973) Approximation Theory, Proceedings, International Symposia, pp. 31-100. , Academic Press, New YorkFernandez, Interpolation of 2n Banach spaces (1979) Studia Math., 65, pp. 175-201D. L. Fernandez, Interpolation of Sobolev-Nikol'skǐi spaces, to appearD. L. Fernandez, On Besov-Nikol'skǐi spaces, to appearD. L. Fernandez, On the interpolation of 2n Banach spaces, II, in preparationJohnen, Über Sätze von M. Zamansky und S. B. Steckin und ihre Umkehrungen auf der n-dimensionale Torus (1969) J. Approx. Theory, 2, pp. 97-110Lions, Peetre, Sur une classe d'espaces d'interpolation (1964) Publ. Math. IHES, 19, pp. 5-68Lizorkin, Multipliers of Fourier integrals and bounds of convolution in spaces with mixed norms Applications (1970) Mathematics of the USSR-Izvestiya, 4, pp. 225-255Lozorkin, Nikol'skii, A classification of differentiable functions in some fundamental spaces with dominant mixed derivatives (1965) Proc. Steklov Inst. Math., 77, pp. 160-187Nikol'skǐi, Functions with a dominant mixed derivatives satisfaying a multiple Hölder condition (1973) Amer. Math. Soc. Transl., 102, pp. 27-51. , (2)Nikol'skǐi, (1975) Approximation of Functions of Several Variables and Imbedding Theorems, , Springer-Verlag, Berlin/HeidelbergPeetre, Espaces d'inionterpolation et théorème de Soboleff (1966) Annales de l’institut Fourier, 16, pp. 279-317Peetre, A Theory of Interpolation of Normed Spaces (1968) Notas de Matemática No. 39, , IMPA, Rio de JaneiroSparr, Interpolation of several Banach spaces (1974) Annali di Matematica Pura ed Applicata, Series 4, 99, pp. 241-316Yoshikawa, Sur la théorie d'espaces d'interpolation—les espaces de moyenne de plusieurs espaces de Banach (1970) J. Fac. Sci. Univ. Tokyo, 16, pp. 407-468