Bernstein polynomials. 2nd ed.

*(English)*Zbl 0989.41504
New York, NY: Chelsea Publishing. x, 134 p. (1986).

From the preface: After the trigonometric integrals, Bernstein polynomials are the most important and interesting concrete operators on a space of continuous functions. Since the appearance of the first edition of this book (Univ. Toronto Press, Toronto, Ont., 1953; Zbl 0051.05001), the interest in this subject has continued. In an appendix we have summed up a few of the most important papers that have appeared since 1953. I am very grateful to Chelsea Publishing Company for their suggestion that a second edition of my book be published.

Some of the theorems of the book were original results at the time. I will mention here: new proofs of the theorems of F. Riesz (page 63) and of Halperin (page 71); the interpolation theorem for operators on page 78 (the same proof also gives the more general theorem of Mityagin and CaldĂ©ron); the new form of Okada’s theorem on page 119; and—perhaps most important—the new exposition of Bernstein’s theory in the complex field, on pages 88-117.

Some of the theorems of the book were original results at the time. I will mention here: new proofs of the theorems of F. Riesz (page 63) and of Halperin (page 71); the interpolation theorem for operators on page 78 (the same proof also gives the more general theorem of Mityagin and CaldĂ©ron); the new form of Okada’s theorem on page 119; and—perhaps most important—the new exposition of Bernstein’s theory in the complex field, on pages 88-117.