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By N. K. Bary

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Extra info for A Treatise on Trigonometric Series. Volume 1

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It is evident from all this that if the curve is given by a formula, then being determinable in some small interval, it is automatically determinable everywhere eiset- Therefore they did not doubt that the second category of curves was wider than the first, since they could not consider, for example, a broken line to be "continuous", but merely composed of sections of continuous lines. e. represented by a formula, this would signify that any kind of "geometrical" curve is a "continuous" curve which appeared to be incredible.

We have THEOREM. Ifq is defined by the equation 1 1 - + - = 1, p q then any linear functional in LP [a, b] has the form b U(J) = j f(x)g(x)dx, q a where g(x) eL [a,b], (See Lyusternik and Sobolev, réf. A18, p. 170). 2). This last statement is proved very easily. 4) if||/llLP

A22, p. 85). 35 BERNSTEIN'S I N E Q U A L I T Y WEIERSTRASS'S THEOREM. Iff(x) is continuous on the entire infinite axis andf(x + 2n) = f(x)for any x, then for any ε > 0 a trigonometric polynomial T(x) is found such that \f(x) - T(x)\ < ε, - oo < x < + oo. We shall prove this theorem in Chapter I, § 27. We refer to it now because it is impossible to prove later theorems without it. § 23. Bernstein [1] holds for trigonometric polynomials: BERNSTEIN'S THEOREM. If T„(x) is a trigonometric polynomial of order not higher than n and \Tn(x)\