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By N. K. Bary

ISBN-10: 1483199169

ISBN-13: 9781483199160

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Extra resources for A Treatise on Trigonometric Series. Volume 1

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In fact, let us suppose v __ \g(x)]*~1 signgfr) ff nx) 1 ~ \\gwr« Since p(q — 1) = q, we immediately derive \um f(x)lP = li^L u\\h ' and then integrating over [a, b] we at once obtain b J a that is \f(x)\'dx=l, 11/IL·. = 1. Now we note that b b U(J) = jf(x)g(x)dx j \g\"dx ,b = y ^ r ^ r r = J \g\'dx)" u = ||^|| L „ and the theorem is proved. § 21. Convergence in norm in the spaces LP We will recall a number of properties of functions belonging to LP [a, b] aip > 1. (1) In order for the sequence of functions fn{x) e LP [a, b] to converge to function f(x) e LP [a, b] in norm, that is, Hm\\fn-f\\LP[a>b] = 0, it is necessary and sufficient that for any ε > 0 a value N must be found such that Wfn -fmWisMi < e for n>N,m>N.

14) that ω(1)(<5,/) ->0 at <5->0, which was required to be proved. Note. Later the following will be required. THEOREM. If fix) is of bounded variation in [0, 2π], then ω(1)(<5,/) = 0(d). 15) for nondecreasing/(χ). But if this is so, then for A > 0 the function/(x + A) —fix) is non-negative in [0, 2π — A]. Therefore 2π 2π-Α 2π J l / ( * + A) - / ( x ) | dx = j [f(x + A) - / ( * ) ] rfx + J \f(x + A) - / ( * ) | rf* 0 0 2%-h 2n-h < 2MA + j [f{x + A) - f(x)] dx, o MODULI OF CONTINUITY AND SMOOTHNESS 41 where M is the upper bound of fix); moreover 2%—h 2π 2n—h In h J [fix + A) - / ( * ) ] dx = I fix) dx - J f{x) dx = \ fix) dx - I f{x) dx.

The converse hypothesis does not occur. § 16. Riemann-Stieltjes integral Let/(x) and g(x) be defined and finite in [a9 b]. Le us subdivide the interval [a, b] by the points x0 = a < x± < x2 < ··· < xn = b. Let ff be any point in [xh xi+1]. 1) 30 THEORY OF SETS AND THEORY OF FUNCTIONS If σ tends to a limit as max (χΛ+ι — xk) -» 0 and this limit does not depend on either the method of dividing the interval or on the choice of the points | Λ , then this limit is denoted by b jf(x)dg(x) a and is called the Riemann-Stieltjes integral off(x) with respect to g(x).

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A Treatise on Trigonometric Series. Volume 1 by N. K. Bary

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