Sentence examples for bounded variation such that from inspiring English sources

Then there exists a sequence of maps convergent pointwise on I to a map f : I → K of bounded variation such that.

( H 0 ) A is an increasing function of bounded variation such that G ( s ) ≥ 0 for s ∈ [ 0, 1 ] and 0 ≤ C < 1, where is defined by (2.9).

Theorem 1.3 Let G : [ a, b ] → R be a function of bounded variation such that G ( a ) ≤ G ( x ) ≤ G ( b ), G ( b ) > G ( a ), (2).

Using the Riesz representation theorem, there exists a 2 × 2 matrix function η , θ ∈ [ − 1, 0 ] → R 2 × 2 whose elements are of bounded variation, such that L μ ϕ = ∫ − 1 0 d η , ϕ ∈ C ( [ − 1, 0 ], R 2 ).

Theorem 3.2 Let G : [ a, b ] → R be a function of bounded variation such that (2) holds, and let f be a continuous function with nondecreasing increments of order three on [ c, d ] ⊂ R k.

By the Riesz representation theorem, there exists a (3times3) matrix function (eta(theta,mu)) ((-1leqslantthetaleqslant0)), whose elements are of bounded variation such that L_{mu} phi)= int_{-1}^{0} deta(theta,mu phi(theta), quad mbox{for } phiin mathcal{C}bigl [-1,0],mathC}bigl [-1,0]gr).

where is a function with bounded variation on such that.

Let be fixed, be continuous and monotonic with, be a function of bounded variation, and such that condition (3.2) is satisfied.

(iv) If f is monotonic nondecreasing and u is of bounded variation and such that the Riemann-Stieltjes integral ∫ a b f ( t ) d u ( t ) exists, then | T g ( f ; u ) | ≤ [ f ( b ) − f ( a ) − Q ( f ) ] ⋁ a b ( u ), (4.4).

Suppose that functions P ( t ) ∈ C ( T, ∞ ), Q ( t ) ∈ C 1 ( T, ∞ ) are real, and there exists a non-negative continuous function k ( t ) of bounded variation on ( T, ∞ ) such that v f ( v ) > 0 for v ≠ 0, (1.14). 2 P ( t ) + Q ′ ( t ) 2 Q ( t ) ≥ k ( t ) Q ( t ), t ≥ T, (1.15).

By the Riesz representation theorem there exists a function (eta(theta,mu)) of bounded variation for (thetain[-1,0]) such that L_{mu}{phi}= int_{-1}^{0}}etahrm{d}}ethetaeta,mu phi(theta) quad mbox{for } {phi}in{C[-1,0]}.