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- D E Edmunds, R Kerman, L Pick
- 1999

Let m and n be positive integers with n 2 and 1 m n ? 1. We study rearrangement-invariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain in R n , with Lebesgue measure jj, there corresponds C = C(jj) > 0 for which one has the Sobolev imbedding inequality % R ? u (jjt) C% D ? jr m uj (jjt) ; u 2 C m 0 ((); involving the… (More)

- David E. Edmunds, Petr Gurka, Jan Lang
- Journal of Approximation Theory
- 2012

Various properties of the generalized trigonometric functions sinp,q are established. In particular, it is shown that those functions can approximate functions from every space Lr (0, 1) (1 < r < ∞) provided that p and q ′ (p, q ∈ (1, ∞)) are not too far apart (in fact we prove that these functions form a basis in every space Lr (0, 1)). An addition formula… (More)

- David E. Edmunds, Jan Lang
- Journal of Approximation Theory
- 2014

- David E. Edmunds, Jan Lang
- Journal of Approximation Theory
- 2013

The Hardy inequality ∫ Ω |u(x)|pd(x)−p dx c ∫ Ω |∇u(x)|p dx with d(x) = dist(x, ∂Ω) holds for u ∈ C∞ 0 (Ω) if Ω ⊂ n is an open set with a sufficiently smooth boundary and if 1 < p < ∞. P.Haj lasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained… (More)

- David E Edmunds, Petr Gurka, Jan Lang
- Proceedings. Mathematical, physical, and…
- 2014

We show that essentially the speed of decay of the Fourier sine coefficients of a function in a Lebesgue space is comparable to that of the corresponding coefficients with respect to the basis formed by the generalized sine functions sin p,q .

Some three-dimensional analogues of the plane Darboux problems for hyperbolic equations with degeneracy are investigated. In 1954, Protter initiated the study of such threedimensional problems, and it is now well known that for an infinite number of smooth right-hand sides these problems have solutions with a strong power-type singularity on the… (More)

- David E. Edmunds, Yu. Netrusov
- Journal of Approximation Theory
- 2014

- D E Edmunds, J Lang
- 2008

Let I = [a, b] ⊂ R, let 1 < q, p < ∞, let u and v be positive functions with u ∈ Lp′(I), v ∈ Lq(I) and let T : Lp(I) → Lq(I) be the Hardy-type operator given by

- David E. Edmunds
- 1994