Polynomial asymptotics and approximation of Sobolev functions

• Autorzy: Piotr Hajłasz , Agnieszka Kałamajska
• Języki publikacji: EN

Abstrakty

EN

We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.

Słowa kluczowe

EN

Sobolev space; Beppo Levi space; approximation; polynomial asymptotics; density of $C_0^∞$ functions

Kategorie tematyczne

• 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
• 41A10: Approximation by polynomials
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 41A99: None of the above, but in this section
• 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 113
• Numer: 1
• Strony: 55-64

Daty

wydano

1995

otrzymano

1993-09-10

poprawiono

1994-08-16

Twórcy

autor

Piotr Hajłasz

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

autor

Agnieszka Kałamajska

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] H. Aikawa, On weighted Beppo Levi functions-integral representations and behavior at infinity, Analysis 9 (1989), 323-346.
• [2] O. V. Besov, The behaviour of differentiable functions at infinity and density of $C^∞_0$ functions, Trudy Mat. Inst. Steklov. 105 (1969), 3-14 (in Russian).
• [3] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Moscow, Nauka, 1975 (in Russian).
• [4] J. Deny and J.-L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953-1954), 305-370.
• [5] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29-81.
• [6] P. I. Lizorkin, On the behaviour at infinity of functions from the Liouville class. On Riesz potentials of an arbitrary order, Trudy Mat. Inst. Steklov. 150 (1979), 174-197 (in Russian).
• [7] V. M. Maz'ya, Sobolev Spaces, Springer, 1985.
• [8] Y. Mizuta, Integral representations of Beppo Levi functions of higher order, Hiroshima Math. J. 4 (1974), 375-396.
• [9] O. Nikodym, Sur une classe de fonctions considérées dans le problème de Dirichlet, Fund. Math. 21 (1933), 129-150.
• [10] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959), 115-162.
• [11] V. N. Sedov, On functions tending to a polynomial at infinity, in: Imbedding Theorems and Their Applications (Proc. Sympos. Imbedding Theorems, Baku, 1966), Moscow, 1970, 204-212 (in Russian).
• [12] K. T. Smith, Formulas to represent functions by their derivatives, Math. Ann. 188 (1970), 53-77.
• [13] S. L. Sobolev, The density of $C^∞_0$ functions in the $L^(m)_p$ space, Dokl. Akad. Nauk SSSR 149 (1963), 40-43 (in Russian); English transl.: Soviet Math. Dokl. 4 (1963), 313-316.
• [14] S. L. Sobolev, The density of $C^∞_0$ finite functions in the $L^(m)_p$ space, Sibirsk. Mat. Zh. 4 (1963), 673-682 (in Russian).
• [15] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.