Banach frames in the affine synthesis problem

Journal Article · Sat Oct 31 04:00:00 EDT 2009 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2009V200N09ABEH004041; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21301491

Terekhin, Pavel A ^[1]

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1. Saratov State University, Saratov (Russian Federation)

We consider the problem of representing functions f element of L{sup p}(R{sup d}) by a series in elements of the affine system. The corresponding representation theorems are established on the basis of the frame inequalities for the Fourier coefficients (g,{psi}{sub j,k})={integral}{sub R{sup d}}g(x){psi}{sub j,k}(x) dx of functions g element of L{sup q}(R{sup d}), 1/p+1/q=1, where is the norm in some Banach space of number families {l_brace}y{sub j,k}{r_brace} and 0<A{<=}B<{infinity} are constants. In particular, it is proved that if the integral of a function {psi} element of L{sup 1} intersection L{sup p}(R{sup d}), 1<p<{infinity}, is nonzero, so {integral}{sub R{sup d}}{psi}(x) dx{ne}0 and the system of translates {l_brace}{psi}(x-bk):k element of Z{sup d}{r_brace} is p-Besselian in the space L{sup p}(R{sup d}), then for any function f element of L{sup p}(R{sup d}) we have the representation f={sigma}{sub jisan} {sub element} {sub of} {sub N}{sigma}{sub kisan} {sub element}{sub of}{sub Z{sup d}}c{sub j,k}{psi}{sub j,k}, where the coefficients satisfy the condition. Bibliography: 19 titles.

• OSTI ID: 21301491
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 9 Vol. 200; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
BANACH SPACE
FUNCTIONS
INTEGRALS