Direct and converse theorems in problems of approximation by vectors of finite degree
- Institute of Mathematics of Ukrainian National Academy of Sciences, Kiev (Ukraine)
Let A be a linear operator in a complex Banach space X with domain D(A) and a non-empty resolvent set. An element g element of D{sub {infinity}}(A):= intersection{sub j=0,1,...}D(A{sup j}) is called a vector of degree at most {zeta}(>0) with respect to A if ||A{sup j}g||{sub X}{<=}c(g){zeta}{sup j}, j=0,1,.... The set of vectors of degree at most {zeta} is denoted by G{sub {zeta}}(A). The quantity E{sub {zeta}}(f,A){sub X}=inf{sub gelement} {sub of{sub G}{sub {zeta}}}{sub (A)}||f-g||{sub X} is introduced and estimated in terms of the K-functional K({zeta}{sup -r},f;X,D(A{sup r}))=inf{sub gelementof} {sub D(A{sup r})}(||f-g||{sub X}+{zeta}{sup -r}||A{sup r}f||{sub X}) (the direct theorem). An estimate of this K-functional in terms of E{sub {zeta}}(f,A){sub X} and ||f||{sub x} is established (the converse theorem). Using the estimates obtained, necessary and sufficient conditions for the following properties are found in terms of E{sub {zeta}}(f,A){sub X}: 1) f element of D{sub {infinity}}(A); 2) the series e{sup zA}f:={sigma}{sub r=0}{sup {infinity}}(z{sup r}A{sup r}f)/(r{exclamation_point}) converges in some disc; 3) the series e{sup zA}f converges in the entire complex plane. The growth order and the type of the entire function e{sup zA}f are calculated in terms of E{sub {zeta}}(f,A){sub X}.
- OSTI ID:
- 21202772
- Journal Information:
- Sbornik. Mathematics, Vol. 189, Issue 4; Other Information: DOI: 10.1070/SM1998v189n04ABEH000312; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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