Infinite Hankel matrices and generalized caretheodory-fejer and I. schur Problems

In this chapter we study various properties of the spacesH 1 H 2 andH ∞in preparation for our study of Toeplitz operators in the following chapter. Due to the availability of several excellent accounts of this subject (see Notes), we do not attempt a comprehensive treatment and proceed in the main using the techniques which we have already introduced.

The Hardy Spaces

On New Generalizations of Hilbert's Inequality and Their Applications

Hilbert-type inequalities are divided three parts: Hilbert’s inequalities (1908), Hardy-Hilbert-type inequalities (1934) and Yang-Hilbert-type inequalities (2009). In this paper, we give a summary of the development of the theory of Hilbert-type inequalities during the past 110 years.

/pdf/a-survey-on-the-study-of-hilbert-type-inequalities-4meccm20st.pdf

A survey on the study of Hilbert-type inequalities

T(x) = A(-) = xlog x+ bx+ o(x) nfix n with b a constant, theni (1) c1x < A(x) < c2x, where c1 and c2 are positive constants. This gives Tschebyschef's theorem if Mwe take A(x) = A(x) and use the fact that log [x]! = x log x x + O(log x). Landau states that if only the condition T(x) = x log x + O(x) is assumed, then (1) does not follow, a remark whose incorrectness was proved by H. N. Shapiro [4]. Landau then goes on to prove [3, pp. 598-604] that if Received by the editors June 14, 1957 and, in revised form, November 25, 1957 and February 24, 1958. 1 Here and in the sequel, (x), A(n), ,(n), and @(x) denote the usual functions of prime number theory. 693

/pdf/on-a-tauberian-theorem-of-landau-1ipnmh02jk.pdf

On a Tauberian theorem of Landau

A Note on Hilbert's Inequality

Improper Integrals as Limits of Sums

Some Tauberian Theorems Connected with the Prime Number Theorem

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A Note on Hilbert's Inequality

Improper Integrals as Limits of Sums

Some Tauberian Theorems Connected with the Prime Number Theorem