Extensions of the Dirichlet-Jordan criterion to a general class of orthogonal polynomial expansions
Hardy and Littlewood proved that if \(f\in Lip(1/p,p)\) (\(p\geq 1)\), then the Fourier series of f converges to s at any point x at which \(\int^{t}_{0}(f(x+n)+f(x-n)-2s) du=o(t).\) They also proved that \(f\in Lip(1,1)\) if and only if f is equivalent to a function of bounded variation, so that the case \(p=1\) of the previous result reduces to the Dirichlet-Jordan criterion. Here we consider the Fourier series of a function f with respect to a sequence of polynomials orthonormal with respect to a given weight function, whose support is the whole real line; we restrict ourselves, however, to the case in which f is continuous. The weight function will be denoted by \(w^ 2\), where we suppose that \(w\geq 0\). It was proved by \textit{G. Freud} [Acta Math. Acad. Sci. Hung. 25, 109- 122 (1974; Zbl 0287.42015)] that if f is of bounded variation on any finite interval and if \(\int^{\infty}_{-\infty}w | df|<\infty\) then (under appropriate conditions on w) we have \((1)\quad \| w(f-s_ n)\| \to 0\) as \(n\to \infty\), where we use the sup norm, and where \(s_ n\) denotes the \(n^{th}\) partial sum of the Fourier series. The object of the present paper is to obtain a result which generalises the theorem of Freud in a manner similar to that in which the result of Hardy and Littlewood generalises the Dirichlet-Jordan criterion. The assumption of Hardy and Littlewood that \(f\in Lip(1/p,p)\) now has to be replaced by a more complicated condition involving w. Under this condition (and under suitable restrictions on w) it is proved that (1) holds. Moreover, in the case \(p=1\), it is shown that the condition is satisfied if and only if f is equivalent to a function satisfying the conditions of Freud.
scientific article; zbMATH DE number 3867959