Abstract. We show that if a function $f$ defined on the real line ${\bf R}$ belongs to the Besov space $B^{1/p}_{p,1},$ where $1\le p< \infty,$ then its Dirichlet integral $s_T(f,x)$ converges uniformly as $T\to\infty.$ In the case where $1\le p\le2$, we even prove that $s_T(f,x)$ converges absolutely. It follows easily that any function $f$ in $B^{1/p}_{p,1}, 1\le p\le2,$ is a Fourier transform on $L^1({\bf R}).$ The counterparts of our results were proved by A. M. Garsia (1976) for functions defined on the unit circle ${\bf T}.$ Our proofs are basically different from those given there.
AMS Subject Classification
(1991): 42A38, 46E35, 41A17
Keyword(s):
Besov space,
modulus of continuity,
Fourier transform,
Dirichlet integral,
Riesz mean,
Hardy-Littlewood inequality,
L^1.,
multiplier of Fourier transform on
Received September 15, 1993. (Registered under 5586/2009.)
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