Problem 19 in the SIAM Activity Group on Orthogonal Polynomials and Special Functions Newsletter 8 (3) (1998), pp. 11–12

19. Uniform bounds for shifted Jacobi multiplier sequences. For Fourier series the following is immediate: Suppose the real or complex sequence {m_{k}} generates a bounded operator on L^{p}(T), 1 ≤ p ≤ ∞, i.e., for polynomial f

∥ ∑ m_{k}f̂_{k}e^{ikφ}∥_{L^{p}(T)} ≤ ∥m∥_{M^{p}(T)}∥ ∑ f̂_{k}e^{ikφ}∥_{L^{p}(T)} ,

then one has for the shifted sequence {m_{k+j}}_{k∈Z} that

sup_{j∈N_{0}} ∥{m_{k+j}}∥_{M^{p}(T)} ≤ C ∥m∥_{M^{p}(T)} , 1 ≤ p ≤ ∞ . (1)

Looking at cosine expansions on L^{p}(0, π) one easily derives the analog of (1) via the addition formula

cos(k ± j)θ = cos kθ cos jθ ∓ sin kθ sin jθ

provided the periodic Hilbert transform is bounded, i.e., for 1 < p < ∞. More generally, by Muckenhoupt’s transplantation theorem [2, Theorem 1.6],

( ∫_{0}^{π} | ∑ m_{k+j} a_{k} P_{k}^{(α,β)}(cos θ)|^{p}sin^{2α+1} \frac{θ}{2} cos^{2β+1} \frac{θ}{2} dθ)^{1/p}

≡ ( ∫_{0}^{π} | ∑ m_{k+j} b_{k} ϕ_{k}^{(α,β)}(cos θ)|^{p}w_{α,β,p}(θ) dθ)^{1/p}

≈ ( ∫_{0}^{π} | ∑ m_{k+j} b_{k} cos kθ|^{p}w_{α,β,p}(θ) dθ)^{1/p},

where P_{k}^{(α,β)} are the Jacobi polynomials, ϕ_{k}^{(α,β)}(cos θ) are the orthonormalized Jacobi functions with respect to dθ, and

w_{α,β,p}(θ) = sin^{(2−p)(α+1/2)} \frac{θ}{2} cos^{(2−p)(β+1/2)} \frac{θ}{2} .

Therefore, the above argument for cosine expansions also applies to Jacobi expansions provided the periodic Hilbert transform is bounded with respect to the weight function w_{α,β,p}; hence, the analog of (1) holds for Jacobi expansions when

\frac{2α + 2}{α + 3/2} < p < \frac{2α + 2}{α + 1/2} , α ≥ β ≥ −\frac{1}{2} .

(i) Can the above p-range be extended? By Muckenhoupt [2, (1.3)], a fixed shift is bounded for all p, 1 < p < ∞ .

(ii) Consider the corresponding problem for Laguerre expansions (for the appropriate setting see [1]); a fixed shift is easily seen to be bounded for all p ≥ 1 .

Both questions are of course trivial for p = 2 since ℓ^{∞} = M^{2} by Parseval’s formula.

References

[1] Gasper, G. and W. Trebels: On necessary multiplier conditions for Laguerre expansions, Canad. J. Math. 43 (1991), 1228 – 1242.

[2] Muckenhoupt, B.: Transplantation Theorems and Multiplier Theorems for Jacobi Series, Memoirs Amer. Math. Soc., Vol. 64, No. 356, Providence, R.I., 1986.

(Submitted on May 19, 1998)

George Gasper

(george@math.nwu.edu)

Walter Trebels

(trebels@mathematik.tu-darmstadt.de)