ON BESSEL AND GRÜSS INEQUALITIES FOR ORTHORNORMAL FAMILIES IN INNER PRODUCT SPACES

S.S. DRAGOMIR

Abstract. A new counterpart of Bessel’s inequality for orthornormal families in real or complex inner product spaces is obtained. Applications for some Grüss type results are also provided.

1. Introduction

In the recent paper [2], the following refinement of the Grüss inequality was proved.

Theorem 1. Let (H, ⟨·, ·⟩) be an inner product space over 𝕂 (𝕂 = ℝ, ℂ) and e ∈ H, ∥e∥ = 1. If ϕ, Φ, γ, Γ are real or complex numbers and x, y are vectors in H such that either

(1.1)  Re ⟨Φe − x, x − ϕe⟩ ≥ 0 and Re ⟨Γe − y, y − γe⟩ ≥ 0

or, equivalently,

(1.2)  ∥x − \frac{ϕ + Φ}{2}e∥ ≤ \frac{1}{2} |Φ − ϕ| , ∥y − \frac{γ + Γ}{2}e∥ ≤ \frac{1}{2} |Γ − γ| ,

hold, then we have the following refinement of the Grüss inequality

|⟨x, y⟩ − ⟨x, e⟩ ⟨e, y⟩| (1.3)

≤ \frac{1}{4} |Φ − ϕ| |Γ − γ| − [Re ⟨Φe − x, x − ϕe⟩]^{\frac{1}{2}} [Re ⟨Γe − y, y − γe⟩]^{\frac{1}{2}}

≤ \frac{1}{4} |Φ − ϕ| |Γ − γ| .

The constant \frac{1}{4} is best possible in both inequalities.

Note that the inequality between the first and last term in (1.3) was firstly established in [1].

A generalisation of the above result for finite families of orthornormal vectors has been pointed out in [3].

Date: May 19, 2003.

2000 Mathematics Subject Classification. 26D15, 46C05.

Key words and phrases. Bessel’s inequality, Grüss’ inequality, Inner product, Lebesgue integral.

Theorem 2. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I, ϕ_{i}, Φ_{i}, γ_{i}, Γ_{i} ∈ 𝕂 (𝕂 = ℝ, ℂ) , i ∈ F and x, y ∈ H. If

(1.4)  Re ⟨∑_{i=1}^{n} Φ_{i}e_{i} − x, x − ∑_{i=1}^{n} ϕ_{i}e_{i}⟩ ≥ 0,

Re ⟨∑_{i=1}^{n} Γ_{i}e_{i} − y, y − ∑_{i=1}^{n} γ_{i}e_{i}⟩ ≥ 0,

or, equivalently,

(1.5)  ∥x − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

∥y − ∑_{i∈F} \frac{Γ_{i} + γ_{i}}{2} e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}} ,

hold, then we have the inequalities

|⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩| (1.6)

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

− [Re ⟨∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}⟩]^{\frac{1}{2}} [Re ⟨∑_{i∈F} Γ_{i}e_{i} − y, y − ∑_{i∈F} γ_{i}e_{i}⟩]^{\frac{1}{2}}

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}} .

The constant \frac{1}{4} is best possible.

Remark 1. We note that the inequality between the first term and the last term for real inner products under the assumption (1.4), has been proved by N. Ujević in [4].

The following corollary provides a counterpart for the well known Bessel’s inequality in real or complex inner product spaces (see also [3]).

Corollary 1. With the above assumptions for {e_{i}}_{i∈I} , F, ϕ_{i}, Φ_{i} and x, one has the inequalities

(1.7)  0 ≤ ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − Re ⟨∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}⟩

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ,

with \frac{1}{4} as the best possible constant in both inequalities.

It is the main aim of this paper to point out a different counterpart for the Bessel and Grüss inequalities stated above. Some related results are also outlined.

2. A Counterpart of Bessel’s Inequality

The following lemma holds.

Lemma 1. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I, λ_{i} ∈ 𝕂, i ∈ F, r > 0 and x ∈ H. If

(2.1)  ∥x − ∑_{i∈F} λ_{i}e_{i}∥ ≤ r,

then we have the inequality

(2.2)  0 ≤ ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2} ≤ r^{2} − ∑_{i∈F} |λ_{i} − ⟨x, e_{i}⟩|^{2} .

Proof. Consider

I_{1} := ∥x − ∑_{i∈F} λ_{i}e_{i}∥^{2} = ⟨x − ∑_{i∈F} λ_{i}e_{i}, x − ∑_{j∈F} λ_{j}e_{j}⟩

= ∥x∥^{2} − ∑_{i∈F} λ_{i}\overline{⟨x, e_{i}⟩} − ∑_{i∈F} \overline{λ_{i}} ⟨x, e_{i}⟩ + ∑_{i∈F} ∑_{j∈F} λ_{i}\overline{λ_{j}} ⟨e_{i}, e_{j}⟩

= ∥x∥^{2} − ∑_{i∈F} λ_{i}\overline{⟨x, e_{i}⟩} − ∑_{i∈F} \overline{λ_{i}} ⟨x, e_{i}⟩ + ∑_{i∈F} |λ_{i}|^{2}

and

I_{2} := ∑_{i∈F} |λ_{i} − ⟨x, e_{i}⟩|^{2} = ∑_{i∈F} (λ_{i} − ⟨x, e_{i}⟩) (\overline{λ_{i}} − \overline{⟨x, e_{i}⟩})

= ∑_{i∈F} [|λ_{i}|^{2} + |⟨x, e_{i}⟩|^{2} − \overline{λ_{i}} ⟨x, e_{i}⟩ − λ_{i}\overline{⟨x, e_{i}⟩}]

= ∑_{i∈F} |λ_{i}|^{2} + ∑_{i∈F} |⟨x, e_{i}⟩|^{2} − ∑_{i∈F} \overline{λ_{i}} ⟨x, e_{i}⟩ − ∑_{i∈F} λ_{i}\overline{⟨x, e_{i}⟩}.

If we subtract I_{2} from I_{1} we deduce an equality that is interesting in its own right

(2.3)  ∥x − ∑_{i∈F} λ_{i}e_{i}∥^{2} − ∑_{i∈F} |λ_{i} − ⟨x, e_{i}⟩|^{2} = ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2} ,

from which we easily deduce (2.2).

The following counterpart of Bessel’s inequality holds.

Theorem 3. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I, ϕ_{i}, Φ_{i}, i ∈ I real or complex numbers. For a x ∈ H, if either

(i) Re ⟨∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}⟩ ≥ 0;

or, equivalently,

(ii) ∥x − ∑_{i∈F} \frac{ϕ_{i}+Φ_{i}}{2} e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ;

holds, then the following counterpart of Bessel’s inequality

(2.4)  0 ≤ ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{ϕ_{i} + Φ_{i}}{2} − ⟨x, e_{i}⟩|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ,

is valid.

The constant \frac{1}{4} is best possible in both inequalities.

Proof. Firstly, we observe that for y, a, A ∈ H, the following are equivalent

(2.5)  Re ⟨A − y, y − a⟩ ≥ 0

and

(2.6)  ∥y − \frac{a + A}{2}∥ ≤ \frac{1}{2} ∥A − a∥ .

Now, for a = ∑_{i∈F} ϕ_{i}e_{i}, A = ∑_{i∈F} Φ_{i}e_{i}, we have

∥A − a∥ = ∥∑_{i∈F} (Φ_{i} − ϕ_{i}) e_{i}∥ = (∥∑_{i∈F} (Φ_{i} − ϕ_{i}) e_{i}∥^{2})^{\frac{1}{2}}

= (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ∥e_{i}∥^{2})^{\frac{1}{2}} = (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

giving, for y = x, the desired equivalence.

Now, if we apply Lemma 1 for λ_{i} = \frac{ϕ_{i}+Φ_{i}}{2} and

r := \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

we deduce the first inequality in (2.4).

Let us prove that \frac{1}{4} is best possible in the second inequality in (2.4). Assume that there is a c > 0 such that

(2.7)  0 ≤ ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2} ≤ c ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{ϕ_{i} + Φ_{i}}{2} − ⟨x, e_{i}⟩|^{2} ,

provided that ϕ_{i}, Φ_{i}, x and F satisfy (i) and (ii).

We choose F = \{1\} , e_{1} = e = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}) ∈ ℝ^{2}, x = (x_{1}, x_{2}) ∈ ℝ^{2}, Φ_{1} = Φ =

m > 0, ϕ_{1} = ϕ = −m, H = ℝ^{2} to get from (2.7) that

(2.8)  0 ≤ x_{1}^{2} + x_{2}^{2} − \frac{(x_{1} + x_{2})^{2}}{2}

≤ 4cm^{2} − \frac{(x_{1} + x_{2})^{2}}{2} ,

provided

(2.9)  0 ≤ ⟨me − x, x + me⟩

= (\frac{m}{\sqrt{2}} − x_{1}) (x_{1} + \frac{m}{\sqrt{2}}) + (\frac{m}{\sqrt{2}} − x_{2}) (x_{2} + \frac{m}{\sqrt{2}}) .

From (2.8) we get

(2.10)  x_{1}^{2} + x_{2}^{2} ≤ 4cm^{2}

provided (2.9) holds.

If we choose x_{1} = \frac{m}{\sqrt{2}}, x_{2} = −\frac{m}{\sqrt{2}}, then (2.9) is fulfilled and by (2.10) we get m^{2} ≤ 4cm^{2}, giving c ≥ \frac{1}{4}.

Remark 2. If F = {1} , e_{1} = 1, ∥e∥ = 1 and for ϕ, Φ ∈ 𝕂 and x ∈ H one has either

(2.11)  Re ⟨Φe − x, x − ϕe⟩ ≥ 0

or, equivalently,

(2.12)  ∥x − \frac{ϕ + Φ}{2}e∥ ≤ \frac{1}{2} |Φ − ϕ| ,

then

(2.13)  0 ≤ ∥x∥^{2} − |⟨x, e⟩|^{2}

≤ \frac{1}{4} |Φ − ϕ|^{2} − |\frac{ϕ + Φ}{2} − ⟨x, e⟩|^{2} ≤ \frac{1}{4} |Φ − ϕ|^{2} .

The constant \frac{1}{4} is best possible in both inequalities.

Remark 3. It is important to compare the bounds provided by Corollary 1 and Theorem 3.

For this purpose, consider

B_{1} (x, e, ϕ, Φ) := \frac{1}{4} (Φ − ϕ)^{2} − ⟨Φe − x, x − ϕe⟩

and

B_{2} (x, e, ϕ, Φ) := \frac{1}{4} (Φ − ϕ)^{2} − (\frac{ϕ + Φ}{2} − ⟨x, e⟩)^{2} ,

where H is a real inner product, e ∈ H, ∥e∥ = 1, x ∈ H, ϕ, Φ ∈ ℝ with

⟨Φe − x, x − ϕe⟩ ≥ 0,

or equivalently,

∥x − \frac{ϕ + Φ}{2}e∥ ≤ \frac{1}{2} |Φ − ϕ| .

If we choose ϕ = −1, Φ = 1, then we have

B_{1} (x, e) = 1 − ⟨e − x, x + e⟩ = 1 − (∥e∥^{2} − ∥x∥^{2}) = ∥x∥^{2} ,
B_{2} (x, e) = 1 − ⟨x, e⟩^{2} ,

provided ∥x∥ ≤ 1.

Choose x = ke, with 0 < k ≤ 1. Then we get

B_{1} (k) = k^{2}, B_{2} (k) = 1 − k^{2},

which shows that B_{1} (k) > B_{2} (k) if 0 < k < \frac{\sqrt{2}}{2} and B_{1} (k) < B_{2} (k) if \frac{\sqrt{2}}{2} < k ≤ 1.

We may state the following proposition.

Proposition 1. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I, ϕ_{i}, Φ_{i} ∈ 𝕂 (i ∈ F). If x ∈ H either satisfies (i), or, equivalently, (ii) of Theorem 3, then the upper bounds

B_{1} (x, e, ϕ, Φ, F) := \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − Re ⟨∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}⟩ ,

B_{2} (x, e, ϕ, Φ, F) := \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{ϕ_{i} + Φ_{i}}{2} − ⟨x, e_{i}⟩|^{2} ,

for the Bessel’s difference B_{s} (x, e, F) := ∥x∥^{2}−∑_{i∈F} |⟨x, e_{i}⟩|^{2} , cannot be compared in general.

3. A Refinement of the Grüss Inequality

The following result holds.

Theorem 4. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I, ϕ_{i}, Φ_{i}, γ_{i}, Γ_{i} ∈ 𝕂, i ∈ F and x, y ∈ H. If either

(3.1)  Re ⟨∑_{i∈F} Φ_{i}e_{i} − x, x − ∑_{i∈F} ϕ_{i}e_{i}⟩ ≥ 0,

Re ⟨∑_{i∈F} Γ_{i}e_{i} − y, y − ∑_{i∈F} γ_{i}e_{i}⟩ ≥ 0,

or, equivalently,

(3.2)  ∥x − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

∥y − ∑_{i∈F} \frac{Γ_{i} + γ_{i}}{2} e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}} ,

hold, then we have the inequalities

(3.3)  0 ≤ |⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩|

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

− ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨x, e_{i}⟩| |\frac{Γ_{i} + γ_{i}}{2} − ⟨y, e_{i}⟩|

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}} .

The constant \frac{1}{4} is best possible.

Proof. Using Schwartz’s inequality in the inner product space (H, ⟨·, ·⟩) one has

(3.4) |⟨x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}, y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i}⟩|^{2}

≤ ∥x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}∥^{2} ∥y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i}∥^{2}

and since a simple calculation shows that

⟨x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}, y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i}⟩ = ⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩

and

∥x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}∥^{2} = ∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2}

for any x, y ∈ H, then by (3.4) and by the counterpart of Bessel’s inequality in Theorem 3, we have

|⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩|^{2} (3.5)

≤ (∥x∥^{2} − ∑_{i∈F} |⟨x, e_{i}⟩|^{2}) (∥y∥^{2} − ∑_{i∈F} |⟨y, e_{i}⟩|^{2})

≤ [\frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨x, e_{i}⟩|^{2}]

× [\frac{1}{4} ∑_{i∈F} |Γ_{i} − γ_{i}|^{2} − ∑_{i∈F} |\frac{Γ_{i} + γ_{i}}{2} − ⟨y, e_{i}⟩|^{2}]

:= K.

Using Aczél’s inequality for real numbers, i.e., we recall that

(3.6)  (a^{2} − ∑_{i∈F} a_{i}^{2}) (b^{2} − ∑_{i∈F} b_{i}^{2}) ≤ (ab − ∑_{i∈F} a_{i}b_{i})^{2} ,

provided that a, b, a_{i}, b_{i} > 0, i ∈ F, we may state that

(3.7) K ≤ [\frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

− ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨x, e_{i}⟩| |\frac{Γ_{i} + γ_{i}}{2} − ⟨y, e_{i}⟩|]^{2} .

Using (3.5) and (3.7) we conclude that

(3.8) |⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩|^{2} ≤ [\frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} · (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

− ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨x, e_{i}⟩| |\frac{Γ_{i} + γ_{i}}{2} − ⟨y, e_{i}⟩|]^{2} .

Taking the square root in (3.8) and taking into account that the quantity in the last square brackets is nonnegative (see for example (2.4)), we deduce the second inequality in (3.3).

The fact that \frac{1}{4} is the best possible constant follows by Theorem 3 and we omit the details.

The following corollary may be stated.

Corollary 2. Let e ∈ H, ∥e∥ = 1, ϕ, Φ, γ, Γ ∈ 𝕂 and x, y ∈ H such that either

(3.9)  Re ⟨Φe − x, x − ϕe⟩ ≥ 0 and Re ⟨Γe − y, y − γe⟩ ≥ 0

or, equivalently,

(3.10)  ∥x − \frac{ϕ + Φ}{2}e∥ ≤ \frac{1}{2} |Φ − ϕ| , ∥y − \frac{γ + Γ}{2}e∥ ≤ \frac{1}{2} |Γ − γ| .

Then we have the following refinement of Grüss’ inequality

(3.11)  0 ≤ |⟨x, y⟩ − ⟨x, e⟩ ⟨e, y⟩|

≤ \frac{1}{4} |Φ − ϕ| |Γ − γ| − |\frac{ϕ + Φ}{2} − ⟨x, e⟩| |\frac{γ + Γ}{2} − ⟨y, e⟩|

≤ \frac{1}{4} |Φ − ϕ| |Γ − γ| .

The constant \frac{1}{4} is best possible in both inequalities.

4. Some Companion Inequalities

The following companion of the Grüss inequality also holds.

Theorem 5. Let {e_{i}}_{i∈I} be a family of orthornormal vectors in H, F a finite part of I and ϕ_{i}, Φ_{i} ∈ 𝕂, i ∈ F, x, y ∈ H and λ ∈ (0, 1) , such that either

(4.1)  Re ⟨∑_{i∈F} Φ_{i}e_{i} − (λx + (1 − λ) y) , λx + (1 − λ) y − ∑_{i∈F} ϕ_{i}e_{i}⟩ ≥ 0

or, equivalently,

(4.2)  ∥λx + (1 − λ) y − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

BESSEL AND GRÜSS INEQUALITIES FOR ORTHORNORMAL FAMILIES 9

holds. Then we have the inequality

Re [⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩] (4.3)

≤ \frac{1}{16} · \frac{1}{λ (1 − λ)} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2}

− \frac{1}{4} \frac{1}{λ (1 − λ)} ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨λx + (1 − λ) y, e_{i}⟩|^{2}

≤ \frac{1}{16} · \frac{1}{λ (1 − λ)} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} .

The constant \frac{1}{16} is the best possible constant in (4.3) in the sense that it cannot be replaced by a smaller constant. Proof. We know that for any z, u ∈ H, one has

Re ⟨z, u⟩ ≤ \frac{1}{4} ∥z + u∥^{2} .

Then for any a, b ∈ H and λ ∈ (0, 1) one has

(4.4)  Re ⟨a, b⟩ ≤ \frac{1}{4λ (1 − λ)} ∥λa + (1 − λ) b∥^{2} .

Since

⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ = ⟨x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}, y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i}⟩ ,

for any x, y ∈ H, then, by (4.4), we get

Re [⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩] (4.5)

= Re [⟨x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}, y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i}⟩]

≤ \frac{1}{4λ (1 − λ)} ∥λ (x − ∑_{i∈F} ⟨x, e_{i}⟩ e_{i}) + (1 − λ) (y − ∑_{i∈F} ⟨y, e_{i}⟩ e_{i})∥^{2}

= \frac{1}{4λ (1 − λ)} ∥λx + (1 − λ) y − ∑_{i∈F} ⟨λx + (1 − λ) y, e_{i}⟩ e_{i}∥^{2}

= \frac{1}{4λ (1 − λ)} [∥λx + (1 − λ) y∥^{2} − ∑_{i∈F} |⟨λx + (1 − λ) y, e_{i}⟩|^{2}] .

If we apply the counterpart of Bessel’s inequality in Theorem 3 for λx + (1 − λ) y, we may state that

∥λx + (1 − λ) y∥^{2} − ∑_{i∈F} |⟨λx + (1 − λ) y, e_{i}⟩|^{2} (4.6)

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨λx + (1 − λ) y, e_{i}⟩|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} .

Now, by making use of (4.5) and (4.6), we deduce (4.3).

The fact that \frac{1}{16} is the best possible constant in (4.3) follows by the fact that if in (4.1) we choose x = y, then it becomes (i) of Theorem 3, implying for λ = \frac{1}{2} (2.4), for which, we have shown that \frac{1}{4} was the best constant.

Remark 4. In practical applications we may use only the inequality between the first and the last term in (4.3).

Remark 5. If in Theorem 5, we choose λ = \frac{1}{2}, then we get

Re [⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩] (4.7)

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ⟨\frac{x + y}{2} , e_{i}⟩|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ,

provided

Re ⟨∑_{i∈F} Φ_{i}e_{i} − \frac{x + y}{2} , \frac{x + y}{2} − ∑_{i∈F} ϕ_{i}e_{i}⟩ ≥ 0

or, equivalently,

∥\frac{x + y}{2} − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} .

We note that (4.7) is a refinement of the corresponding result in [3].

Corollary 3. With the assumptions of Theorem 5 and if

(4.8)  Re ⟨∑_{i∈F} Φ_{i}e_{i} − (λx ± (1 − λ) y) , λx ± (1 − λ) y − ∑_{i∈F} ϕ_{i}e_{i}⟩ ≥ 0

or, equivalently

(4.9)  ∥λx ± (1 − λ) y − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · e_{i}∥ ≤ \frac{1}{2} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} ,

then we have the inequality

(4.10)  |Re [⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩]| ≤ \frac{1}{16} · \frac{1}{λ (1 − λ)} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} .

The constant \frac{1}{16} is best possible in (4.10).

Remark 6. If H is a real inner product space and m_{i}, M_{i} ∈ ℝ with the property

(4.11)  ⟨∑_{i∈F} M_{i}e_{i} − (λx ± (1 − λ) y) , λx ± (1 − λ) y − ∑_{i∈F} m_{i}e_{i}⟩ ≥ 0

or, equivalently,

(4.12)  ∥λx ± (1 − λ) y − ∑_{i∈F} \frac{M_{i} + m_{i}}{2} · e_{i}∥ ≤ \frac{1}{2} [∑_{i∈F} (M_{i} − m_{i})^{2}]^{\frac{1}{2}} ,

then we have the Grüss type inequality

(4.13)  |⟨x, y⟩ − ∑_{i∈F} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩| ≤ \frac{1}{16} · \frac{1}{λ (1 − λ)} ∑_{i∈F} (M_{i} − m_{i})^{2} .

5. Integral Inequalities

Let (Ω, Σ, μ) be a measure space consisting of a set Ω, a σ−algebra of parts Σ and a countably additive and positive measure μ on Σ with values in ℝ ∪ {∞} . Let ρ ≥ 0 be a μ−measurable function on Ω. Denote by L_{ρ}^{2} (Ω, 𝕂) the Hilbert space of all real or complex valued functions defined on Ω and 2 − ρ−integrable on Ω, i.e.,

(5.1)  ∫_{Ω} ρ (s) |f (s)|^{2} dμ (s) < ∞.

Consider the family {f_{i}}_{i∈I} of functions in L_{ρ}^{2} (Ω, 𝕂) with the properties that

(5.2)  ∫_{Ω} ρ (s) f_{i} (s) \overline{f_{j}} (s) dμ (s) = δ_{ij}, i, j ∈ I,

where δ_{ij} is 0 if i ≠ j and δ_{ij} = 1 if i = j. {f_{i}}_{i∈I} is an orthornormal family in L_{ρ}^{2} (Ω, 𝕂) .

The following proposition holds.

Proposition 2. Let {f_{i}}_{i∈I} be an orthornormal family of functions in L_{ρ}^{2} (Ω, 𝕂) , F a finite subset of I, ϕ_{i}, Φ_{i} ∈ 𝕂 (i ∈ F) and f ∈ L_{ρ}^{2} (Ω, 𝕂) , so that either

(5.3) ∫_{Ω} ρ (s) Re [(∑_{i∈F} Φ_{i}f_{i} (s) − f (s)) (\overline{f} (s) − ∑_{i∈F} \overline{ϕ_{i}} \overline{f_{i}} (s))] dμ (s) ≥ 0

or, equivalently,

(5.4)  ∫_{Ω} ρ (s) |f (s) − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} · f_{i} (s)|^{2} dμ (s) ≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} .

12 S.S. DRAGOMIR

Then we have the inequality

(5.5)  0 ≤ ∫_{Ω} ρ (s) |f (s)|^{2} dμ (s) − ∑_{i∈F} |∫_{Ω} ρ (s) f (s) \overline{f_{i}} (s) dμ (s)|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} − ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ∫_{Ω} ρ (s) f (s) \overline{f_{i}} (s) dμ (s)|^{2}

≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} .

The constant \frac{1}{4} is best possible in both inequalities.

The proof follows by Theorem 3 applied for the Hilbert space L_{ρ}^{2} (Ω, 𝕂) and the orthornormal family {f_{i}}_{i∈I} .

The following Grüss type inequality also holds. Proposition 3. Let {f_{i}}_{i∈I} and F be as in Proposition 2. If ϕ_{i}, Φ_{i}, γ_{i}, Γ_{i} ∈ 𝕂 (i ∈ F) and f, g ∈ L_{ρ}^{2} (Ω, 𝕂) so that either

(5.6) ∫_{Ω} ρ (s) Re [(∑_{i∈F} Φ_{i}f_{i} (s) − f (s)) (\overline{f} (s) − ∑_{i∈F} \overline{ϕ_{i}} \overline{f_{i}} (s))] dμ (s) ≥ 0,
∫_{Ω} ρ (s) Re [(∑_{i∈F} Γ_{i}f_{i} (s) − g (s)) (\overline{g} (s) − ∑_{i∈F} \overline{γ_{i}} \overline{f_{i}} (s))] dμ (s) ≥ 0,

or, equivalently,

(5.7)  ∫_{Ω} ρ (s) |f (s) − ∑_{i∈F} \frac{Φ_{i} + ϕ_{i}}{2} f_{i} (s)|^{2} dμ (s) ≤ \frac{1}{4} ∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2} ,
∫_{Ω} ρ (s) |g (s) − ∑_{i∈F} \frac{Γ_{i} + γ_{i}}{2} f_{i} (s)|^{2} dμ (s) ≤ \frac{1}{4} ∑_{i∈F} |Γ_{i} − γ_{i}|^{2} ,

then we have the inequalities

(5.8) |∫_{Ω} ρ (s) f (s) \overline{g (s)}dμ (s)

− ∑_{i∈F} ∫_{Ω} ρ (s) f (s) \overline{f_{i}} (s) dμ (s) ∫_{Ω} ρ (s) f_{i} (s) \overline{g (s)}dμ (s)|

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}}

− ∑_{i∈F} |\frac{Φ_{i} + ϕ_{i}}{2} − ∫_{Ω} ρ (s) f (s) \overline{f_{i}} (s) dμ (s)| |\frac{Γ_{i} + γ_{i}}{2} − ∫_{Ω} ρ (s) g (s) \overline{f_{i}} (s) dμ (s)|

≤ \frac{1}{4} (∑_{i∈F} |Φ_{i} − ϕ_{i}|^{2})^{\frac{1}{2}} (∑_{i∈F} |Γ_{i} − γ_{i}|^{2})^{\frac{1}{2}} .

The constant \frac{1}{4} is the best possible.

BESSEL AND GRÜSS INEQUALITIES FOR ORTHORNORMAL FAMILIES 13

The proof follows by Theorem 4 and we omit the details. Remark 7. Similar results may be stated if one applies the inequalities in Section 4. We omit the details.

In the case of real spaces, the following corollaries provide much simpler sufficient conditions for the counterpart of Bessel’s inequality (5.5) or for the Grüss type inequality (5.8) to hold. Corollary 4. Let {f_{i}}_{i∈I} be an orthornormal family of functions in the real Hilbert space L_{ρ}^{2} (Ω) , F a finite part of I, M_{i}, m_{i} ∈ ℝ (i ∈ F) and f ∈ L_{ρ}^{2} (Ω) so that

(5.9)  ∑_{i∈F} m_{i}f_{i} (s) ≤ f (s) ≤ ∑_{i∈F} M_{i}f_{i} (s) for μ − a.e. s ∈ Ω,

then we have the inequalities

(5.10)  0 ≤ ∫_{Ω} ρ (s) f^{2} (s) dμ (s) − ∑_{i∈F} [∫_{Ω} ρ (s) f (s) f_{i} (s) dμ (s)]^{2}

≤ \frac{1}{4} ∑_{i∈F} (M_{i} − m_{i})^{2} − ∑_{i∈F} [\frac{M_{i} + m_{i}}{2} − ∫_{Ω} ρ (s) f (s) f_{i} (s) dμ (s)]^{2}

≤ \frac{1}{4} ∑_{i∈F} (M_{i} − m_{i})^{2} .

The constant \frac{1}{4} is best possible. Corollary 5. Let {f_{i}}_{i∈I} and F be as in Corollary 4. If M_{i}, m_{i}, N_{i}, n_{i} ∈ ℝ (i ∈ F) and f, g ∈ L_{ρ}^{2} (Ω) such that

(5.11)  ∑_{i∈F} m_{i}f_{i} (s) ≤ f (s) ≤ ∑_{i∈F} M_{i}f_{i} (s)

and

∑_{i∈F} n_{i}f_{i} (s) ≤ g (s) ≤ ∑_{i∈F} N_{i}f_{i} (s) for μ − a.e. s ∈ Ω.

Then we have the inequalities |∫_{Ω} ρ (s) f (s) g (s) dμ (s) (5.12)

− ∑_{i∈F} ∫_{Ω} ρ (s) f (s) f_{i} (s) dμ (s) ∫_{Ω} ρ (s) g (s) f_{i} (s) dμ (s)|

≤ \frac{1}{4} (∑_{i∈F} (M_{i} − m_{i})^{2})^{\frac{1}{2}} (∑_{i∈F} (N_{i} − n_{i})^{2})^{\frac{1}{2}}

− ∑_{i∈F} |\frac{M_{i} + m_{i}}{2} − ∫_{Ω} ρ (s) f (s) f_{i} (s) dμ (s)|

× |\frac{N_{i} + n_{i}}{2} − ∫_{Ω} ρ (s) g (s) f_{i} (s) dμ (s)|

≤ \frac{1}{4} (∑_{i∈F} (M_{i} − m_{i})^{2})^{\frac{1}{2}} (∑_{i∈F} (N_{i} − n_{i})^{2})^{\frac{1}{2}} .

References

[1] S.S. DRAGOMIR, A generalisation of Grüss’ inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74-82.

[2] S.S. DRAGOMIR, Some Grüss’ type inequalities in inner product spaces, J. Ineq. Pure & Appl. Math., to appear. [ONLINE: http://jipam.vu.edu.au]

[3] S.S. DRAGOMIR, A counterpart of Bessel’s inequality in inner product spaces and some Grüss type related results, RGMIA Res. Rep. Coll., 6(2003), Supplement, Article 10. [ON

   LINE http://rgmia.vu.edu.au/v6(E).html]

[4] N. UJEVIĆ, A generalisation of Grüss inequality in prehilbertian spaces, Math. Ineq. & Appl., to appear.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, MCMC 8001, Victoria, Australia.

   E-mail address: sever.dragomir@vu.edu.au
   URL: http://rgmia.vu.edu.au/SSDragomirWeb.html